Common fixed points for -weakly contractive mappings in generalized metric spaces
© Işık and Türkoğlu; licensee Springer 2013
Received: 1 August 2012
Accepted: 29 April 2013
Published: 16 May 2013
We establish some common fixed point theorems for mappings satisfying a -weakly contractive condition in generalized metric spaces. Presented theorems extend and generalize many existing results in the literature.
MSC: Primary 54H25; secondary 47H10.
1 Introduction and preliminaries
In 2000, Branciari  introduced the concept of a generalized metric space where the triangle inequality of a metric space was replaced by an inequality involving three terms instead of two. As such, any metric space is a generalized metric space, but the converse is not true . He proved the Banach fixed point theorem in such a space. After that, many fixed point results have been established for this interesting space. For more, the reader can refer to [2–12].
It is also known that common fixed point theorems are generalizations of fixed point theorems. Recently, many researchers have interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems. In a recent paper, Choudhury and Kundu  established the -weak contraction principle to coincidence point and common fixed point results in partially ordered metric spaces.
The purpose of this paper is to extend the results in  to the set of generalized metric spaces.
Definition 1 ()
if and only if ,
(the rectangular inequality).
Then is called a generalized metric space (or for short g.m.s.).
Definition 2 ()
We say that is a g.m.s. convergent to x if and only if as . We denote this by .
We say that is a g.m.s. Cauchy sequence if and only if for each there exists a natural number such that for all .
is called a complete g.m.s. if every g.m.s. Cauchy sequence is g.m.s. convergent in X.
We denote by Ψ the set of functions satisfying the following hypotheses:
(ψ 1) ψ is continuous and monotone nondecreasing,
(ψ 2) if and only if .
We denote by Φ the set of functions satisfying the following hypotheses:
(α 1) α is continuous,
(α 2) if and only if .
We denote by Γ the set of functions satisfying the following hypotheses:
(β 1) β is lower semi-continuous,
(β 2) if and only if .
2 Main results
Definition 3 ()
Let X be a non-empty set and let . The mappings T, F are said to be weakly compatible if they commute at their coincidence points, that is, if for some implies that .
(ii) By (i) the sequence is non-increasing, hence there is such that as . Letting in (2.1), using the lower semi-continuity of β and the continuities of ψ and α, we obtain , which by (2.2) implies that . □
for all , where , , and satisfy condition (2.2). Then T and F have a unique coincidence point in X. Moreover, if T and F are weakly compatible, then T and F have a unique common fixed point.
Therefore, w is a point of coincidence of T and F. The uniqueness of the point of coincidence is a consequence of condition (2.3).
From (2.2), . Then, by (2.12) and (2.13), we have . Consequently, w is the unique common fixed point of T and F. □
Denote by Λ the set of functions satisfying the following hypotheses:
() γ is a Lebesgue-integrable mapping on each compact of .
We have the following result.
for all , where and satisfy condition (2.2). If T and F are weakly compatible, then T and F have a unique fixed point.
Proof Follows from Theorem 1 by taking , and . □
Taking for in Theorem 2, we obtain the following result.
for all , where and and satisfy condition (2.2). If T and F are weakly compatible, then T and F have a unique fixed point.
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