Overall approach to Mizoguchi--Takahashi type fixed point results

Abstract

In this work, inspired by the recent technique of Jleli and Samet, we give a new generalization of the wellknown Mizoguchi-Takahashi fixed point theorem, which is the closest answer to Reich's conjecture about the existence of fixed points of multivalued mappings on complete metric spaces. We also provide a nontrivial example showing that our result is a proper generalization of the Mizoguchi-Takahashi result.

The influence of theta-function to the class of MWP operators

“…Thus, every θ-contraction mapping on a metric space is continuous. Afterwards, many researches were conducted on a variety of generalizations, extensions and applications of the result of Jleli and Samet (See [4,5,6,10,15]). Hançer et al [10] also extended the concept of θ-contraction to multivalued case.…”

“…Hançer et al [10] also extended the concept of θ-contraction to multivalued case. Moreover in these directions, Durmaz and Altun [5] and Mınak and Altun [15] presented the following concepts: Let (X, d) be a metric space and T : X → CB(X) be given a mapping. Then, (i) T is said to be a multivalued almost θ-contraction with θ ∈ Θ [5] if there exist two constants k ∈ (0, 1) and λ ≥ 0 such that…”

“…(ii) T is said to be a multivalued nonlinear θ-contraction with θ ∈ Θ [15] if there exists a function k :…”

“…MN Ω (X)). If we examine the proof of Theorem 2.1 in [5] and the proof of Theorem 8 in [15], we can infer the following theorems, respectively: Theorem 3. If (X, d) is a complete metric space, then…”