Journal of Function Spaces

Volume 2015, Article ID 136306, 13 pages

http://dx.doi.org/10.1155/2015/136306

## New Dependent Fixed Point Theorems for Suzuki Type -Contractions

^{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

Received 30 December 2014; Revised 6 March 2015; Accepted 9 March 2015

Academic Editor: Calogero Vetro

Copyright © 2015 M. A. Kutbi 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 new concepts of an -*GF*-contractive nonself-mapping, a weak -*GF*-contractive nonself-mapping, a generalized -*GF*-contractive nonself-mapping, and Suzuki type *GF*-contractions and establish the existence of
*PPF* dependent fixed point theorems for such kind of contractive nonself-mappings in the Razumikhin class. As applications of our results, we derive some *PPF* dependent fixed point theorems for *GF*-contractive nonself-mappings whenever the range space is endowed with a graph or a partial order. The obtained results generalize, extend, and modify some *PPF* dependent fixed point results in the literature.

#### 1. Introduction

It is well known that the contraction mapping principle, formulated and proved in the PhD dissertation of Banach, has laid the foundation of metric fixed point theory for contraction mappings on complete metric spaces. Since then, Banach’s fixed point theorem has been generalized, improved, and extended in several directions; see the papers (see [1–11] and references therein). Bernfeld et al. [12] introduced the concept of fixed point for mappings that have different domains and ranges, which is called dependent fixed point or the fixed point with dependence. Also, they introduced the notion of Banach type contraction for nonself-mapping and established the existence of dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings (see [13]). The dependent fixed point theorems are useful for proving the solutions of nonlinear functional differential and integral equations which may depend on the past history, present data, and future consideration (see [14–16]). However, as proved in a recent paper by Cho et al. [17], the starting conditions [imposed by the problem setting] relative to the ambient Razumikhin class may be converted into starting conditions relative to the constant class ; so, ultimately, we may arrange for these dependent fixed point results holding over .

On the other hand, Samet et al. [18] introduced and studied --contractive mappings in complete metric spaces and provided applications of the results to ordinary differential equations. More recently, Salimi et al. [19] modified the notions of --contractive and -admissible mappings and established fixed point theorems to modify the results in [18].

Consistent with Wardowski [20], we denote by the set of all functions satisfying the following conditions:) is strictly increasing;()for all sequence , if and only if ;()there exists such that .

A mapping is called an *-contraction* if there exists such that, for all , we have

For example, from the -contraction of , we have the following.

Define a mapping by for all . Then it is clear that satisfies ()–(). Each mapping satisfying is an -contraction such that for all with . Clearly, for all with , the inequality also holds and this implies that is a Banach contraction.

Further, consistent with Hussain and Salimi [21], we denote by the set of all functions satisfying the following.()For all with there exists such that .

*Example 1 (see [21]). *If where and , then .

*Example 2 (see [21]). *If where and , then .

*Example 3 (see [21]). *If where and , then .

In this paper, motivated by the works of Hussain and Salimi, Samet et al., and Wardowski we introduce the concept of a 0--admissible nonself-mapping, a --contractive nonself-mapping, a weak --contractive nonself-mapping, a generalized --contractive nonself-mapping, and Suzuki type -contractive nonself-mapping and establish the existence of dependent fixed point theorems for such kinds of contractive nonself-mappings in the Razumikhin class. As applications of our theorems, we deduce some dependent fixed point theorems for -contractive nonself-mapping and integral type contractions whenever the range space is endowed with a graph or a partial order.

#### 2. Preliminaries

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

Here, denotes the set of all natural numbers; in addition, for each , we put .

*Definition 4 (see [12]). *A mapping is said to be a dependent fixed point or a fixed point with dependence of mapping if for some .

Motivated by results of Agarwal et al. [22], Ćirić et al. [23], and Hussain et al. [15], we give the following notion which is suitable for our main results.

*Definition 5. *Let and and . is called a 0--admissible nonself-mapping if, for any ,

*Example 6. *Let be a real Banach space with usual norm and . Define by for all and bythen is an --admissible nonself-mapping.

*Definition 7. *Let and be two nonself-mappings and .(i) is called a --contraction if there exist and such that, for all with , (ii) is called a* weak *-contraction if there exist and such that, for all with , (iii) is called a generalized -contraction if there exist and such that, for all with ,

*Definition 8 (see [17]). *The Razumikhin or minimal class (attached to ) is defined asAlso, denote, for simplicity,It will be referred as the constant Razumikhin class. To get a useful representation for this subclass, we need a lot of preliminary facts. For each , let denote the constant function of , defined as Note that, by this definition,whence . We now claim thator, in other words, the constant Razumikhin class is just the subclass of all constant functions in . In fact, the right to left inclusion is clear. For the left to right inclusion, it will suffice noting that any constant function in may be written as and this ends our argument.

The following properties of this subclass are almost immediate, so, we do not give details.

Proposition 9 (see [17]). *Under the above conventions, *(i)*, ,*(ii)*, , ,*(iii)*, ,*(iv)*the mapping is an algebraic and topological isomorphism between and .*

*Definition 10. *Let be a nonself-mapping.(i) is called *-starting*, if there exists such that .(ii) is called *-starting*, if there exists such that .

Clearly, if is -starting, then it is also -starting. The reciprocal assertion is also true, under certain regularity conditions upon . Precisely, we have the following.

*Proposition 11. Let the nonself-mapping be such that (i) is 0--admissible;(ii) is -starting.Then, is -starting.*

*Proof. *By (ii) there exist , such that Since , we may consider the element from the constant Razumikhin class ; this, by definition, means thatThe condition upon becomes Since is 0--admissible, this yields or, equivalently, This ends the argument.

*The usefulness of this result is to be judged from the observations below. To state them, two more concepts involving the Razumikhin class are needed.*

*Definition 12. *(i) The class is said to be algebraically closed with respect to difference if whenever .

(ii) The class is said to be* topologically closed* if it is closed with respect to the topology on generated by the norm .

*3. Main Results*

*We start with the following proposition which will be crucial to our main results.*

*Proposition 13. Let , , , and be such that(i) is 0--admissible,(ii) is a generalized --contraction,(iii)there exists such that .In addition, assume that(Ze) has no dependent fixed points in (, for all ).Then, there exist a sequence in , a , and a , such that (c1) and for all ;(c2) as ;(c3) (hence, ), for all .*

*Proof. *By a previous observation, condition (iii) may be written as (iv)there exists such that .Let be such an element. Since , we may consider the element from the constant Razumikhin class ; this, by definition, means thatFurther, since , we may consider the element from the constant Razumikhin class ; this, by definition, means thatThe process may continue indefinitely; it gives us a sequence in the constant Razumikhin class , withBy the algebraic-topological properties of the constant Razumikhin class , it follows thatfor all . Since is 0--admissible andthenAgain since is 0--admissible, thenBy continuing this process, we havefor all ; and this proves the conclusion (c1).

By the imposed condition on nonself-mapping , a relation like is impossible, so that, Now since is generalized --contraction, so we haveOn the other hand,Now since , so from () there exists such that,Since , so by (30) and (32) we obtainTherefore,which implies thatNow, if , then we havewhich is a contradiction. Therefore, we have and soHence which implies , since . Further, there exists such thatFrom (38), it follows thatBy taking limit as we haveTherefore, there exists such thatfor all . This implies thatfor all . Now, for all , we obtain Since , the series converges. Therefore, as , which tells us that is a Cauchy sequence. Completeness of ensures that there exists such that as . As a consequence, conclusion (c2) holds too. Finally, assume that conclusion (c3) is not true. ConsiderThis tells us that there exists an infinite sequence in , with Passing to limit as we get , which is contradiction to the imposed hypothesis. Hence, the conclusion (c3) holds too and the proof is complete.

*Theorem 14. Let , , , and be such that(i) is 0--admissible,(ii) is an --contraction,(iii)if is a sequence in such that as and for all , then for all ,(iv)there exists such that .Then has a dependent fixed point in .*

*Proof. *Since is strictly increasing function, every --contraction is a generalized --contraction. Thus all conditions of Proposition 13 hold and hence there exist a sequence in , a and such that(c1), for all ,(c2) as ,(c3) (hence, ), for all .Since is an --contraction, we have, , which impliessince . Now we obtainSince , we getand henceTaking limit as in the above inequality, we have ; that is, . This completes the proof.

*Example 15. *Let be a Banach space where and and denote the set of all continuous -valued functions on equipped with the supremum norm defined byDefine , , , and byLet . Then, and and so and ; that is, . Therefore is 0--admissible mapping. Let be a sequence in such that as and for all . Now since for all and as , so ; that is, for all . Clearly, .

Let and . Then , , , andTherefore,Otherwise, implies thatHence, is an --contraction and all conditions of Theorem 14 hold. Thus has a dependent fixed point. Here, is dependent fixed point of .

Further, define and for all . Clearly, . Note thatThus and hence, in this case, is not algebraically closed with respect to difference. Therefore it is interesting to notice that almost all results on dependent fixed points established in [15, 22, 23] cannot be applied to this example.

*If in Theorem 14 we take for all , then we derive following result.*

*Corollary 16. Let the nonself-mapping and be such that there exist and such that, for all with ,Then has a dependent fixed point in .*

*Theorem 17. Let the nonself-mapping , , and the be such that(i) is 0--admissible,(ii) is generalized --contraction such that and are continuous,(iii)if is a sequence in such that as and for all , then for all ,(iv)there exists such that .Then has a dependent fixed point in .*

*Proof. *Since is generalized --contraction, it follows from Proposition 13 that there exist a sequence in , a , and a such that(c1), for all ,(c2) as ,(c3) (hence, ), for all .Since is generalized --contraction and for all , we obtainNow, since and are continuous, by taking limit as , in the above inequality, we obtainwhich is a contradiction. Therefore, we have ; that is, . This completes the proof.

*If in Theorem 17 we take for all , then we derive the following corollary.*

*Corollary 18. Let the nonself-mapping , , and be such that, for all with , one hasThen has a dependent fixed point in .*

*Similarly, we can prove the following theorem.*

*Theorem 19. Let the nonself-mapping , , and be such that(i) is --admissible;(ii) is a weak --contraction and is continuous;(iii)if is a sequence in such that as and for all , then for all ;(iv)there exists such that .Then has a dependent fixed point in .*

*If in Theorem 19 we take for all , then we derive the following corollary.*

*Corollary 20. Let the nonself-mapping , , and be such that, for all with , one hasThen has a dependent fixed point in .*

*Now, motivated by Suzuki [24], we prove the following Suzuki type results for --contractions in Razumikhin class.*

*Theorem 21. Let the nonself-mapping and be such that(i) is 0--admissible;(ii)there exist and such that, for all with and , one has(iii)if is a sequence in such that as and for all , then for all ;(iv)there exists such that .Then has a dependent fixed point in .*

*Proof. *By previous observation, condition (iv) may be written as(v)there exists such that .Since , we may consider the element from the constant Razumikhin class ; this, by definition, means thatFurther, since , we may consider the element from the constant Razumikhin class ; this, by definition, means thatThe process may continue indefinitely; it gives us a sequence in , withBy the algebraic-topological properties of the constant Razumikhin class , it follows thatfor all . Since is 0--admissible andsoAgain since is a 0--admissible nonself-mapping, soBy continuing this process, we havefor all . By the imposed condition about our nonself-mapping , a relation like is impossible, so that, we must have Now, we haveSo, by (63), we haveand thenOn the other hand there exists