a fixed point theorem in a generalized metric space

SOOCHOW JOURNAL OF MATHEMATICS

Volume 33, No. 1, pp. 33-39, January 2007

A FIXED POINT THEOREM IN A GENERALIZED METRIC SPACE BY PRATULANANDA DAS AND LAKSHMI KANTA DEY

Abstract. We prove a fixed point theorem for uniformly locally contractive mappings in a generalized metric space, a concept recently introduced in [1].

1. Introduction If (X, d) is a complete metric space and T : X → X is a contraction i.e. d(T x, T y) ≤ α · d(x, y), 0 < α < 1 for all x, y ∈ X then the widely known Banach’s contraction mapping principle tells that T has a unique fixed point in X. A lot of generalizations of this theorem have been done, mostly by relaxing the contraction condition and sometimes by withdrawing the requirement of completeness or even both. Recently a very interesting generalization of the concept of metric space was obtained by Branciari [1], by replacing the triangle inequality of a metric space by a more general inequality. Correspondingly, the Banach’s contraction mapping principle was extended to the case of this generalized metric space. Under the situation, it is reasonable to consider if other important fixed point theorems can be obtained in such a space. Some works have already been done in this respect ([2], [4]). In this paper we take uniformly locally contractive mappings and show that they can have unique fixed point under some general condition in a generalized metric space. Received July 21, 2005; revised January 5, 2006. AMS Subject Classification. Primary 54H25; secondary 47H10. Key words. generalized metric space, uniformly locally contractive mapping, ǫ-chainable, fixed point. 33

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2. Preliminaries Let R+ denote the set of all non-negative real numbers and N denote the set of all positive integers. Definition 1.([1]) Let X be a set and d : X 2 → R+ be a mapping such that for all x, y ∈ X and for all distinct points z, w ∈ X each of them different from x and y, one has (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x), (iii) d(x, y) ≤ d(x, z) + d(z, w) + d(w, y) then we will say that (X, d) is a generalized metric space (or shortly g.m.s). Any metric space is a g.m.s but the converse is not true([1], see also Example 1). As in a metric space, a topology can be defined in a g.m.s X with the help of the neighbourhood basis given by B = {B(x, r); x ∈ X, r ∈ R+ \ {0}} where B(x, r) = {y ∈ X; d(x, y) < r} is the open ball with centre x and radius r. Definition 2.([1]) Let (X, d) be a g.m.s. A sequence xn , n ∈ N in X is said to be a Cauchy sequence if for all ǫ > 0 there exists a natural number nǫ ∈ N such that for all m, n ∈ N, n ≥ nǫ one has d(xn , xn+m ) < ǫ. (X, d) is called complete if every Cauchy sequence is convergent in X. Definition 3.([1]) Let T be a mapping of a g.m.s (X, d) into itself. (X, d) is said to be T -orbitally complete if and only if every Cauchy sequence which is contained in {x, T x, T 2 x, T 3 x, . . .} for some x ∈ X converges in X. A T -orbitally complete g.m.s may not be complete ([4]). Throughout the paper by X we will mean a generalized metric space. 3. Main Result In this section we prove our main result. We first introduce the following definitions, adapted after the case of usual metric spaces ([3]). Definition 4. A g.m.s X is said to be ǫ-chainable if for any two points

A FIXED POINT THEOREM IN A GENERALIZED METRIC SPACE

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a, b ∈ X there exists a finite set of points a = x0 , x1 , . . . , xn−1 , xn = b such that d(xi−1 , xi ) ≤ ǫ for i = 1, 2, 3, . . . , n where ǫ > 0. Definition 5. A mapping T : X → X is called locally contractive if for every x ∈ X there exists an ǫx > 0 and λx ∈ [0, 1) such that for all p, q ∈ {y; d(x, y) ≤ ǫx } the relation d(T (p), T (q)) ≤ λx d(p, q) holds. Definition 6. T : X → X is called (ǫ, λ) uniformly locally contractive if it is locally contractive at all points x ∈ X and ǫ, λ do not depend on x i.e. d(x, y) < ǫ ⇒ d(T x, T y) < λd(x, y) for all x, y ∈ X. Note 1. From the definition it is clear that a uniformly locally contractive mapping is continuous (in the usual sense). Theorem 1. If T is an (ǫ, λ) uniformly locally contractive mapping defined on a T - orbitally complete, 2ǫ -chainable g.m.s X satisfying the following condition (A) for all x, y, z ∈ X, d(x, y) < 2ǫ and d(y, z) < 2ǫ implies d(x, z) < ǫ then T has a unique fixed point in X. Proof. We prove the theorem in three steps. Step I. Let x ∈ X. Since X is 2ǫ -chainable, we can find finite number of points x = x0 , x1 , x2 , . . . , xn−1 , xn = T x such that d(xi−1 , xi ) < 2ǫ for all i = 1, 2, . . . , n. Without any loss of generality we can assume that the points x1 , x2 , . . . , xn−1 are distinct (and different from x and T x if n > 2 ). We shall show that d(x, T x)
2. Consider two cases. Case-I. First let n be odd and n = 2m + 1 (say) where m ≥ 1. Now d(x, T x) ≤ d(x, x1 ) + d(x1 , x2 ) + · · · + d(x2m , T x) ǫ nǫ < (2m + 1) = . 2 2

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Case-II. Let n be even and n = 2m (say) where m ≥ 2. Then d(x, T x) ≤ d(x, x2 ) + d(x2 , x3 ) + · · · + d(x2m−1 , T x) ǫ (by(A)) < ǫ + (2m − 2) 2 nǫ = . 2 Again since T is (ǫ, λ) uniformly locally contractive, d(T xi−1 , T xi ) < λd(xi−1 , xi )