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Random fixed point theorem on a Ćirić-type contractive mapping and its consequence

M Saha1 and Anamika Ganguly2*

Author Affiliations

1 Department of Mathematics, The University of Burdwan, Burdwan, West Bengal, 713104, India

2 Burdwan Railway Balika Vidyapith High School, Khalasipara, Burdwan, West Bengal, 713101, India

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Fixed Point Theory and Applications 2012, 2012:209  doi:10.1186/1687-1812-2012-209

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/209


Received:25 June 2012
Accepted:6 November 2012
Published:22 November 2012

© 2012 Saha and Ganguly; 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

Purpose

The main purpose of this paper is to prove a random fixed point theorem in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings.

Results

The main finding of this paper is the identification of some random fixed point theorems and the relevant application with appropriate supporting examples.

Conclusion

A random fixed point theorem is useful to determine the existence of a solution in a Banach space of a random nonlinear integral equation.

MSC: 47H10, 60H25.

Keywords:
complete probability measure space; random variable; random solution; random fixed point equation; Bochner integral

1 Introduction

The application of fixed point theory in different branches of mathematics, statistics, engineering and economics relating to problems associated with approximation theory, theory of differential equations, theory of integral equations, etc. has been recognized in the existing literature [1,2] and [3]. Progress in the study on fixed points of non-expansive mappings, contractive mappings in various spaces like a metric space, a Banach space, a fuzzy metric space, a cone metric space etc. has been saturated at large. After the initial impetus given by the Prague school of Probability in 1950s, considerable attention has been given to the study of random fixed point theorems. This arises because of the significance of fixed point theorems in probabilistic functional analysis and probabilistic models along with several applications. Issues relating to measurability of solutions, probabilistic and statistical aspects of random solutions have arisen due to the introduction of randomness. It is no denying the fact that random fixed point theorems are stochastic generalizations of classical fixed point theorems that have been described as deterministic results.

Špaček [4] and Hanš [5,6] first proved random fixed point theorems for random contraction mappings on separable complete metric spaces. The article by Bharucha-Reid [7] in 1976 attracted the attention of several mathematicians and led to the development of this theory. Špaček’s and Hanš’s theorems have been extended to multivalued contraction mappings by Itoh [8]. A random version of Schaduer’s fixed point theorem on an atomic probability measure space has been provided by Mukherjee [9]. The results of this work have been generalized by Bharucha-Reid [1,7] on a general probability measure space. Itoh [8] obtained random fixed point theorems with an application to random differential equations in Banach spaces. Several random fixed point theorems including random analogue of the classical results based on Rothe [10] have been obtained by Sehgal and Waters [11]. Kumam in a series of papers (see [12-25]) proved some remarkable results on random fixed point theorems. In a couple of papers [17] and [19], he along with his coauthor proved some random fixed point theorems for multivalued non-expansive non-self operators in the framework of Banach spaces satisfying inwardness conditions. In another paper, Kumam and Plubtieng [13] proved some random coincidence points and random common fixed point theorems for nonlinear multivalued random operators. They also proved the existence of a random coincidence point for a pair of reciprocally continuous and compatible single-valued and multivalued operators. Saha [26], Saha and Debnath [27] in their works established some random fixed point theorems over a separable Banach space and a separable Hilbert space. On the other hand, Padgett [28] applied a random fixed point theorem to prove the existence of a solution in a Banach space of a random nonlinear integral equation. Achari [29], Saha and Dey [30] developed this new area of application.

Banach’s contraction principle [31] is one of the pivotal results of nonlinear analysis. It has been the source of metric fixed point theory and its significance rests in its vast applicability in different branches of mathematics. In the general setting of a complete metric space, this theorem runs as follows (see Theorem 2.1 [32] or Theorem 1.2.2 [33]).

Theorem 1.1 (Banach’s contraction principle)

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M1">View MathML</a>be a complete metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M2">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M3">View MathML</a>be a mapping such that for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M5">View MathML</a>.

Thenfhas a unique fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M6">View MathML</a>, and for each<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M8">View MathML</a>.

On the other hand, Greguš [34] proved the following fixed point theorem.

Theorem 1.2LetXbe a Banach space, Cbe a closed convex subset ofXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9">View MathML</a>be a mapping satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M10">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M12">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M13">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M14">View MathML</a>. ThenThas a unique fixed point.

During the eighties, many theorems which are closely related to the Greguš theorem have appeared in several literatures (see [35-38] and [39]). Also, Ćirić [40] dealt with a class of mappings (not necessarily continuous) which are defined on a metric space and proved the following fixed point theorem which is a double generalization of Greguš [34].

Theorem 1.3LetCbe a closed convex subset of a complete convex metric spaceXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9">View MathML</a>be a mapping satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M16">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M12">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M18">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M19">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11">View MathML</a>. ThenThas a unique fixed point.

In light of Theorem 1.3, the following theorem has been proved by Ćirić [40].

Theorem 1.4LetCbe a closed convex metric spaceXand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M9">View MathML</a>be a mapping satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M22">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M11">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M24">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M25">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M26">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M27">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M28">View MathML</a>. ThenThas a unique fixed point.

Ćirić also introduced several contractive operators on metric spaces and proved many fixed point theorems on such operators. Inspired by Ćirić’s contractive operators, many researchers have obtained fixed point theorems on Ćirić’s operators in different settings. In this context, Karapinar [41] proved some non-unique fixed point theorems on Ćirić-type contractive operators in cone metric spaces. Also Karapinar et al.[42] proved a fixed point theorem on a metric space for a class of maps that satisfy the Ćirić-type contractive condition.

In this paper, our main objective is to prove some random fixed point theorems in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings. The results are stochastic generalizations of deterministic fixed point theorems of Ćirić [40]. The result obtained in this paper will also be useful in application to a random nonlinear integral equation. Also, we have introduced some appropriate supporting examples.

In order to make the paper self-contained, we state some important definitions and an example that are available in Joshi and Bose [3] and Debnath and Mikusinski [2].

2 Preliminaries

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M29">View MathML</a> be a separable Banach space, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M30">View MathML</a> is a σ-algebra of Borel subsets of X, and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31">View MathML</a> denote a complete probability measure space with measure μ and β be a σ-algebra of subsets of Ω. For more details, one can see Joshi and Bose [3].

Definition 2.1 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> is said to be an X-valued random variable if the inverse image under the mapping x of every Borel set B of X belongs to β, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M33">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M34">View MathML</a>.

Definition 2.2 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> is said to be a finitely-valued random variable if it is constant on each finite number of disjoint sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M36">View MathML</a> and is equal to 0 on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M37">View MathML</a>. x is called a simple random variable if it is finitely valued and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M38">View MathML</a>.

Definition 2.3 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> is said to be a strong random variable if there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M40">View MathML</a> of simple random variables which converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41">View MathML</a> almost surely, that is, there exists a set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M42">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M44">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M45">View MathML</a>.

Definition 2.4 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> is said to be a weak random variable if the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M47">View MathML</a> is a real-valued random variable for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M48">View MathML</a>, the space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M49">View MathML</a> denoting the first normed dual space of X.

In a separable Banach space X, the notions of strong and weak random variables <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> (see Corollary 1 of Joshi and Bose [3]) coincide, and in respect of such a space X, x is termed as a random variable.

We recall the following results.

Theorem 2.5 (see Theorem 6.1.2(a) of Joshi and Bose [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M51">View MathML</a>be strong random variables andα, βbe constants. Then the following statements hold:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M52">View MathML</a>is a strong random variable.

(b) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M53">View MathML</a>is a real-valued random variable and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M54">View MathML</a>is a strong random variable, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M55">View MathML</a>is a strong random variable.

(c) If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M40">View MathML</a>is a sequence of strong random variables converging strongly to<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41">View MathML</a>almost surely, i.e., if there exists a set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M42">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M43">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M60">View MathML</a>for every<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M61">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M54">View MathML</a>is a strong random variable.

Remark 2.6 If X is a separable Banach space, then every strong and also weak random variable is measurable in the sense of Definition 2.1.

Let Y be another Banach space. We also need the following definitions as cited in Joshi and Bose [3].

Definition 2.7 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63">View MathML</a> is said to be a random mapping if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M64">View MathML</a> is a Y-valued random variable for every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7">View MathML</a>.

Definition 2.8 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63">View MathML</a> is said to be a continuous random mapping if the set of all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a> for which <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M68">View MathML</a> is a continuous function of x has measure one.

Definition 2.9 A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63">View MathML</a> is said to be demi-continuous at the <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M7">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M71">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M72">View MathML</a> almost surely.

Theorem 2.10 (see Theorem 6.2.2 of Joshi and Bose [3])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63">View MathML</a>be a demi-continuous random mapping where a Banach spaceYis separable. Then, for anyX-valued random variablex, the function<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M74">View MathML</a>is aY-valued random variable.

Remark 2.11 (see [3])

Since a continuous random mapping is a demi-continuous random mapping, Theorem 2.5 is also true for a continuous random mapping.

We shall also recall the following definitions as seen in Joshi and Bose [3].

Definition 2.12 An equation of the type <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M75">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M76">View MathML</a> is a random mapping, is called a random fixed point equation.

Definition 2.13 Any mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M32">View MathML</a> which satisfies the random fixed point equation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M78">View MathML</a> almost surely is said to be a wide sense solution of the fixed point equation.

Definition 2.14 Any X-valued random variable <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41">View MathML</a> which satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M80">View MathML</a> is said to be a random solution of the fixed point equation or a random fixed point of F.

Remark 2.15 A random solution is a wide sense solution of the fixed point equation. But the converse is not necessarily true. This is evident from the following example as found under Remark 1 in the work of Joshi and Bose [3].

Example 2.16 Let X be the set of all real numbers and let E be a non-measurable subset of X. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M63">View MathML</a> be a random mapping defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M82">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>.

In this case, the real-valued function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M41">View MathML</a>, defined as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M85">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, is a random fixed point of F. However, the real-valued function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M87">View MathML</a> defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M88">View MathML</a>

is a wide sense solution of the fixed point equation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M89">View MathML</a> without being a random fixed point of F.

3 Main results

Theorem 3.1LetXbe a separable Banach space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31">View MathML</a>be a complete probability measure space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91">View MathML</a>be a continuous random operator such that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, Tsatisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M93">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M94">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M96">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97">View MathML</a>are real-valued random variables such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M98">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M99">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M100">View MathML</a>almost surely.

Then there exists a unique random fixed point ofTinX.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M101">View MathML</a>

Let S be a countable dense subset of X. We now prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M103">View MathML</a>

Then for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M104">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M105">View MathML</a>

(3.1)

Since S is dense in X, given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M106">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M107">View MathML</a>), there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M109">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M107">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M94">View MathML</a>.

Note that

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

We now examine the following cases.

Case I:

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M117">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M118">View MathML</a>

(3.7)

Using (3.2), (3.3), (3.7), we get by routine calculation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M119">View MathML</a>

(3.8)

Since for a particular <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121">View MathML</a> is a continuous function of x, so for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M106">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M124">View MathML</a>) such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M125">View MathML</a>

(3.9)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M126">View MathML</a>

(3.10)

Now choose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M127">View MathML</a>

(3.11)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M128">View MathML</a>

(3.12)

For such a choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130">View MathML</a> by (3.8), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M131">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122">View MathML</a> is arbitrary, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M133">View MathML</a>

(3.13)

Case II:

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M134">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M135">View MathML</a>

(3.14)

Using (3.2), (3.4), (3.5), (3.14), by routine calculation, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M136">View MathML</a>

(3.15)

Since for a particular <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121">View MathML</a> is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130">View MathML</a>, we get by (3.15)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M141">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122">View MathML</a> is arbitrary, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M143">View MathML</a>

(3.16)

Case III:

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M144">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M145">View MathML</a>

(3.17)

Using (3.3), (3.6), (3.17), by a routine check-up, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M146">View MathML</a>

(3.18)

Since for a particular <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121">View MathML</a> is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130">View MathML</a>, we get from relation (3.18)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M151">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122">View MathML</a> is arbitrary, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M153">View MathML</a>

(3.19)

Case IV:

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M154">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M155">View MathML</a>

(3.20)

Using (3.4), (3.5), (3.6) and (3.20), we get by a routine check-up

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M156">View MathML</a>

(3.21)

Since for a particular <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121">View MathML</a> is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M129">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M130">View MathML</a>, we get by (3.21)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M161">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M122">View MathML</a> is arbitrary, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M163">View MathML</a>

(3.22)

Combining (3.13), (3.16), (3.19) and (3.22), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M164">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M165">View MathML</a>, which implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M166">View MathML</a>

Also,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M167">View MathML</a>

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M168">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M169">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M170">View MathML</a>.

So, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M171">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M121">View MathML</a> is a deterministic operator due to Ćirić [40]. Hence, T has a unique fixed point in X. □

Theorem 3.2LetXbe a separable Banach space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M31">View MathML</a>be a complete probability measure space. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91">View MathML</a>be a continuous random operator such that for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, Tsatisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M176">View MathML</a>

(3.23)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M4">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M96">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97">View MathML</a>are real-valued random variables such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M181">View MathML</a>

(3.24)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M182">View MathML</a>

(3.25)

thenThas a unique random fixed point inX.

Proof Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M183">View MathML</a>.

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M184">View MathML</a> and we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M185">View MathML</a>

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M186">View MathML</a>, the relation (3.23), (3.24) and (3.25) would imply Theorem 3.1 with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M184">View MathML</a>. Therefore, we can apply Theorem 3.1 and consequently T has a unique random fixed point in X. □

We now give a couple of examples in support of Theorem 3.1 and Theorem 3.2.

Example 3.3 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M188">View MathML</a> with the usual norm of reals.

Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M189">View MathML</a> and let β be a σ-algebra of Lebesgue measurable sets of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M190">View MathML</a>.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M191">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M193">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>.

By a routine check-up, we see that the condition of Theorem 3.1 is satisfied whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M195">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M196">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M197">View MathML</a>. The function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M199">View MathML</a> is a unique random fixed point of T.

By considering E and Ω as above, we take <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M200">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M196">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M202">View MathML</a>. We see that condition (3.23) of Theorem 3.2 is satisfied and the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M199">View MathML</a> is the unique random fixed point of T.

Example 3.4 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M205">View MathML</a> with the usual norm of reals. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M206">View MathML</a>. β be a σ-algebra of Lebesgue measurable sets of ℝ.

Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M91">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M208">View MathML</a>.

All the conditions of Theorem 3.1 and Theorem 3.2 are satisfied. In both of the cases, we see that the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M198">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M210">View MathML</a> is the unique random fixed point of T.

4 Application to a random nonlinear integral equation

Here we apply Theorem 3.1 to prove the existence of a solution in a Banach space of a random nonlinear integral equation of the following form:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M211">View MathML</a>

(4.1)

where

(i) S is a locally compact metric space with a metric d on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M212">View MathML</a> equipped with a complete σ-finite measure <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213">View MathML</a> defined on the collection of Borel subsets of S;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M67">View MathML</a>, where ω is the supporting element of a set of probability measure space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M215">View MathML</a>;

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M216">View MathML</a> is the unknown vector-valued random variable for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217">View MathML</a>;

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M218">View MathML</a> is the stochastic free term defined for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M219">View MathML</a>;

(v) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220">View MathML</a> is the stochastic kernel defined for t and s in S and

(vi) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M221">View MathML</a> is a vector-valued function of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217">View MathML</a> and x.

The integral in equation (4.1) is interpreted as a Bochner integral [43].

We shall further assume that S is the union of a decreasing sequence of countable family of compact sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M223">View MathML</a> such that for any other compact set in S there is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M224">View MathML</a> which contains it (see [44]).

Definition 4.1 We define the space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225">View MathML</a> to be the space of all continuous functions from S into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M226">View MathML</a> with the topology of uniform convergence on compact sets of S, that is, for each fixed <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M228">View MathML</a> is a vector-valued random variable such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M229">View MathML</a>

It may be noted that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a> is a locally convex space (see [43]) whose topology is defined by a countable family of semi-norms given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M231">View MathML</a>

Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a> is complete relative to this topology since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M226">View MathML</a> is complete.

We further define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M234">View MathML</a> to be the Banach space of all bounded continuous functions from S into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M235">View MathML</a> with the norm

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M236">View MathML</a>

Here the space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M237">View MathML</a> is the space of all second-order vector-valued functions defined on S which are bounded and continuous in mean square. We will consider the function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M218">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M239">View MathML</a> to be in the space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a> with respect to the stochastic kernel. We assume that for each pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M241">View MathML</a> and denote the norm by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M242">View MathML</a>

Let us suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220">View MathML</a> is such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M244">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213">View MathML</a>-integrable with respect to s for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M247">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a>, and let there exist a real-valued function G defined <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213">View MathML</a>-a.e. on S, so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M250">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213">View MathML</a>-integrable so that for each pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M252">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M253">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M213">View MathML</a>-a.e. Further, for almost all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M255">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M220">View MathML</a> will be continuous in t from S into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M257">View MathML</a>.

We now define the random integral operator T on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M259">View MathML</a>

(4.2)

where the integral is a Bochner integral. Moreover, we have that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M217">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M261">View MathML</a> and that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M262">View MathML</a> is continuous in mean square by the Lebesgue dominated convergence theorem. So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M263">View MathML</a>.

Definition 4.2 (see [29] and [45])

Let B and D be two Banach spaces. The pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M264">View MathML</a> is said to be admissible with respect to a random operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M265">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M266">View MathML</a>.

Lemma 4.3 (see [28])

The linear operatorTdefined by (4.2) is continuous from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M267">View MathML</a>into itself.

Lemma 4.4 (see [45] and [28])

IfTis a continuous linear operator from<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a>into itself and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M269">View MathML</a>are Banach spaces stronger than<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M264">View MathML</a>is admissible with respect toT, thenTis continuous fromBintoD.

Remark 4.5 (see [28])

The operator T defined by (4.2) is a bounded linear operator from B into D.

A random solution of equation (4.1) will mean a function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M228">View MathML</a> in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M230">View MathML</a> which satisfies equation (4.1) μ-a.e.

We are now in a position to prove the following theorem.

Theorem 4.6We consider the stochastic integral equation (4.1) subject to the following conditions:

(a) BandDare Banach spaces stronger than<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M225">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M275">View MathML</a>is admissible with respect to the integral operator defined by (4.2);

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M276">View MathML</a>is an operator from the set<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M277">View MathML</a>into the spaceBsatisfying

(4.3)

for<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M279">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M95">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M281">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M97">View MathML</a>are real-valued random variables where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M283">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M284">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M285">View MathML</a>almost surely, and

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M286">View MathML</a>.

Then there exists a unique random solution of (4.1) in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287">View MathML</a>, provided<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M288">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M289">View MathML</a>

Proof Define the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290">View MathML</a> from <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287">View MathML</a> into D by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M292">View MathML</a>

Now

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M293">View MathML</a>

Then from (4.3) of this theorem,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M294">View MathML</a>

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M295">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M296">View MathML</a>

(4.4)

So, by (4.4)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M297">View MathML</a>

(4.5)

Again, suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M298">View MathML</a>

So,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M299">View MathML</a>

(4.6)

Therefore, by (4.6)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M300">View MathML</a>

(4.7)

Also, suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M301">View MathML</a>

Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M302">View MathML</a>

(4.8)

Therefore, by (4.8)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M303">View MathML</a>

(4.9)

Suppose

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M304">View MathML</a>

So,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M305">View MathML</a>

(4.10)

Therefore, by (4.10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M306">View MathML</a>

(4.11)

Then by (4.5), (4.7), (4.9) and (4.11), we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M307">View MathML</a>.

Then for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M308">View MathML</a>. We have by condition (b)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M310">View MathML</a>.

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290">View MathML</a> is a random contractive nonlinear operator on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M287">View MathML</a>. Hence, by Theorem 3.1, there exists a random fixed point of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M290">View MathML</a>, which is the random solution of equation (4.1). □

Example 4.7 Consider the following nonlinear stochastic integral equation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M314">View MathML</a>

Comparing with (4.1), we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M315">View MathML</a>

By routine calculation, it is easy to show that (4.3) is satisfied with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M316">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M317">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M318">View MathML</a>.

Comparing with integral operator equation (4.2), we see that the norm of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M265">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M320">View MathML</a>.

Also, we see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/209/mathml/M321">View MathML</a>. So, all the conditions of Theorem 4.6 are satisfied and hence there exists a random fixed point of the integral operator T satisfying (4.2).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both the authors are jointly responsible for the present research work and have equally contributed to the research work. Both the authors have read and approved the final manuscript.

Acknowledgements

Authors remain grateful to the honorable reviewers for their kind suggestions for improvement of our paper.

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