Several fixed point theorems concerning -distance
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Using the notion of τ-distance, we prove several fixed point theorems, which are generalizations of fixed point theorems by Kannan, Meir-Keeler, Edelstein, and Nadler. We also discuss the properties of τ-distance. 1. Introduction In 1922, Banach proved the following famous fixed point theorem [1]. Let (X,d) be a complete metric space. Let T be a contractive mapping on X, that is, there exists r ∈ [0,1) satisfying d(Tx,T y) ≤ rd(x, y)
(1.1)
for all x, y ∈ X. Then there exists a unique fixed point x0 ∈ X of T. This theorem, called the Banach contraction principle, is a forceful tool in nonlinear analysis. This principle has many applications and is extended by several authors: Caristi [2], Edelstein [5], Ekeland [6, 7], Meir and Keeler [14], Nadler [15], and others. These theorems are also extended; see [4, 9, 10, 13, 23, 25, 26, 27] and others. In [20], the author introduced the notion of τ-distance and extended the Banach contraction principle, Caristi’s fixed point theorem, and Ekeland’s ε-variational principle. In 1969, Kannan proved the following fixed point theorem [12]. Let (X,d) be a complete metric space. Let T be a Kannan mapping on X, that is, there exists α ∈ [0,1/2) such that
d(Tx,T y) ≤ α d(Tx,x) + d(T y, y)
(1.2)
for all x, y ∈ X. Then there exists a unique fixed point x0 ∈ X of T. We note that Kannan’s fixed point theorem is not an extension of the Banach contraction principle. We also know that a metric space X is complete if and only if every Kannan mapping has a fixed point, while there exists a metric space X such that X is not complete and every contractive mapping on X has a fixed point; see [3, 17]. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 195–209 2000 Mathematics Subject Classification: 54H25, 54E50 URL: http://dx.doi.org/10.1155/S168718200431003X
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Fixed point theorems concerning τ-distance
In this paper, using the notion of τ-distance, we prove several fixed point theorems, which are generalizations of fixed point theorems by Kannan, Meir-Keeler, Edelstein, and Nadler. We also discuss the properties of τ-distance. 2. τ-distance Throughout this paper, we denote by N the set of all positive integers. In this section, we discuss some properties of τ-distance. Let (X,d) be a metric space. Then a function p from X × X into [0, ∞) is called a τ-distance on X [20] if there exists a function η from X × [0, ∞) into [0, ∞) and the following are satisfied: (τ1) p(x,z) ≤ p(x, y) + p(y,z) for all x, y,z ∈ X; (τ2) η(x,0) = 0 and η(x,t) ≥ t for all x ∈ X and t ∈ [0, ∞), and η is concave and continuous in its second variable; (τ3) limn xn = x and limn sup{η(zn , p(zn ,xm )) : m ≥ n} = 0 imply p(w,x) ≤ liminf n p(w,xn ) for all w ∈ X; (τ4) limn sup{ p(xn , ym ) : m ≥ n} = 0 and limn η(xn ,tn ) = 0 imply limn η(yn ,tn ) = 0; (τ5) limn η(zn , p(zn ,xn )) = 0 and limn η(zn , p(zn , yn )) = 0 imply limn d(xn , yn ) = 0. We may replace (τ2) by the following (τ2) (see [20]): (τ2) inf {η(x,t) : t > 0} = 0 for all x ∈ X, and η is nondecreasing
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