Abstract and Applied Analysis

Volume 2013 (2013), Article ID 827205, 9 pages

http://dx.doi.org/10.1155/2013/827205

## New Fixed Point Results with PPF Dependence in Banach Spaces Endowed with a Graph

^{1}Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran^{3}Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran^{4}Department of Mathematics, GDCW, Bosan Road, Multan, Pakistan

Received 30 October 2013; Accepted 21 November 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 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

We introduce the concept of an -admissible non-self-mappings with respect to and establish the existence of PPF dependent fixed and coincidence point theorems for --contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for -contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed point results in the literature. Several interesting consequences of our theorems are also provided.

#### 1. Introduction and Preliminaries

In nonlinear functional analysis, one of the most significant research areas is fixed point theory. On the other hand, fixed point theory has an application in distinct branches of mathematics and also in different sciences, such as engineering, computer science, and economics. In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. This celebrated result have been generalized and improved by many authors in the context of different abstract spaces for various operators (see [1–31] and references therein). In 1997, Bernfeld et al. [5] introduced the concept of fixed point for mappings that have different domains and ranges, which is called PPF dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach type contraction for non-self-mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contractions (see [17]). The PPF dependent fixed point theorems are useful for proving the solutions of nonlinear functional differential and integral equations which may depend upon the past history, present data, and future consideration (see [9]). On the other hand, Samet et al. [22] introduced the concept of -admissible self-mappings and proved fixed point results for -admissible contractive mappings in complete metric spaces and provided application of the obtained results to ordinary differential equations. More recently, Salimi et al. [24] modified the notions of --contractive and -admissible mappings and established fixed point theorems to generalize the results in [22]. In this paper, we introduce the concept of an -admissible non-self-mapping with respect to and establish the existence of PPF dependent fixed and coincidence point theorems for --contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for -contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed results in the literature. Several interesting consequences of our theorems are also provided.

Throughout this paper, we assume that is a Banach space, denotes a closed interval in , and denotes the set of all continuous -valued functions on equipped with the supremum norm defined by

For a fixed element , the Razumikhin or minimal class of functions in is defined by Clearly, every constant function from to belongs to .

*Definition 1. *Let be the Razumikhin class, then(i)the class is algebraically closed with respect to difference, if whenever , ;(ii)the class is topologically closed if it is closed with respect to the topology on generated by the norm .

*Definition 2 (see [5]). *A mapping is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping if for some .

*Definition 3 (see [17]). *Let and let . A point is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of and if for some .

*Definition 4 (see [5]). *The mapping is called a Banach type contraction if there exists such that
for all , .

Samet et al. [22] defined the notion of -admissible mappings as follows.

*Definition 5. *Let be a self-mapping on and let be a function. We say that is an -admissible mapping if

In [22] the authors considered the family of nondecreasing functions such that for each , where is the th iterate of .

Salimi et al. [24] modified and generalized the notions of --contractive mappings and -admissible mappings as follows.

*Definition 6 (see [24]). *Let be a self-mapping on and , be two functions. We say that is an -admissible mapping with respect to if
Note that if we take , then this definition reduces to Definition 5. Also, if we take, , then we say that is an -subadmissible mapping.

The following result is a proper generalization of the above-mentioned results.

Theorem 7 (see [24]). *Let be a complete metric space and let be an -admissible mapping. Assume that
**
where and
**
Also, suppose that the following assertions hold: *(i)*there exists such that ,*(ii)*either is continuous or for any sequence in with for all and as , we have for all .**Then has a fixed point.*

For more details on modified --contractive mappings and related fixed point results we refer the reader to [8, 13, 14, 25, 26].

#### 2. PPF Dependent Fixed and Coincidence Point Results

First we define the notion of non-self -admissible mapping with respect to as follows.

*Definition 8. *Let and let , , . We say that is an -admissible non-self-mapping with respect to if for , ,
Note that if we take , then we say is an -admissible non-self-mapping. Also, if we take , then we say that is an -subadmissible non-self-mapping.

*Example 9. *Let be a real Banach space with usual norm and let . Define by for all and , by
. Then, is an -admissible mapping with respect to . In fact, if , then and so, . That is, which implies that .

Denote with the family of nondecreasing functions such that for all , where is the th iterate of .

The following Remark is obvious.

*Remark 10. *If , then for all .

*Definition 11. *Let , , be three mappings and . Then, (i) is an --contractive non-self-mapping if
(ii) is a modified --contractive non-self-mapping if
where and

The following theorem is our first main result in this section.

Theorem 12. *Let , , be three mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is an -admissible non-self-mapping with respect to ;*(iii)* is an --contractive non-self-mapping;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

*Proof. *Let, . Since , there exists such that . Choose such that,
By continuing this process, by induction, we can build a sequence in such that

Since is algebraically closed with respect to difference, it follows that
If there exists such that , then is a PPF dependent fixed point of and we have nothing to prove. Hence we assume that for all .

Since is an -admissible non-self-mapping with respect to and
so,
By continuing this process we get
for all . Then from (10) we get
where
which implies that
Now, if , then
which is a contradiction. Hence,
for all . So,
for all .

Fix , then there exists such that
Let , with . By triangular inequality we get
Consequently, . Hence is a Cauchy sequence in . By the completeness of , converges to a point , that is, , as . Since is topologically closed, we deduce that . From (iv) we have for all . By (10) we have
where
Taking limit as in the above inequality we get
Therefore, . That is, . This implies that is a PPF dependent fixed point of in .

If in Theorem 12 we take for all , , then we deduce the following corollary.

Corollary 13. *Let and be two mappings satisfy that the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is an -admissible non-self-mapping;*(iii)* is a modified --contractive non-self-mapping;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

We now introduce the notion of -admissible mapping with respect to for the pair of maps as follows.

*Definition 14. *Let , , and let , . We say that the pair is an -admissible with respect to , if for , ,
Note that if we take , then we say that the pair is an -admissible mapping. Also, if we take , then we say that the pair is an -subadmissible mapping.

Now we introduce the notion of --contractiveness for the pair as follows.

*Definition 15. *Let , , and , . Then, (i)we say that the pair is an --contractive if
(ii)we say that the pair is a modified --contractive if
where and

Theorem 16. *Let , , , be four mappings satisfying the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is an -admissible with respect to ;*(iii)*the pair is an --contractive;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**Then, and have a PPF dependent coincidence point.*

*Proof. *As , so there exists such that and is one-to-one. Since , we can define the mapping by for all . Since is one-to-one, then is well defined. Let

Therefore, by (31) we have
where
This shows that is an --contractive non-self-mapping. Further, all other conditions of Theorem 12 hold true for . Thus, there exists PPF dependent fixed point of ; that is, . Since , so there exists such that . Thus,

That is, is a PPF dependent coincidence point of and .

Corollary 17. *Let , , be three mappings satisfying the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is an -admissible;*(iii)*the pair is a modified --contractive;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

#### 3. Some Results in Banach Spaces Endowed with a Graph

Consistent with Jachymski [15], let be a metric space where for all and denotes the diagonal of the Cartesian product of . 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 [16, page 309]) 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 more details [6, 11, 15]).

*Definition 18 (see [15]). *Let be a metric space endowed with a graph . We say that a self-mapping is a Banach -contraction or simply a -contraction if preserves the edges of ; that is,
and decreases weights of the edges of in the following way:

Theorem 19. *Let and endowed with a graph . Suppose that the following assertions hold true: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*if , then ;*(iii)*assume that
*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

*Proof. *Define by
First, we prove that is an -admissible non-self-mapping. Assume that . Then, we have . From (ii), we have ; that is, . Thus is an -admissible non-self-mapping. From (v) there exists such that . Let, be a sequence in such that as and for all . Then, for all . Thus, from (iv) we get, for all . That is, for all . Therefore all conditions of Corollary 13 hold true and has a PPF dependent fixed point.

Similarly as an application of Corollary 17, we can prove the following Theorem.

Theorem 20. *Let , , and endowed with a graph . Suppose that the following assertions hold true: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*if , then ;*(iii)*assume that
for , , where ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

The study of existence of fixed points in partially ordered sets has been initiated by Ran and Reurings [27] with applications to matrix equations. Agarwal et al. [1, 2], Ćirić et al. [7], and Hussain et al. [11, 12] presented some new results for nonlinear contractions in partially ordered Banach and metric spaces with applications. Here as an application of our results we deduce some new PPF dependent fixed and coincidence point results whenever the range space is endowed with a partial order.

*Definition 21. *Let , and endowed with a partial order . We say that is a -increasing non-self-mapping if for , with we have .

*Definition 22. *Let , , and endowed with a partial order . We say that the pair is -increasing if for , with we have .

Theorem 23. *Let and endowed with a partial order . Suppose that the following assertions holds true: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is a -increasing non-self-mapping;*(iii)*Assume that
holds for all , with where ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

*Proof. *Define by
First, we prove that is an -admissible non-self-mapping. Assume that . Then, we have . Since is -increasing, we get ; that is, . Thus is an -admissible non-self-mapping. From (v) there exists such that . That is, . Let be a sequence in such that as and for all . Then, for all . Thus, from (iv) we get for all . That is, for all . Therefore all conditions of Corollary 13 hold true and has a PPF dependent fixed point.

Similarly we can prove following Theorem.

Theorem 24. *Let , , and endowed with a partial order . Suppose that the following assertions hold true:*(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is a -increasing mapping;*(iii)*assume that
holds for all , with , where ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

#### 4. Further Consequences

##### 4.1. Consequences of Corollary 13

Theorem 25. *Let and be two mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is an -admissible mapping;*(iii)*assume that
holds for all , , where ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

*Proof. *Let . Hence, from (iii) we have
That is, all conditions of Corollary 13 are satisfied and has a PPF dependent fixed point.

Similarly we can prove the following results.

Theorem 26. *Let and be two mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is an -admissible mapping;*(iii)*assume that
holds for all , , where and ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, has a PPF dependent fixed point.*

Theorem 27. *Let and be two mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)* is an -admissible mapping;*(iii)*assume that
holds for all , , where and ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**Then, has a PPF dependent fixed point.*

##### 4.2. Consequences of Corollary 17

Theorem 28. *Let , and be three mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is an -admissible;*(iii)*assume that
holds for all , , where ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

Theorem 29. *Let , and be three mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is an -admissible;*(iii)*assume that
holds for all , , where and ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

Theorem 30. *Let , and be three mappings that satisfy the following assertions: *(i)*there exists such that is topologically closed and algebraically closed with respect to difference;*(ii)*the pair is an -admissible;*(iii)*assume that
holds for all , , where and ;*(iv)*if is a sequence in such that as and for all , then for all ;*(v)*there exists such that .**
Then, and have a PPF dependent coincidence point.*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

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