Abstract
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.
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1 Introduction and preliminaries
In 2000, Branciari [1] 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 [1]. 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 [13] 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 [13] to the set of generalized metric spaces.
Definition 1 ([1])
Let X be a non-empty set and let be a mapping such that for all and for all distinct points , each of them different from x and y, one has
-
(i)
if and only if ,
-
(ii)
,
-
(iii)
(the rectangular inequality).
Then is called a generalized metric space (or for short g.m.s.).
Definition 2 ([1])
Let be a g.m.s., let be a sequence in X and .
-
(i)
We say that is a g.m.s. convergent to x if and only if as . We denote this by .
-
(ii)
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 .
-
(iii)
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 ([14])
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 .
Lemma 1 Let be a sequence of non-negative real numbers. If
for all , where , , and
then the following hold:
-
(i)
if ,
-
(ii)
as .
Proof (i) Let, if possible, for some . Then, using the monotone property of ψ and (2.1), we have
which implies that by (2.2), a contradiction with . Therefore, for all ,
(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 . □
Theorem 1 Let be a Hausdorff and complete g.m.s. and let be self-mappings such that , and FX is a closed subspace of X, and that the following condition holds:
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.
Proof Let be an arbitrary point in X. Since , we can define the sequence in X by
Substituting and for every in (2.3), using (2.4), we have
By (ii) of Lemma 1, we obtain that
Next we prove that is a g.m.s. Cauchy sequence. Suppose that is not a g.m.s. Cauchy sequence. Then there exists , for which we can find subsequences and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (2.6). Then
Now, using (2.6), (2.7) and the rectangular inequality, we have
Letting in the above inequality, using (2.5) with , we obtain
Again, the rectangular inequality gives us
Taking in the above inequalities and using (2.5) and (2.8), we get
Substituting and in (2.3), we have
Letting in (2.10) and using the lower semi-continuity of β and the continuities of ψ and α, we obtain
which implies that by (2.2), a contradiction with . It then follows that is a g.m.s. Cauchy sequence, and hence is convergent in the complete g.m.s. . Since FX is closed and by (2.4), for all , we have that there exists for which
We can find y in X such that . From (2.3), we get
On taking limit as and using (2.11), we have
which implies that , and . Then we obtain
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).
Now, we show that there exists a common fixed point of T and F. Since T and F are weakly compatible, by (2.12), we have that , and
If , then y is a common fixed point. If , then by (2.3) we have
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 .
() For every , we have
We have the following result.
Theorem 2 Let be a Hausdorff and complete g.m.s. and let be self-mappings such that , and FX is a closed subspace of X, and that the following condition holds:
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.
Corollary 1 Let be a Hausdorff and complete g.m.s. and let be self-mappings such that , and FX is a closed subspace of X, and that the following condition holds:
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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Işık, H., Türkoğlu, D. Common fixed points for -weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl 2013, 131 (2013). https://doi.org/10.1186/1687-1812-2013-131
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DOI: https://doi.org/10.1186/1687-1812-2013-131