This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

PPF dependent fixed point theorems for an αc-admissible non-self mapping in the Razumikhin class

Ravi P Agarwal12, Poom Kumam3* and Wutiphol Sintunavarat4*

Author Affiliations

1 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX, 78363, USA

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Bangkok, 10140, Thailand

4 Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand

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Fixed Point Theory and Applications 2013, 2013:280  doi:10.1186/1687-1812-2013-280


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/280


Received:2 April 2013
Accepted:3 September 2013
Published:8 November 2013

© 2013 Agarwal et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we introduce the concept of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible non-self mappings and prove the existence and convergence of the past-present-future (briefly, PPF) dependent fixed point theorems for such mappings in the Razumikhin class. We use these results to prove the PPF dependent fixed point of Bernfeld et al. (Appl. Anal. 6:271-280, 1977) and also apply our results to PPF dependent coincidence point theorems.

MSC: 47H09, 47H10.

Keywords:
PPF fixed points; Razumikhin classes; rational type contraction

1 Introduction

The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. The theory can be applied to solve many problem in real world, for example: equilibrium problems, variational inequalities and optimization problems. A very powerful tool in fixed point theory is the Banach fixed point theorem or Banach’s contraction principle for a single-valued mapping. It is no surprise that there is a great number of generalizations of this principle. Several mathematicians have gone in several directions modifying Banach’s contractive condition, changing the space or extending a single-valued mapping to a multivalued mapping (see [1-10]).

One of the most interesting results is the extension of Banach’s contraction principle in case of non-self mappings. In 1997, Bernfeld et al.[11] introduced the concept of fixed point for mappings that have different domains and ranges, the so called past-present-future (briefly, PPF) dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach-type contraction for a non-self mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach-type contraction mappings. These results are useful for proving the solutions of nonlinear functional differential and integral equations which may depend upon the past history, present data and future consideration. Several PPF dependence fixed point theorems have been proved by many researchers (see [12-15]).

On the other hand, Samet et al.[16] were first to introduce the concept of α-admissible self-mappings and they proved the existence of fixed point results using contractive conditions involving an α-admissible mapping in complete metric spaces. They also gave some examples and applications to ordinary differential equations of the obtained results. Subsequently, there are a number of results proved for contraction mappings via the concept of α-admissible mapping in metric spaces and other spaces (see [17-19] and references therein).

To the best of our knowledge, there has been no discussion so far concerning the PPF dependent fixed point theorems via α-admissible mappings. In this paper, we introduce the concept of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible non-self mappings and establish the existence and convergence of PPF dependent fixed point theorems for contraction mappings involving <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible non-self mappings in the Razumikhin class. Furthermore, we apply our results to the existence of PPF dependent fixed point theorems in [11] and also apply to PPF dependent coincidence point theorems.

2 Preliminaries

Throughout this paper, E denotes a Banach space with the norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M5">View MathML</a>, I denotes a closed interval <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M6">View MathML</a> in ℝ, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M7">View MathML</a> denotes the set of all continuous E-valued functions on I equipped with the supremum norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M8">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M9">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M10">View MathML</a>.

For a fixed element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11">View MathML</a>, the Razumikhin or minimal class of functions in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M13">View MathML</a>

It is easy to see that the constant function is one of the mapping in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. The class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is said to be algebraically closed with respect to difference if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M16">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M17">View MathML</a>. Also, we say that the class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is topologically closed if it is closed with respect to the topology on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a> generated by the norm <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M8">View MathML</a>.

Definition 2.1 (Bernfeld et al.[11])

A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M10">View MathML</a> is said to be a PPF dependent fixed point or a fixed point with PPF dependence of the non-self mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M22">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M23">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11">View MathML</a>.

Definition 2.2 (Bernfeld et al.[11])

The mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a> is called a Banach-type contraction if there exists a real number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M27">View MathML</a>

(2.1)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>.

Definition 2.3 (Samet et al.[16])

Let X be a nonempty set, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M29">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M30">View MathML</a>. We say that T is an α-admissible mapping if it satisfies the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M31">View MathML</a>

Example 2.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M32">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M35">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M36">View MathML</a>

Then T is α-admissible.

Example 2.5 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M32">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M41">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M42">View MathML</a>

Then T is α-admissible.

Remark 2.6 In the setting of Examples 2.4 and 2.5, every nondecreasing self-mapping T is ß-admissible.

Example 2.7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M43">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M33">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M46">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M47">View MathML</a>

Then T is α-admissible.

3 PPF dependent fixed point theorems for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible mappings

First of all, we introduce the concept of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible non-self mappings.

Definition 3.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M52">View MathML</a>. We say that T is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible mapping if for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M54">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M55">View MathML</a>

Example 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M56">View MathML</a> be real Banach spaces with usual norms and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M57">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M58">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M59">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M60">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M61">View MathML</a>

Then T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M62">View MathML</a>-admissible.

Next, we prove the following result for a PPF dependent fixed point.

Theorem 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64">View MathML</a>be two mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) Tis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M69">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, thenThas a unique PPF dependent fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a>.

Moreover, for a fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, if a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>of iterates ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M87">View MathML</a>

(3.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>converges to a PPF dependent fixed point ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91">View MathML</a> be a point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M96">View MathML</a>. Choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M97">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M98">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M99">View MathML</a> and by hypothesis, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100">View MathML</a>. This implies that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M102">View MathML</a>. Thus, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M103">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M104">View MathML</a>

By continuing this process, by induction, we can construct the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M107">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

It follows from the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is algebraically closed with respect to difference that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M110">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Since T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113">View MathML</a>, we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M114">View MathML</a>

By continuing this process, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Next, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M120">View MathML</a>

By repeating the above relation, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M121">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124">View MathML</a>, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M125">View MathML</a>

This implies that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a>. By the completeness of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129">View MathML</a> converges to a limit point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M130">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is topologically closed, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M133">View MathML</a>.

Now we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T. By (d), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M135">View MathML</a>. From assumption (c), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M136">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>. Taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a> in the above inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M139">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M140">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Finally, we prove the uniqueness of a PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145">View MathML</a> be two PPF dependent fixed points of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148">View MathML</a>. Now we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M149">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M150">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M151">View MathML</a> and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152">View MathML</a>. Therefore, T has a unique PPF dependent fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. This completes the proof. □

Theorem 3.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64">View MathML</a>be two mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) Tis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M160">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M162">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, thenThas a unique PPF dependent fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a>.

Moreover, for a fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, if a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>of iterates ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M179">View MathML</a>

(3.2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>converges to a PPF dependent fixed point ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91">View MathML</a> be a point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M96">View MathML</a>. Now, we choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M189">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M190">View MathML</a>

From the fact that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100">View MathML</a>, we obtain that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M102">View MathML</a>. Thus, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M194">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M195">View MathML</a>

By continuing this process, we can construct the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M198">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

By algebraic closedness with respect to difference of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M201">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Since T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M205">View MathML</a>

By repeating this process and by induction, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M206">View MathML</a>

(3.3)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Next, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M211">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M212">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>. Repeated application of the above relation yields

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M214">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124">View MathML</a>, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M218">View MathML</a>

This implies that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is topologically closed and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a> is complete, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129">View MathML</a> converges to a limit point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M224">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131">View MathML</a>.

Now we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T. By (3.3) and assumption (d), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M227">View MathML</a>. From assumption (c), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M228">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>. Taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a> in the above inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M231">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M232">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Finally, we prove the uniqueness of a PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145">View MathML</a> be two PPF dependent fixed points of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148">View MathML</a>. By assumption (c), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M241">View MathML</a>

and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M242">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M243">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M151">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152">View MathML</a>. Therefore, T has a unique PPF dependent fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. This completes the proof. □

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64">View MathML</a>be two mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) Tis<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M253">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M255">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M73">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M75">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M77">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, thenThas a unique PPF dependent fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a>in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a>.

Moreover, for a fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>, if a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>of iterates ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M272">View MathML</a>

(3.4)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>converges to a PPF dependent fixed point ofTin <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Proof For fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M91">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M79">View MathML</a>. Here we construct the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M94">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M95">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M283">View MathML</a>. Choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M97">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M285">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M100">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M288">View MathML</a>. By the same argument, we can choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M103">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M290">View MathML</a>

By induction, we produce the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M293">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

We also obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M295">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is algebraically closed with respect to difference.

Since T is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M113">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M300">View MathML</a>

By continuing this process, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M115">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Next, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M306">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M307">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M308">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>. By repeating this inequality, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M310">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M123">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M124">View MathML</a>, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M314">View MathML</a>

This implies that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M126">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M92">View MathML</a>.

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is topologically closed, by the completeness of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M129">View MathML</a> converges to a limit point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M224">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131">View MathML</a>.

Now we prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M323">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M131">View MathML</a>, by using condition (d), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a>. From condition (c), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M327">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>.

Since the exponential function is a real continuous function, we can take the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a> in the above inequality, and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M330">View MathML</a>

This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M232">View MathML</a> and hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> is a PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Finally, we prove the uniqueness of PPF dependent fixed point of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M80">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M145">View MathML</a> be two PPF dependent fixed points of T in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M148">View MathML</a>. Now we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M340">View MathML</a>

and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M341">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M150">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M343">View MathML</a> and then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M152">View MathML</a>. Therefore, T has a unique PPF dependent fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. This completes the proof. □

Remark 3.6 If the Razumikhin class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a> is not topologically closed, then the limit of the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a> in Theorems 3.3, 3.4 and 3.5 may be outside of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>, which may not be unique.

4 Consequences

In this section, we show that many existing results in the literature can be deduced from and applied easily to our theorems.

4.1 Banach contraction theorem

By applying Theorems 3.3, 3.4 and 3.5, we obtain the following results.

Theorem 4.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, and there exists a real number<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M350">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M351">View MathML</a>

(4.1)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is topologically closed and algebraically closed with respect to difference, thenThas a unique PPF dependent fixed point in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Moreover, for a fixed<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>, if a sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>of iterates ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M359">View MathML</a>

(4.2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M76">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M71">View MathML</a>converges to a PPF dependent fixed point ofTin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64">View MathML</a> be the mapping defined by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M364">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M365">View MathML</a>. Then T is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible mapping. It is easy to show that all the hypotheses of Theorems 3.3, 3.4 and 3.5 are satisfied. Consequently, T has a unique PPF dependent fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>. □

4.2 PPF dependent coincidence point theorems

In this section, we discuss some relation between PPF dependent fixed point results and PPF dependent coincidence point results. First, we give the concept of PPF dependent coincidence point.

Definition 4.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>. A point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M370">View MathML</a> is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M371">View MathML</a> for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M11">View MathML</a>.

Definition 4.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M374">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M64">View MathML</a>. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377">View MathML</a> is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible pair if for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M54">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M380">View MathML</a>

Remark 4.4 It easy to see that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377">View MathML</a> is an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible pair and S is an identity mapping, then T is also an <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible mapping.

Now, we indicate that Theorem 3.3 can be utilized to derive a PPF dependent coincidence point theorem.

Theorem 4.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386">View MathML</a>be three mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M392">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400">View MathML</a>.

(e) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M401">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403">View MathML</a>, thenSandThave a PPF dependent coincidence pointωin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405">View MathML</a>.

Proof Consider the mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368">View MathML</a>. We obtain that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M407">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M408">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M409">View MathML</a> is one-to-one. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M410">View MathML</a>, we can define a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M411">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M412">View MathML</a>

(4.3)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M413">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M409">View MathML</a> is one-to-one, then is well defined.

From (4.3) and condition (c), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M416">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M417">View MathML</a>. This shows that satisfies condition (c) of Theorem 3.3.

Now, we use Theorem 3.3 with a mapping , then there exists a unique PPF dependent fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M420">View MathML</a> of , that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M422">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M423">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M420">View MathML</a>, we can find <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M425">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M426">View MathML</a>. Therefore, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M427">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M428">View MathML</a>

This implies that ω is a PPF dependent coincidence point of T and S. This completes the proof. □

Similarly, we can apply Theorems 3.4 and 3.5 to the Theorems 4.6 and 4.7. Then, in order to avoid repetition, the proof is omitted.

Theorem 4.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386">View MathML</a>be three mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M437">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M162">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M74">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400">View MathML</a>.

(e) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M401">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403">View MathML</a>, thenSandThave a PPF dependent coincidence pointωin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405">View MathML</a>.

Theorem 4.7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M368">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M386">View MathML</a>be three mappings satisfying the following conditions:

(a) There exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M50">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M388">View MathML</a>is topologically closed and algebraically closed with respect to difference.

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M377">View MathML</a>is<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M1">View MathML</a>-admissible.

(c) For all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M460">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M70">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M462">View MathML</a>.

(d) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M394">View MathML</a>is a sequence in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M12">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M396">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M466">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M398">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M399">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M400">View MathML</a>.

(e) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M470">View MathML</a>.

If there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M78">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M403">View MathML</a>, thenSandThave a PPF dependent coincidence pointωin<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M14">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/280/mathml/M405">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC project No. NRU56000508).

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