Journal of Function Spaces
Volume 2015 (2015), Article ID 136306, 13 pages
http://dx.doi.org/10.1155/2015/136306

Research Article

New Dependent Fixed Point Theorems for Suzuki Type -Contractions

M. A. Kutbi,¹ N. Hussain,¹ and S. Khaleghizadeh²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department 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 by

then 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 as

Also, 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 that

or, 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 that

The 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 that

Further, since , we may consider the element from the constant Razumikhin class ; this, by definition, means that

The process may continue indefinitely; it gives us a sequence in the constant Razumikhin class , with

By the algebraic-topological properties of the constant Razumikhin class , it follows that

for all . Since is 0--admissible and

then

Again since is 0--admissible, then

By continuing this process, we have

for 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 have

On the other hand,

Now since , so from () there exists such that,

Since , so by (30) and (32) we obtain

Therefore,

which implies that

Now, if , then we have

which is a contradiction. Therefore, we have

and so

Hence which implies , since . Further, there exists such that

From (38), it follows that

By taking limit as we have

Therefore, there exists such that

for all . This implies that

for 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. Consider

This 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 implies

since . Now we obtain

Since , we get

and hence

Taking 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 by

Define , , , and by

Let . 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 , ,  , and

Therefore,

Otherwise, implies that

Hence, 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 that

Thus 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 obtain

Now, since and are continuous, by taking limit as , in the above inequality, we obtain

which 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 has

Then 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 has

Then 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 that

Further, since , we may consider the element from the constant Razumikhin class ; this, by definition, means that

The process may continue indefinitely; it gives us a sequence in , with

By the algebraic-topological properties of the constant Razumikhin class , it follows that

for all . Since is 0--admissible and

so

Again since is a 0--admissible nonself-mapping, so

By continuing this process, we have

for all . By the imposed condition about our nonself-mapping , a relation like

is impossible, so that, we must have

Now, we have

So, by (63), we have

and then

On the other hand there exists such that

and so

which implies

for all . Again as in proof of Proposition 13, we can deduce that is a Cauchy sequence in and there exists such that as .
Also, from (iii) we have for all .
From (79), we get

Now, since , we have

for all . Suppose that there exists such that

Then, from (81), it follows that

which is a contradiction. Hence either

or

holds for all . First, suppose that

holds for all . Therefore, from (63), we have

which implies

Now, since , we get

and so

Taking limit as in the above inequality, we get ; that is, . By the similar method, we can deduce when

Hence it follows that is a dependent fixed point of in . This completes the proof.

Corollary 22. Let the nonself-mapping , , and be such that for all with and

holds. Then

() has a dependent fixed point in ;() has a unique dependent fixed point in .

Proof. By taking in Theorem 21,

we deduce that has a dependent fixed point in . For the uniqueness, suppose that and are two dependent fixed points of in such that . So, we have

which implies from (92) that

Thus,

On the other hand, there exists, such that

and so

which is a contradiction. Hence . This completes the proof.

4. Some Results in Banach Spaces Endowed with a Graph

Consistent with Jachymski [7], 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 [25], p. 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 [4, 6, 7]).

Definition 23 (see [7]). Let be a metric space endowed with a graph . One says 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:

Definition 24. Let be a nonself-mapping and , where is endowed with a graph :

(i) is called a graphic -contraction if there exist and such that, for all with and , one has

(ii) is called a weak graphic -contraction if there exist and such that, for all with and , one has

(iii) is called a generalized graphic -contraction if there exist and such that, for all with and , one has

Theorem 25. Let be a nonself-mapping and , where is endowed with a graph . Suppose that the following assertions hold:

(i)if , then ;(ii) is a graphic -contraction, for some ;(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. Define by

First, we prove that is 0--admissible nonself-mapping. Assume that . Then we have . From (i), we have ; that is, . Thus is an 0--admissible nonself-mapping. From (iv), there exists such that . Let be a sequence in such that as and for all . Then for all . Thus from (iii), we get for all . That is, for all .
Now, if , then . Hence, from definition of graphic -contraction, we have

Otherwise, . Then

Therefore, for all , we have

and so all the conditions of Theorem 14 hold and has a dependent fixed point. This completes the proof.

Similarly, we can prove the following theorems.

Theorem 26. Let be a nonself-mapping and , where is endowed with a graph . Suppose that the following assertions hold:

(i)if , then ;(ii) is a graphic weak -contraction, such that 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 .

Theorem 27. Let be a nonself-mapping and , where is endowed with a graph . Suppose that the following assertions hold:

(i)if , then ;(ii) is a graphic 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 .

5. Some Results in Banach Spaces Endowed with a Partial Order

The study of existence of fixed points in partially ordered sets has been established by Ran and Reurings [11] and Nieto and Rodrıguez-Lopez [26] with applications to matrix and differential equations. Agarwal et al. [1], Ćirić et al. [5], and Hussain et al. [6, 27] and many other authors obtained some new fixed point results for nonlinear contractions in partially ordered Banach and metric spaces with some applications. In this section, as an application of our results we derive some new dependent fixed and coincidence point results whenever the range space is endowed with partial order.

Definition 28 (see [15]). Let , , and be endowed with a partial order . One says that is a -increasing nonself-mapping if for with one has .

Definition 29. Let be a nonself-mapping and , where is endowed with a partial order :

(i) is called ordered -contraction if there exist and such that, for all with and , one has

(ii) is called weak ordered -contraction if there exist and such that, for all with and , one has

(iii) is called generalized ordered -contraction if there exist and such that, for all with and , one has

Theorem 30. Let be a nonself-mapping and , where is endowed with a partially order . Suppose that the following conditions hold:

(i) is -increasing;(ii) is an ordered -contraction such that 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 .

Proof. Define by

First, we prove that is an 0--admissible nonself-mapping. Assume that . Then we have . Since is -increasing, we get , that is, . Hence, is a 0--admissible nonself-mapping. From (iv), there exists such that . That is, . Let be a sequence in such that as and for all . Then for all . Thus, from (iii), we get for all . That is, for all . Therefore, all the conditions of Theorem 14 hold and has a dependent fixed point. This completes the proof.

Similarly, we can prove the following results.

Theorem 31. Let be a nonself-mapping and , where is endowed with a partially order . Suppose that the following conditions hold:

(i) is -increasing;(ii) is an ordered weak -contraction, such that 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 .

Theorem 32. Let be a nonself-mapping and , where is endowed with a partially order . Suppose that the following assertions hold:

(i) is -increasing;(ii) is an ordered 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 .

Conflict of Interests

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

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR and KAU for financial support. The authors are grateful to the anonymous referees for their useful comments and suggestions.

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