tailieunhanh - Overall approach to Mizoguchi–Takahashi type fixed point results

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. | Turk J Math (2016) 40: 895 – 904 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Overall approach to Mizoguchi–Takahashi type fixed point results ˙ G¨ ulhan MIKNAK∗, Ishak ALTUN Department of Mathematics, Faculty of Science and Arts, Kırıkkale University, Yah¸sihan, Kırıkkale, Turkey Received: • Accepted/Published Online: • Final Version: 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. Key words: Fixed point, multivalued mappings, Mizoguchi–Takahasi result, θ -contraction 1. Introduction and preliminaries In 1922, Banach established the most famous fundamental fixed point theorem, called the Banach contraction principle, for metric fixed point theory. This principle is a very powerful test for the existence and uniqueness of the solution of considerable problems arising in mathematics and has played an important role in various fields of applied mathematical analysis. The Banach contraction principle asserts that if (X, d) is a complete metric space and T : X → X is a contraction mapping, that is, there exists L ∈ [0, 1) such that d(T x, T y) ≤ Ld(x, y) for all x, y ∈ X , then there exists a unique x ∈ X such that x = T x . This principle has been extended and generalized in many ways (see [3, 4, 11, 16, 25]). In 1969, Nadler [19] initiated the idea for multivalued contraction mapping and extended the Banach contraction principle to multivalued mappings and afterwards proved the following result: Theorem 1 (Nadler [19]) Let (X, d) be a complete metric space and T : X → CB(X) be