Abstract
The purpose of this paper is to prove some existence theorems for fixed point problem by using a generalization of metric distance, namely u-distance. Consequently, some special cases are discussed and an interesting example is also provided. Presented results are generalizations of the important results due to Ume (Fixed Point Theory Appl 2010(397150), 21 pp, 2010) and Suzuki and Takahashi (Topol Methods Nonlinear Anal 8, 371-382, 1996).2010 Mathematics Subject Classification: 47H09, 47H10.
Highlights
Introduction and preliminariesLet (X, d) be a metric space
Following the Banach contraction principle, Nadler Jr. [2] established the fixed point result for multi-valued contraction maps, which in turn is a generalization of the Banach contraction principle
By using concepts of τ-distance, he proved some results on fixed point problems and showed that the class of w-distance is properly contained in the class of τ-distance
Summary
Introduction and preliminariesLet (X, d) be a metric space. A mapping T: X ® X is said to be contraction if there exists r Î [0, 1) such that d(T(x), T(y)) ≤ rd(x, y), ∀x, y ∈ X. (1:1)In 1922, Banach [1] proved that if (X, d) is a complete metric space and the mapping T satisfies (1.1), T has a unique fixed point, that is T(u) = u for some u Î X. In 1922, Banach [1] proved that if (X, d) is a complete metric space and the mapping T satisfies (1.1), T has a unique fixed point, that is T(u) = u for some u Î X. We will prove some fixed point theorems in metric spaces by using such a u-distance concept. Let (X, d) be a metric space and T: X ® 2X be a mapping.
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