Common fixed points for maps on metric space with w-distance

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Abstract

The purpose of this paper is to generalize and to unify fixed point theorems of Das and Naik, Ćirić, Jungck and Ume in terms of a w-distance on complete metric space.

Section snippets

Introduction and preliminaries

The purpose of this paper is to generalize and to unify fixed point theorems of Jungck [9], Ćirić [2], Das and Naik [5] and Ume [17] in terms of a w-distance on complete metric space.

Let X be a metric space with metric d. Then a function p : X × X → [0, ∞) is called a w-distance on X if the following are satisfied:

  • (1)

    p(x, z) ⩽ p(x, y) + p(y, z), for any x, y, z ∈ X,

  • (2)

    for any x ∈ X, p(x, ·) : X → [0, ∞) is lower semicontinuous,

  • (3)

    for any ϵ > 0, there exist δ > 0 such that p(z, x) ⩽ δ and p(z, y) ⩽ δ imply d(x, y) ⩽ ϵ.

Let us recall that a

Auxiliary results

In this section we gather some auxiliary results, which will be used in the next sections.

Lemma 2.1

Let X be a metric space with metric d and let f, g be mappings from X into itself which commutes, f or g is continuous, and f and g satisfy (3). Then for every y ∈ X with f(y) ≠ g(y),inf{d(fx,y)+d(fx,gx):x∈X}>0.

Proof 1

Suppose that f is continuous and there exists z ∈ X with f(z) ≠ g(z) and inf{d(fx, z) + d(fx, gx):x ∈ X} = 0. Then there exists a sequence {xn} in X such thatlimn→∞[d(fxn,z)+d(fxn,gxn)]=0.Since, d(fxn, z) → 0 and d(fxn

Main results for self mappings

In this section we study quasicontraction type self mappings on metric spaces with w-distance.

Theorem 3.1

Let X be a complete metric space with metric d and let p be a w-distance on X. Let f, g : X → X commutes, satisfy (1), (7), and let

  • (i)

    for every y ∈ X with f(y) ≠ g(y),inf{p(fx,y)+p(fx,gx):x∈X}>0.

Then f and g have a common unique fixed point u in X and p(u, u) = 0.

Proof 5

Starting with an arbitrary x0 ∈ X, we construct the sequence {xn} in X such that g(x0) = f(x1). Having defined xn ∈ X, let xn+1 be such that g(xn) = f(xn+1). We set

Non-self mappings

Let us point out that in many applications of the fixed point theorems, a mapping of a closed subset K of a Banach space X is not generally a self mapping of K into K but into X, or to check the invariance condition T(K) ⊂ K is very difficult. Hence, it is of great interest to find sufficient conditions on non self mapping which will provide the existence of a fixed point.

In 1970, Takahashi [16] introduced the definition of convexity in metric space and generalized same important fixed point

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