Journal of Function Spaces
Volume 2016 (2016), Article ID 6514920, 7 pages
http://dx.doi.org/10.1155/2016/6514920

Research Article

Fixed Point Theorems for Generalized Mizoguchi-Takahashi Graphic Contractions

Nawab Hussain,¹ Jamshaid Ahmad,² and Marwan Amin Kutbi¹

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department of Mathematics, University of Sargodha (Subcampus Bhakkar), Bhakkar, Pakistan

Received 9 April 2016; Revised 27 June 2016; Accepted 28 June 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Nawab 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.

Remarkable feature of contractions is associated with the concept Mizoguchi-Takahashi function. For the purpose of extension and modification of classical ideas related with Mizoguchi-Takahashi contraction, we define generalized Mizoguchi-Takahashi -contractions and establish some generalized fixed point theorems regarding these contractions in this paper. Some applications to the construction of a fixed point of multivalued mappings in -chainable metric space are also discussed.

1. Introduction

In 1922, Banach [1] presented a revolutionary contraction principle (namely, Banach contraction principle) in which Picard iteration process was used for the evaluation of a fixed point. This principle guarantees the existence and uniqueness of fixed points of certain self-mappings on metric spaces and provides a constructive method to find those fixed points. The Banach contraction principle was also used to establish the existence of a unique solution for a nonlinear integral equation [2]. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool in solving existence problems in many directions. Several authors have obtained various extensions and generalizations of Banach’s theorem by defining a variety of contractive type conditions for self- and non-self-mappings on metric spaces.

Following the Banach contraction principle, Nadler [3] introduced the concept of multivalued contractions using the Hausdorff metric and established that a multivalued contraction possesses a fixed point in a complete metric space (see [4–14]). Subsequently, in 1969, Mizoguchi and Takahashi [15] proved the following famous result as a generalization of Nadler’s fixed point theorem [3].

Theorem 1 (see [15]). Let be a complete metric space and let be a multivalued mapping. Assume that

for all , where is such that

for all Then, has a fixed point.

We denote by the set of all functions satisfying inequalities (1) and (2).

Recently, Javahernia et al. [16] generalized the abovementioned function by introducing the notion of generalized Mizoguchi-Takahashi function in such a way.

Definition 2 (see [16]). A function is called a generalized Mizoguchi-Takahashi function (shortly, generalized -function) if the following conditions hold:

for all ;for any bounded sequence and any nonincreasing sequence , one has

Consistent with Javahernia et al. [16], we denote by the set of all functions satisfying conditions -.

On the other hand, Jachymski [17] proved some fixed point results in metric spaces endowed with a graph and generalized simultaneously Banach’s contraction principle from metric and partially ordered metric spaces in 2008. Consistent with Jachymski, 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 has no parallel edges and so we can identify with the pair . Moreover, we may treat as a weighted graph (see [17]) 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 each A graph is connected if there is a path between any two vertices. is weakly connected if is connected.

In 2010, Beg et al. [18] obtained sufficient conditions for the existence of a fixed point of multivalued mapping in the complete metric space endowed with a graph. For more details in this direction, we refer the reader to [19–24].

Recently, Sultana and Vetrivel [25] introduced the notion of Mizoguchi-Takahashi -contraction in this way.

Definition 3 (see [25]). Let be metric space endowed with a graph and let be a multivalued mapping. Then, is called a Mizoguchi-Takahashi -contraction if

(1)for all , with and ,

 where ;(2) and are such that ; then,

They established the following fixed point theorem as main result regarding Mizoguchi-Takahashi -contraction.

Theorem 4 (see [25]). Let be a complete metric space endowed with a graph and a Mizoguchi-Takahashi -contraction. Assume that there exist and such that

(1);(2)for any sequence , if and for all , then, there is a subsequence such that for each

Then, has a fixed point and the sequence converges to the fixed point of .

The purpose of this paper is to define generalized Mizoguchi-Takahashi -contractions and prove some generalized fixed point theorems for these contractions.

The following lemma of Sultana and Vetrivel [25] for single-valued mappings is crucial for some of our proofs.

Lemma 5. Let be a single-valued mapping satisfying

for all with Let be two points such that there is a path from to . Then, there exist and such that

for all , where

2. Main Results

We start this section with the definition of a generalized Mizoguchi-Takahashi -contraction.

Definition 6. Let be metric space endowed with a graph and . Then, is said to be generalized Mizoguchi-Takahashi -contraction if

(1)for all , with and ,

 where ;(2) and are such that ; then,

The following remark illustrates our main definition.

Remark 7. Taking in Definition 6, we get Definition 3 which is the main definition of [25].

Now, we state our main result of this section.

Theorem 8. Let be a complete metric space endowed with a graph and a generalized Mizoguchi-Takahashi -contraction. Assume that there exist and such that

(1);(2)for any sequence , if and for all , then there is a subsequence such that for each

Then, has a fixed point and the sequence converges to a fixed point of .

Proof. Let Then, there is a path in from to ; that is, , , and for each Select a positive integer such that

Since , it follows from (7) that

and, also, since , by the definition of Hausdorff metric, there exists such that

From (9), we have

It follows from (8) and (11) that Thus, Select a positive integer such that

Since , it follows from (7) that

Since , by the definition of Hausdorff metric, there exists such that

From (13), we have

It follows from (12) and (15) that Thus, Continuing the abovementioned procedure, we have

and, for a chosen positive integer for each ,

Since , it follows from (7) that

Since , by the definition of Hausdorff metric, there exists such that

that is, for each Set Then, there is a path in from to ; that is, , and for each Hence, Select a positive integer such that

Since , it follows from (7) that

and, also, , which gives, by the definition of Hausdorff metric, the notion that there exists such that

It follows from (20) and (22) that and so Select a positive integer such that

Since and , it follows from (7) that

Since , by the definition of Hausdorff metric, there exists such that

It follows from (23) and (25) that and so Continuing the abovementioned procedure, we have

and, for a chosen positive integer for each ,

As , from (7), we have

Since , by the definition of Hausdorff metric, there exists such that

Hence, for each Set Then, there is a path in from to ; that is, , and for each Hence, Continuing in this way for each , we construct by generating a path in from to ; that is, , and for each , and, for a chosen positive integer with for , we have

Since and , it follows from (7) that

Since , by the definition of Hausdorff metric, there exists such that

Hence, for , we have For , we denote and From the abovementioned inequality, we have the notion that is a monotone nonincreasing sequence of nonnegative real numbers for each . Let By and inequality (31), it follows that is a bounded sequence. By , we have

Let for each Then, there exist positive integers and real numbers such that for each , where

Hence, for each , we have

for all , where and For and , we have

Let us denote the second term of the last inequality by to get

where and are nonnegative real numbers. Hence, we have

where are nonnegative real numbers. Thus, for each ,

Hence, for each and , we have

Since as , is a Cauchy sequence and so . If

and , where is the cardinal number of , then and If , then one can deduce that there exists such that for all with This means and for each Since , for a sequence with for each , there is a subsequence such that for each Thus, we have

Taking the limit as , in the last inequality, we have Since is closed, it follows that This completes the proof.

Remark 9. Taking in Theorem 8, we get main result of Sultana and Vetrivel’s work which is Theorem 4.

Remark 10. We have Theorem 1 as a corollary to our Theorem 8 by taking , and

Remark 11. Taking in Theorem 8, we can get the main result of [18].

Javahernia et al. [16] introduced the concept of weak lower semicontinuity in the following way.

Definition 12. A function is said to be weak lower semicontinuous function if, for each bounded sequence , one has

Consistent with Javahernia et al. [16], we denote by the set of all functions satisfying the abovementioned condition.

Theorem 13. Let be a complete metric space endowed with a graph and let be a multivalued mapping such that

(1)for all , with and ,

 where is such that , , and ;(2)if and are such that , then

Assume that there exist and such that

(a);(b)for any sequence , if and for each , then there is a subsequence such that for each

Then, has a fixed point and the sequence converges to the fixed point of .

Proof. Define for all Since, for each bounded sequence , we have , we have Thus, we have

This shows that Also, we have

Thus, by Theorem 8, is fixed point of This completes the proof.

3. Applications

As an application of Theorem 8, we now establish the following theorem for mappings satisfying inequality (7) uniformly locally on an -chainable metric space.

Theorem 14. Let be a complete -chainable metric space and let be a multivalued mapping such that, for all with and , there exists satisfying

Then, has a fixed point.

Proof. We consider the graph with and Evidently, and has no parallel edges. Then, one can easily check that is a generalized Mizoguchi-Takahashi -contraction for the abovementioned chosen graph . Let and Since is -chainable, there is a path from to ; that is, , and for each Thus, there exists such that If is a sequence in such that converges to some , then there exists a positive integer such that for each Therefore, we can obtain a subsequence of such that for all Thus, all the conditions of Theorem 8 hold and has a fixed point. This completes the proof.

Corollary 15 (see [25]). Let be a complete -chainable metric space and let be a multivalued mapping such that, for all with and , there exists satisfying

Then, has a fixed point.

Proof. Take in Theorem 14.

Competing 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. The first and third authors, therefore, acknowledge technical and financial support of KAU.

References

S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.

E. Kryeyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, NY, USA, 1978.

S. Nadler Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475–488, 1969.

W. Cholamjiak, P. Cholamjiak, and S. Suantai, “Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems,” Journal of Nonlinear Science and Its Applications, vol. 8, no. 6, pp. 1245–1256, 2015.

N. Hussain and P. Salimi, “Suzuki-Wardowski type fixed point theorems for α-GF-contractions,” Taiwanese Journal of Mathematics, vol. 18, no. 6, pp. 1879–1895, 2014.

N. Hussain, J. Ahmad, L. Ćirić, and A. Azam, “Coincidence point theorems for generalized contractions with application to integral equations,” Fixed Point Theory and Applications, vol. 2015, no. 1, article 78, 2015.

N. Hussain, J. Ahmad, and A. Azam, “On Suzuki-Wardowski type fixed point theorems,” Journal of Nonlinear Science and Its Applications, vol. 8, no. 6, pp. 1095–1111, 2015.

N. Hussain and M. A. Taoudi, “Krasnosel'skii-type fixed point theorems with applications to Volterra integral equations,” Fixed Point Theory and Applications, vol. 2013, no. 1, article 196, 2013.

A. Latif and A. A. N. Abdou, “Multivalued generalized nonlinear contractive maps and fixed points,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1436–1444, 2011.

A. Latif and D. T. Luc, “A general fixed point theorem for multivalued mappings that are not necessarily contractions and applications,” Abstract and Applied Analysis, vol. 2014, Article ID 104762, 5 pages, 2014.

H. K. Pathak, R. P. Agarwal, and Y. J. Cho, “Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions,” Journal of Computational and Applied Mathematics, vol. 283, pp. 201–217, 2015.

R. Saadati, S. M. Vaezpour, and L. B. Ćirić, “Generalized distance and some common fixed point theorems,” Journal of Computational Analysis and Applications, vol. 12, no. 1, pp. 157–162, 2010.

W. Sintunavarat, D. M. Lee, and Y. J. Cho, “Mizoguchi-Takahashi's type common fixed point theorems without -weakly commuting condition and invariant approximations,” Fixed Point Theory and Applications, vol. 2014, no. 1, article 112, 10 pages, 2014.

M. U. Ali, T. Kamran, W. Sintunavarat, and P. Katchang, “Mizoguchi-Takahashi's fixed point theorem with α− functions,” Abstract and Applied Analysis, vol. 2013, Article ID 418798, 4 pages, 2013.

N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 177–188, 1989.

M. Javahernia, A. Razani, and F. Khojasteh, “Common fixed point of the generalized Mizoguchi-Takahashi's type contractions,” Fixed Point Theory and Applications, vol. 2014, no. 1, article 195, 12 pages, 2014.

J. Jachymski, “The contraction principle for mappings on a metric space with a graph,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1359–1373, 2008.

I. Beg, A. R. Butt, and S. Radenović, “The contraction principle for set valued mappings on a metric space with a graph,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1214–1219, 2010.

J. Ahmad, N. Hussain, A. R. Khan, and A. Azam, “Fixed point results for generalized multi-valued contractions,” Journal of Nonlinear Science and Its Applications, vol. 8, no. 6, pp. 909–918, 2015.

F. Bojor, “Fixed point theorems for Reich type contractions on metric spaces with a graph,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 9, pp. 3895–3901, 2012.

R. Espnola and W. A. Kirk, “Fixed point theorems in R-trees with applications to graph theory,” Topology and its Applications, vol. 153, pp. 1046–1055, 2006.

N. Hussain, J. Ahmad, and A. Azam, “Generalized fixed point theorems for multi-valued α-ψ-contractive mappings,” Journal of Inequalities and Applications, vol. 2014, article 348, 2014.

N. Hussain, S. Al-Mezel, and P. Salimi, “Fixed points for ψ-graphic contractions with application to integral equations,” Abstract and Applied Analysis, vol. 2013, Article ID 575869, 11 pages, 2013.

N. Hussain, M. Arshad, A. Shoaib, and Fahimuddin, “Common fixed point results for α-ψ-contractions on a metric space endowed with graph,” Journal of Inequalities and Applications, vol. 2014, article 136, 2014.

A. Sultana and V. Vetrivel, “Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications,” Journal of Mathematical Analysis and Applications, vol. 417, no. 1, pp. 336–344, 2014.