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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 280817, 11 pages

http://dx.doi.org/10.1155/2014/280817

## Fixed Point Theory in -Complete Metric Spaces with Applications

^{1}Department of Mathematics, King Abdulaziz University, P.O. Box 80203, 21589 Jeddah, Saudi Arabia^{2}Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

Received 28 November 2013; Accepted 14 December 2013; Published 6 February 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 N. Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper is to introduce new concepts of --complete metric space and --continuous function and establish fixed point results for modified ---rational contraction mappings in --complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for -graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

*This paper is dedicated to Professor Miodrag Mateljević on the occasion of his 65th birthday*

#### 1. Preliminaries

We know by the Banach contraction principle [1], which is a classical and powerful tool in nonlinear analysis, that a self-mapping on a complete metric space such that for all , where , has a unique fixed point. Since then, the Banach contraction principle has been generalized in several directions (see [2–26] and references cited therein).

In 2008, Suzuki [21] proved the following result that is an interesting generalization of the Banach contraction principle which also characterizes the metric completeness.

Theorem 1. *Let be a complete metric space and let be a self-mapping on . Define a nonincreasing function by
**
Assume that there exists such that
**
for all . Then there exists a unique fixed point of . Moreover, for all .*

In 2012, Samet et al. [19] introduced the concepts of --contractive and -admissible mappings and established various fixed point theorems for such mappings defined on complete metric spaces. Afterwards Salimi et al. [16] and Hussain et al. [7] modified the notions of --contractive and -admissible mappings and established fixed point theorems which are proper generalizations of the recent results in [12, 19].

*Definition 2 (see [19]). *Let be a self-mapping on and let be a function. One says that is an -admissible mapping if

*Definition 3 (see [16]). *Let be a self-mapping on and let be two functions. One says that is an -admissible mapping with respect to if
Note that if we take , then this definition reduces to Definition 2. Also, if we take , then we say that is an -subadmissible mapping.

Here we introduce the notions of --complete metric space and --continuous function and establish fixed point results for modified ---rational contractions in --complete metric spaces which are not necessarily complete. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for -graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

#### 2. Main Results

First, we introduce the notions of --complete metric space and --continuous function.

*Definition 4. *Let be a metric space and . The metric space is said to be --complete if and only if every Cauchy sequence with for all converges in . One says is an -complete metric space when for all and one says is an -complete metric space when for all .

*Example 5. *Let and be a metric function on . Let be a closed subset of . Define by
Clearly, is not a complete metric space, but is an --complete metric space. Indeed, if is a Cauchy sequence in such that for all , then for all . Now, since is a complete metric space, then there exists such that as .

*Remark 6. *Let be a self-mapping on metric space and let be an orbitally -complete. Define by
where is an orbit of a point . Then is an --complete metric space. Indeed, if be a Cauchy sequence, where for all , then . Now, since is an orbitally -complete metric space, then converges in . That is, is an --complete metric space. Also, suppose that ; then . Hence, . That is, . Thus, is an -admissible mapping with respect to .

*Definition 7. *Let be a metric space. Let and . One says is an --continuous mapping on , if for given and sequence with

*Example 8. *Let and be a metric on . Assume that and be defined by
Clearly, is not continuous, but is --continuous on . Indeed, if as and , then and so .

*Remark 9. *Define and as in Remark 6. Let be a an orbitally continuous map on . Then is --continuous on . Indeed if as and for all , so for all , then there exists sequence of positive integer such that as . Now since is an orbitally continuous map on , then as as required.

A function is called Bianchini-Grandolfi gauge function [13, 14, 27] if the following conditions hold:(i)is nondecreasing;(ii)there exist and and a convergent series of nonnegative terms such that for and any .

In some sources, Bianchini-Grandolfi gauge function is known as —comparison function (see e.g., [2]). We denote by the family of Bianchini-Grandolfi gauge functions. The following lemma illustrates the properties of these functions.

Lemma 10 (see [2]). *If , then the following hold: *(i)* converges to as for all ;*(ii)*, for any ;*(iii)* is continuous at ;*(iv)*the series converges for any .*

*Definition 11. *Let be a metric space and let be a self-mapping on . Let

Then, (a)we say is a modified ---rational contraction mapping if
where ;(b)we say is a modified --rational contraction mapping if
where .

The following is our first main result of this section.

Theorem 12. *Let be a metric space and let be a self-mapping on . Also, suppose that are two functions and . Assume that the following assertions hold true:*(i)* is an --complete metric space;*(ii)* is an -admissible mapping with respect to ;*(iii)* is modified ---rational contraction mapping on ;*(iv)* is an --continuous mapping on ;*(v)*there exists such that .**Then has a fixed point.*

*Proof. *Let be such that . Define a sequence in by for all . If for some , then is a fixed point for and the result is proved. Hence, we suppose that for all . Since is -admissible mapping with respect to and , we deduce that . Continuing this process, we get
for all . Now, by (a) we get
where
and so, . Now since is nondecreasing, so from (14), we have
Now, if for some , then
which is a contradiction. Hence, for all we have
By induction, we have
Fix ; there exists such that
Let with . Then by triangular inequality we get
Consequently . Hence is a Cauchy sequence. On the other hand from (13) we know that for all . Now since is an --complete metric space, there is such that as . Also, since is an --continuous mapping, so as . That is, as required.

*Example 13. *Let . We endow with the metric
Define , , and by
Clearly, is not a complete metric space. However, it is an --complete metric space. In fact, if is a Cauchy sequence such that for all , then for all . Now, since is a complete metric space, then the sequence converges in . Let ; then . On the other hand, for all . Then, . That is, is an -admissible mapping with respect to . Let be a sequence, such that as and for all . Then, for all . So, (since for all ). Now, since is continuous on . Then, as . That is, is an --continuous mapping. Clearly, . Let . Now, if or , then which implies which is a contradiction. Then, . Now we consider the following cases: (i)let with ; then,
(ii)let with ; then
(iii)let and ; then
(iv)let , or let , ; then, . That is,
Thus is a modified ---rational contraction mapping. Hence all conditions of Theorem 12 are satisfied and has a fixed point. Here, is fixed point of .

By taking for all in Theorem 12, we obtain the following corollary.

Corollary 14. *Let be a metric space and let be a self-mapping on . Also, suppose that is a function and . Assume that the following assertions hold true: *(i)* is an -complete metric space;*(ii)* is an -admissible mapping;*(iii)* is a modified --rational contraction on ;*(iv)* is an -continuous mapping on ;*(v)*there exists such that .**Then has a fixed point.*

Theorem 15. *Let be a metric space and let be a self-mapping on . Also, suppose that are two functions and . Assume that the following assertions hold true: *(i)* is an --complete metric space;*(ii)* is an -admissible mapping with respect to ;*(iii)* is a modified ---rational contraction on ;*(iv)*there exists such that ;*(v)*if is a sequence in such that with as , then either
holds for all .** Then has a fixed point.*

*Proof. *Let be such that . Define a sequence in by for all . Now as in the proof of Theorem 12 we have for all and there exists such that as . Let . From (v) either
holds for all . Then,
holds for all . Let hold for all . Now from (a) we get
By taking limit as in the above inequality we get
which is a contradiction. Hence, implies . By the similar method we can show that if holds for all .

*Example 16. *Let . We endow with usual metric. Define , , and by
Note that is not a complete metric space. But it is an --complete metric space. Indeed, if is a Cauchy sequence such that for all , then for all . Now, since is a complete metric space, then the sequence converges in . Let ; then . On the other hand, for all . Then, . That is, is an -admissible mapping with respect to . If is a sequence in such that with as . Then, for all . That is,
holds for all . Clearly, . Let, . Now, if or , then , which is a contradiction. So, . Therefore,
Therefore is a modified ---rational contraction mapping. Hence all conditions of Theorem 15 hold and has a fixed point. Here, is a fixed point of .

If in Theorem 15 we take for all , then we obtain the following result.

Corollary 17. *Let be a metric space and let be a self-mapping on . Also, suppose that is a function and . Assume that the following assertions hold true: *(i)* is a -complete metric space;*(ii)* is an -admissible mapping;*(iii)* is a modified --rational contraction mapping on ;*(iv)*there exists such that ;*(v)*if is a sequence in such that with as , then either
holds for all .** Then has a fixed point.*

Corollary 18. *Let be a complete metric space and let be a continuous self-mapping on . Assume that is a modified rational contraction mapping, that is,
**
where . Then has a fixed point.*

Corollary 19. *Let be a complete metric space and let be a continuous self-mapping on . Assume that satisfies the following rational inequality:
**
where and
**
Then has a fixed point.*

#### 3. Consequences

##### 3.1. Suzuki Type Fixed Point Results

From Theorem 12 we deduce the following Suzuki type fixed point result.

Theorem 20. *Let be a complete metric space and let be a continuous self-mapping on . Assume that there exists such that
**
for all , where
**
Then has a unique fixed point.*

*Proof. *Define and by
for all and , where . Clearly, for all . That is, conditions (i)–(v) of Theorem 12 hold true. Let . Then, . Now from (40) we have . That is, is a modified ---rational contraction mapping on . Then all conditions of Theorem 12 hold and has a fixed point. The uniqueness of the fixed point follows easily from (40).

Corollary 21. *Let be a complete metric space and let be a continuous self-mapping on . Assume that there exists such that
**
for all . Then has a unique fixed point.*

Now, we prove the following Suzuki type fixed point theorem without continuity of .

Theorem 22. *Let be a complete metric space and let be a self-mapping on . Define a nonincreasing function by
**
Assume that there exists such that
**
for all . Then has a unique fixed point.*

*Proof. *Define and by
for all and , where . Now, since for all , for all . That is, conditions (i)–(iv) of Theorem 15 hold true. Let be a sequence with as . Since for all , then from (45) we get
for all .

Assume there exists such that
then,
and so by (47) we have
which is a contradiction. Hence, either
holds for all . That is condition (v) of Theorem 15 holds.

Let, . So, . Then from (45) we get . Hence, all conditions of Theorem 15 hold and has a fixed point. The uniqueness of the fixed point follows easily from (45).

##### 3.2. Fixed Point Results in Orbitally -Complete Metric Spaces

Theorem 23. *Let be a metric space and let be a self-mapping on . Suppose the following assertions hold: *(i)* is an orbitally -complete metric space;*(ii)*there exists such that
holds for all for some , where
*(iii)*if is a sequence such that with as , then .** Then has a fixed point.*

*Proof. *Define as in Remark 6. From Remark 6 we know that is an -complete metric space and is an -admissible mapping. Let ; then . Then from (ii) we have
That is, is a modified --rational contraction mapping. Let be a sequence such that with as . So, . From (iii) we have . That is, . Hence, all conditions of Corollary 17 hold and has a fixed point.

Corollary 24. *Let be a metric space and let be a self-mapping on . Suppose the following assertions hold: *(i)* is an orbitally -complete metric space;*(ii)*there exists such that
holds for all for some , where
*(iii)*if is a sequence such that with as , then .** Then has a fixed point.*

##### 3.3. Fixed Point Results for Graphic Contractions

Consistent with Jachymski [11], let be a metric space and let denote the diagonal of the Cartesian product . Consider a directed graph such that the set of its vertices coincides with , and the set of its edges contains all loops; that is, . We assume that has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph (see [11]) by assigning to each edge the distance between its vertices. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that , and for . A graph is connected if there is a path between any two vertices. is weakly connected if is connected (see for details [3, 6, 10, 11]).

Recently, some results have appeared providing sufficient conditions for a mapping to be a Picard operator if is endowed with a graph. The first result in this direction was given by Jachymski [11].

*Definition 25 (see [11]). *We say that a mapping is a Banach -contraction or simply -contraction if preserves edges of ; that is,
and decreases weights of edges of in the following way:

*Definition 26 (see [11]). *A mapping is called -continuous, if given and sequence

Theorem 27. *Let be a metric space endowed with a graph and let be a self-mapping on . Suppose that the following assertions hold: *(i)*for all , ;*(ii)*there exists such that ;*(iii)*there exists such that
for all , where
*(iv)* is -continuous;*(v)*if is a Cauchy sequence in with for all , then is convergent in .** Then has a fixed point.*

*Proof. *Define by
At first we prove that is an -admissible mapping. Let ; then . From (i), we have . That is, . Thus is an -admissible mapping. Let be -continuous on . Then,
That is,
which implies that is -continuous on . From (ii) there exists such that . That is, . Let ; then . Now, from (iii) we have . That is,
Condition (v) implies that is an -complete metric space. Hence, all conditions of Corollary 14 are satisfied and has a fixed point.

Theorem 28. *Let be a complete metric space endowed with a graph and let be a self-mapping on . Suppose that the following assertions hold: *(i)*for all , ;*(ii)*there exists such that ;*(iii)*there exists such that
for all , where
*(iv)* is -continuous.**Then has a fixed point.*

As an application of Corollary 17, we obtain.

Theorem 29. *Let be a metric space endowed with a graph and let be a self-mapping on . Suppose that the following assertions hold: *(i)*for all , ;*(ii)*there exists such that ;*(iii)*there exists such that
for all , where
*(iv)*if is a sequence such that with as , then either
holds for all ;*(v)*if is a Cauchy sequence in with for all , then either is convergent in or is a complete metric space.** Then has a fixed point.*

Let be a partially ordered metric space. Define the graph by

For this graph, condition (i) in Theorem 27 means that is nondecreasing with respect to this order [5]. From Theorems 27–29 we derive the following important results in partially ordered metric spaces.

Theorem 30. *Let be a partially ordered metric space and let be a self-mapping on . Suppose that the following assertions hold: *(i)* is nondecreasing map;*(ii)*there exists such that ;*(iii)*there exists such that
for all , where
*(iv)*either for a given and sequence
or is continuous;*(v)*if is a Cauchy sequence in with for all , then either is convergent in or is a complete metric space.** Then has a fixed point.*

Corollary 31 (Ran and Reurings [15]). *Let be a partially ordered complete metric space and let be a continuous nondecreasing self-mapping such that for some . Assume that
**
holds for all with , where . Then has a fixed point.*

Theorem 32. *Let be a partially ordered metric space and let be a self-mapping on . Suppose that the following assertions hold: *(i)* is nondecreasing map;*(ii)*there exists such that ;*(iii)*there exists such that
for all , where
*(iv)*if is a sequence such that with as , then either
holds for all ;*(v)*if is a Cauchy sequence in with for all , then either is convergent in or is a complete metric space.** Then has a fixed point.*

#### 4. Application to Existence of Solutions of Integral Equations

Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [28–30] and references therein). In this section, we apply our result to the existence of a solution of an integral equation. Let be the set of real continuous functions defined on and let be defined by for all . Then is a complete metric space. Also, assume this metric space endowed with a graph .

Consider the integral equation as follows: and let be defined by We assume that (A) is continuous;(B) is continuous;(C) is continuous;(D)there exists a such that for all