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 thatfor all , where is such thatfor 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) thatand, 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) thatSince , by the definition of Hausdorff metric, there exists such thatFrom (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) thatSince , 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 thatSince , it follows from (7) thatand, 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) thatSince , 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 haveSince , 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) thatSince , 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 haveLet for each Then, there exist positive integers and real numbers such that for each , where Hence, for each , we havefor 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 havewhere 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.