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Fixed point theorems for mappings satisfying contractive conditions of integral type and applications

Zeqing Liu1, Xin Li1, Shin Min Kang2* and Sun Young Cho2*

Author Affiliations

1 Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, People's Republic of China

2 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea

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Fixed Point Theory and Applications 2011, 2011:64  doi:10.1186/1687-1812-2011-64


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2011/1/64


Received:13 April 2011
Accepted:11 October 2011
Published:11 October 2011

© 2011 Liu et al; licensee Springer.

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

Abstract

In this paper, the existence, uniqueness and iterative approximations of fixed points for contractive mappings of integral type in complete metric spaces are established. As applications, the existence, uniqueness and iterative approximations of solutions for a class of functional equations arising in dynamic programming are discussed. The results presented in this paper extend and improve essentially the results of Branciari (A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29, 531-536, 2002), Kannan (Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71-76, 1968) and several known results. Four concrete examples involving the contractive mappings of integral type with uncountably many points are constructed.

2010 Mathematics Subject Classfication: 54H25, 47H10, 49L20, 49L99, 90C39

Keywords:
contractive mappings of integral type; complete metric space; fixed point theorem; functional equation; dynamic programming; bounded solution

1. Introduction

Throughout this paper, we assume that ℝ = (-∞, + ∞), ℝ+ = [0, + ∞), ℕ denotes the set of all positive integers, opt stands for sup or inf, Z and Y are Banach spaces, S Z is the state space, D Y is the decision space, B(S) denotes the Banach space of all bounded real-valued functions on S with norm

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M1">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M2">View MathML</a>

The famous Banach contraction principle is as follows.

Theorem 1.1. ([1]) Let f be a mapping from a complete metric space (X,d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M3">View MathML</a>

(1.1)

where c ∈ (0, 1) is a constant. Then f has a unique fixed point a X such that limn→∞ fn x = a for each x X.

It is well known that the Banach contraction principle has a lot of generalizations and various applications in many directions, see, for example, [2-30] and the references cited therein. In 1962, Rakotch [29] extended the Banach contraction principle with replacing the contraction constant c in (1.1) by a contraction function γ and established the result later.

Theorem 1.2. ([29]) Let f be a mapping from a complete metric space (X, d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M4">View MathML</a>

(1.2)

where γ : ℝ+ → [0,1) is monotonically decreasing. Then f has a unique fixed point a X such that limn→∞ fnx = a for each x X.

In 1968, Kannan [12] generalized the Banach contraction principle from continuous mappings to noncontinuous mappings and proved the following fixed point theorem.

Theorem 1.3. ([12]) Let f be a mapping from a complete metric space (X, d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M5">View MathML</a>

(1.3)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M6">View MathML</a>is a constant. Then f has a unique fixed point in X.

In 2002, Branciari [8] gave an integral version of the Banach contraction principles and showed the following fixed point theorem.

Theorem 1.4. ([8]) Let f be a mapping from a complete metric space (X,d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M7">View MathML</a>

(1.4)

where c ∈ (0, 1) is a constant and φ ∈ Φ. Then f has a unique fixed point a X such that limn→∞fnx = a for each x X.

In recent years, there has been increasing interest in the study of fixed points and common fixed points of mappings satisfying contractive conditions of integral type. The authors [2,3,9-11,28,30] and others continued the study of Branciari. In 2006, Aliouche [2] proved a fixed point theorem using a general contractive condition of integral type in symmetric spaces. In 2007, Djoudi and Aliouche [9] obtained common fixed point theorems of Gregus type for two pairs of weakly compatible mappings satisfying contractive conditions of integral type, and Suzuki [30] proved that Theorem 1.4 previously is a corollary of the Meir-Keeler fixed point theorem and that the Meir-Keeler contractions of integral type are still Meir-Keeler contractions. In 2009, Pathak [28] bore out a general common fixed point theorem of integral φ-type for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces, and Jachymski [10] testified that most contractive conditions of integral type given recently by many authors coincide with classical ones and got a new contractive condition of integral type which is independent of classical ones. However, to the best of our knowledge, the concrete examples constructed in [8,10], which guarantee the existence of fixed points for the contractive mappings of integral type in complete metric spaces, include at most countably many points.

On the other hand, by using various fixed point theorems, the authors [4-7,13-26] studied the existence, uniqueness and iterative approximations of solutions, coincidence solutions and nonnegative solutions for the functional equations arising in dynamic programming below

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M8">View MathML</a>

(1.5)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M9">View MathML</a>

(1.6)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M10">View MathML</a>

(1.7)

where x and y signify the state and decision vectors, respectively, T represents the transformation of the process, and f(x) denotes the optimal return function with the initial state x.

The purposes of this paper are both to study the existence, uniqueness and iterative approximations of fixed points for three classes of contractive mappings of integral type, respectively, under different from or weaker than the conditions in [1-3,8-11,28,30], to construct four examples with uncountably many points to show the superiority of the results presented in this paper and to show solvability of the functional Equation (1.7) in B(S). Our results improve essentially Theorems 1.1-1.4.

2. Lemmas

The following lemmas play important roles in this paper.

Lemma 2.1. Let φ ∈ Φ and {rn}n∈ℕ be a nonnegative sequence with limn→∞rn = a. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M11">View MathML</a>

The proof of Lemma 2.1 follows from Remark 2.1 in [27].

Lemma 2.2. Let φ ∈ Φ and {rn}n∈ℕ be a nonnegative sequence. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M12">View MathML</a>if and only if limn→∞rn = 0.

The proof of Lemma 2.2 follows by Lemma 2.1 in [27].

Lemma 2.3. ([18]) Let E be a set, p and q :E → ℝ be mappings. If optyE p(y) and optyE q(y) are bounded, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M13">View MathML</a>

3. Fixed point theorems for contractive mappings of integral type

In this section, we show the existence, uniqueness and iterative approximations of fixed points for three classes of contractive mappings of integral type. For each x X and n ≥ 0, put dn = d(fnx, fn+1x).

Theorem 3.1. Let f be a mapping from a complete metric space (X,d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M14">View MathML</a>

(3.1)

where φ ∈ Φ and α : ℝ+ → [0, 1) is a function with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M15">View MathML</a>

Then f has a unique fixed point a X such that for each x X, limn→∞ fnx = a.

Proof. Let x be an arbitrary point in X. It follows from (3.1) and (3.2) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M16">View MathML</a>

(3.2)

Now, we show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M17">View MathML</a>

(3.3)

Suppose that (3.4) does not hold. That is, there exists some n0 ∈ ℕ satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M18">View MathML</a>

(3.4)

Since φ ∈ Φ, it follows from (3.2), (3.3) and (3.5) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M19">View MathML</a>

which means that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M20">View MathML</a>

which is a contradiction and hence (3.4) holds. Note that (3.4) yields that the sequence {dn}n∈ℕ is nonincreasing, which implies that there exists a constant c with limn→∞ dn = c ≥ 0.

Next, we show that c = 0. Otherwise c > 0. Taking upper limit in (3.3) and using (3.2), Lemma 2.1 and φ ∈ Φ, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M21">View MathML</a>

which is absurd. Therefore, c = 0, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M22">View MathML</a>

(3.5)

Now, we claim that {fnx}n∈ℕ is a Cauchy sequence. Suppose that {fnx}n∈ℕ is not a Cauchy sequence, which means that there is a constant ε > 0 such that for each positive integer k, there are positive integers m(k) and n(k) with m(k) > n(k) > k such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M23">View MathML</a>

For each positive integer k, let m(k) denote the least integer exceeding n(k) and satisfying the above inequality. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M24">View MathML</a>

(3.6)

Note that ∀k ∈ ℕ

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M25">View MathML</a>

(3.7)

In light of (3.6)-(3.8), we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M26">View MathML</a>

(3.8)

In view of (3.1), we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M27">View MathML</a>

(3.9)

Taking upper limit in (3.10) and by virtue of (3.2), (3.9), Lemma 2.1 and φ ∈ Φ, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M28">View MathML</a>

which is a contradiction. Thus, {fnx}n∈ℕ is a Cauchy sequence. Since (X, d) is a complete metric space, there exists a point a X such that limn→∞ fnx = a. By (3.1), (3.2) and Lemma 2.2, we arrive at

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M29">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M30">View MathML</a>

which together with Lemma 2.2 gives that limn→∞ d(fn+1x, fa) = 0. Consequently, we conclude immediately that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M31">View MathML</a>

which means that a = fa.

Finally, we prove that f has a unique fixed point in X. Suppose that f has another fixed point b X\{a}. It follows from φ ∈ Φ, (3.2) and (3.3) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M32">View MathML</a>

(3.10)

which is a contradiction. This completes the proof.

Theorem 3.2. Let f be a mapping from a complete metric space (X, d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M33">View MathML</a>

(3.11)

where φ ∈ Φ and α, β : ℝ+ → [0, 1) are two functions with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M34">View MathML</a>

(3.12)

Then f has a unique fixed point a X such that for each x X, limn→∞ fnx = a.

Proof. Let x be an arbitrary point in X. By (3.11), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M35">View MathML</a>

which together with (3.12) yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M36">View MathML</a>

As in the proof of Theorem 3.1, we conclude similarly that the sequence {dn}n∈ℕ is nonincreasing and converges to 0.

Next, we show that {fnx}n∈ℕ is a Cauchy sequence. Suppose that {fnx}n∈ℕ is not a Cauchy sequence. It follows that there is a constant ε > 0 such that for each positive integer k, there are positive integers m(k) and n(k) with m(k) > n(k) > k with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M37">View MathML</a>

For each positive integer k, let m(k) denote the least integer exceeding n(k) and satisfying the above inequality. It is easy to verify that (3.7)-(3.9) hold. By means of (3.9), (3.11), (3.12), Lemma 2.1 and φ ∈ Φ, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M38">View MathML</a>

which is a contradiction. Hence, {fnx}n∈ℕ is a Cauchy sequence. Since (X, d) is a complete metric space, there exists a point a X such that limn→∞ fnx = a, which means that limn→∞ d(fn+1x, fa) = d(a, fa). If d(a, fa) ≠ 0, by (3.11), (3.12) and Lemma 2.1, we infer that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M39">View MathML</a>

which is impossible. Thus, d(a, fa) = 0. That is, a = fa.

Finally, we prove that f has a unique fixed point in X. Suppose that f has another fixed point b X\{a}. It follows from φ ∈ Φ and (3.12) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M40">View MathML</a>

which is a contradiction. This completes the proof.

As in the proof of Theorem 3.2, we get similarly the below result.

Theorem 3.3. Let f be a mapping from a complete metric space (X, d) into itself satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M41">View MathML</a>

(3.13)

where φ ∈ Φ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M42">View MathML</a>is a function with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M43">View MathML</a>

(3.14)

Then f has a unique fixed point a X such that for each x X, limn→∞ fnx = a.

4. Remarks and illustrative examples

In this section, by constructing four nontrivial examples with uncountably many points, we discuss and compare the fixed point theorems obtained in Section 3 with the known results in Section 1.

Remark 4.1. If α(t) = c for all t ∈ ℝ+, where c ∈ (0,1) is a constant, then Theorem 3.1 changes into Theorem 1.4; furthermore, if φ(t) = 1 for all t ∈ ℝ+, then Theorem 3.1 brings Theorem 1.1. The following example manifests that Theorem 3.1 extends substantially Theorems 1.1 and 1.4.

Example 4.1. Let X = ℝ+ be endowed with the Euclidean metric d = | · |, f: X X, α: ℝ+ → [0,1) and φ ∈ Φ be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M44">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M45">View MathML</a>

It is obvious that (3.2) holds and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M46">View MathML</a>

That is, the conditions of Theorem 3.1 are fulfilled. It follows from Theorem 3.1 that f has a unique fixed point 0 ∈ X. But, we can neither invoke Theorem 1.1 nor Theorem 1.4 to show the existence of a fixed point of f in X because (1.1) and (1.4) do not hold.

Suppose that (1.1) holds. It follows that there exists a constant c ∈ (0,1) satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M47">View MathML</a>

which gives that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M48">View MathML</a>

which yields that c ≥ 1, which is absurd.

Suppose that (1.4) holds. It follows that there exists some constant c ∈ (0,1) satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M49">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M50">View MathML</a>

which means that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M51">View MathML</a>

which is a contradiction.

Remark 4.2. In case φ(t) = 1 for all t ∈ ℝ+, then Theorem 3.1 reduces to a result, which generalizes Theorem 1.2. The following example reveals that Theorem 3.1 is a proper generalization Theorem 1.2.

Example 4.2. Let X = + be endowed with the Euclidean metric d = | · |, f: X X, α: ℝ+→ [0,1) and φ ∈ Φ be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M52">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M53">View MathML</a>

It is easy to see that (3.2) holds. In order to verify (3.1), we have to consider three possible cases as follows:

Case 1. x, y X with x = y. It is clear that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M54">View MathML</a>

Case 2. x, y X with 0 < |x - y| ≤ 1. Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M55">View MathML</a>

Case 3. x,y X with |x - y| > 1. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M56">View MathML</a>

Hence, (3.1) holds. Consequently, the conditions of Theorem 3.1 are satisfied.

It follows from Theorem 3.1 that f has a unique fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M57">View MathML</a>.

However, Theorem 1.2 is useless in guaranteeing the existence of a fixed point of f in X. Otherwise, suppose that the conditions of Theorem 1.2 are fulfilled. Notice that γ: ℝ+ → [0,1) is monotonically decreasing. It follows that limt→∞+ γ(t) exists and belongs to [γ(1), γ(0)] ⊂ [0,1). Using (1.2), we infer that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M58">View MathML</a>

which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M59">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M60">View MathML</a>

which is impossible.

Remark 4.3. In case φ(t) = 1 and γ(t) = h for all t ∈ ℝ+, then Theorem 3.3 comes into being Theorem 1.3. The below example demonstrates that Theorem 3.3 is indeed a proper extension of Theorem 1.3.

Example 4.3. Let X = [0, 4] be endowed with the Euclidean metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M61">View MathML</a> and φ ∈ Φ be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M62">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M63">View MathML</a>

respectively. It is obvious that (3.14) holds and (3.13) is equivalent to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M64">View MathML</a>

(4.1)

Note that x and y in (4.1) are symmetric, (4.1) holds for all x = y X and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M65">View MathML</a>

(4.2)

In order to verify (3.13), by (4.1) and (4.2) we need only to show that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M66">View MathML</a>

(4.3)

Now, we have to consider the below six possible cases:

Case 1. x, y X with 4 ≥ x > y ≥ 2. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M67">View MathML</a>

Case 2. x, y X with 4 ≥ x ≥ 2 > y ≥ 0 and y ≠ 1. It is clear that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M68">View MathML</a>

Case 3. x; y X with 4 ≥ x ≥ 2 and y = 1. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M69">View MathML</a>

Case 4. x, y X with 2 > x > y ≥ 0, x ≠ 1 and y ≠ 1. Notice that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M70">View MathML</a>

Case 5. x, y X with x = 1 > y ≥ 0. Obviously

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M71">View MathML</a>

Case 6. x, y X with 2 > x > 1 = y. Notice that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M72">View MathML</a>

Hence, (3.13) holds. That is, the conditions of Theorem 3.3 are satisfied. It follows from Theorem 3.3 that f has a unique fixed point in X. However, it is easy to verify that for x0 = 1 and y0 = 0

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M73">View MathML</a>

which yields that (1.3) in Theorem 1.3 does not hold.

Next, we construct an example with uncountably many points to explain Theorem 3.2.

Example 4.4. Let X = [1,3] be endowed with the Euclidean metric d = |; · |, f: X X, α, β: ℝ+ → [0,1) and φ ∈ Φ be defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M74">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M75">View MathML</a>

It is easy to see that (3.12) holds. In order to verify (3.11), we have to consider the below five possible cases:

Case 1. x, y X with 3 ≥ x y ≥ 2. Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M76">View MathML</a>

Case 2. x, y X with x [2,3] and y ∈ [1, 2). It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M77">View MathML</a>

Case 3. x, y X with x, y ∈ [1, 2). Notice that fx = fy = 1. It follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M78">View MathML</a>

Case 4. x, y X with 3 ≥ y > x ≥ 2. Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M79">View MathML</a>

Case 5. x, y X with x ∈ [1, 2) and y [2,3]. Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M80">View MathML</a>

that is, (3.11) holds. Thus, all the conditions of Theorem 3.2 are satisfied. It follows from Theorem 3.2 that f has a unique fixed point 1 ∈ X.

5. Applications

In this section, by using the fixed point theorems obtained in Section 3, we study solvability of the functional Equation (1.7) in B(S).

Theorem 5.1. Let u: S × D → ℝ, T : S × D S, H : S × D × ℝ → ℝ, φ ∈ Φ and α : ℝ+ → [0, 1) satisfy (3.2),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M81">View MathML</a>

(5.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M82">View MathML</a>

(5.2)

Then the functional Equation (1.7) has a unique solution w B(S) and {Anz}n∈ℕ converges to w for each z B(S), where the mapping A is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M83">View MathML</a>

(5.3)

Proof. It follows from (5.1) that there exists M > 0 satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M84">View MathML</a>

(5.4)

It is easy to see that A is a self-mappings in B(S) by (5.3), (5.4) and Lemma 2.3.

Using Theorem 12.34 in [31] and φ ∈ Φ, we conclude that for each ε > 0, there exists δ > 0 satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M85">View MathML</a>

(5.5)

where m(C) denotes the Lebesgue measure of C.

Let x S,h,g B(S). Suppose that optyD = infyD. Clearly, (5.3) implies that there exist y, z D satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M86">View MathML</a>

Put

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M87">View MathML</a>

It is easy to verify that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M88">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M89">View MathML</a>

which yield that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M90">View MathML</a>

(5.6)

Similarly, we infer that (5.6) holds also for optyD = supyD. Combining (5.2), (5.5) and (5.6), we arrive at

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M91">View MathML</a>

which means that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M92">View MathML</a>

letting ε → 0+ in the above inequality, we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M93">View MathML</a>

Thus, Theorem 5.1 follows from Theorem 3.1. This completes the proof.

Remark 5.1. Theorem 5.1 extends and unifies Theorem 2.1 in [7], Theorem 3.1 in [18] and Theorem 3.2 in [25].

Theorem 5.2. Let u : S × D → ℝ, T: S × D S, H : S × D × ℝ → ℝ, φ ∈ Φ and α, β : ℝ+ → [0, 1) satisfy (3.12), (5.1) and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M94">View MathML</a>

(5.7)

Then the functional Equation (1.7) has a unique solution w B(S) and {Anz}n∈ℕ converges to w for each z B(S), where the mapping A is defined by (5.3).

Proof. As in the proof of Theorem 5.1, by (3.12), (5.1), (5.3) and (5.7), we conclude that (5.4)-(5.6) hold and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M95">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M96">View MathML</a>

letting ε→ 0+ in the above inequality, we infer that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M97">View MathML</a>

Thus, Theorem 5.2 follows from Theorem 3.2. This completes the proof.

Theorem 5.3. Let u : S × D → ℝ, T : S × D → S, H:S × D × ℝ → ℝ, φ ∈ Φ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M98">View MathML</a>satisfy (3.14), (5.1) and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M99">View MathML</a>

(5.8)

Then the functional Equation (1.7) has a unique solution w ∈ B(S) and {Anz}n∈ℕ converges to w for each z B(S), where the mapping A is defined by (5.3).

Proof. As in the proof of Theorem 5.1, by (3.14), (5.1), (5.3) and (5.8), we conclude that (5.4)-(5.6) hold and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M100">View MathML</a>

which yields that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M101">View MathML</a>

letting ε → 0+ in the above inequality, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/64/mathml/M102">View MathML</a>

Thus, Theorem 5.3 follows from Theorem 3.3. This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

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

Acknowledgements

The authors are indebted to the referees for their helpful comments.

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