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Common fixed points for (\psi ,\alpha ,\beta )weakly contractive mappings in generalized metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 131 (2013)
Abstract
We establish some common fixed point theorems for mappings satisfying a (\psi ,\alpha ,\beta )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 [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 (\psi ,\alpha ,\beta )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 nonempty set and let d:X\times X\to [0,+\mathrm{\infty}) be a mapping such that for all x,y\in X and for all distinct points u,v\in 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)\le d(x,u)+d(u,v)+d(v,y) (the rectangular inequality).
Then (X,d) is called a generalized metric space (or for short g.m.s.).
Definition 2 ([1])
Let (X,d) be a g.m.s., let \{{x}_{n}\} be a sequence in X and x\in X.

(i)
We say that \{{x}_{n}\} is a g.m.s. convergent to x if and only if d({x}_{n},x)\to 0 as n\to +\mathrm{\infty}. We denote this by {x}_{n}\to x.

(ii)
We say that \{{x}_{n}\} is a g.m.s. Cauchy sequence if and only if for each \epsilon >0 there exists a natural number n(\epsilon ) such that d({x}_{n},{x}_{m})<\epsilon for all n>m>n(\epsilon ).

(iii)
(X,d) 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 \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) satisfying the following hypotheses:
(ψ 1) ψ is continuous and monotone nondecreasing,
(ψ 2) \psi (t)=0 if and only if t=0.
We denote by Φ the set of functions \alpha :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) satisfying the following hypotheses:
(α 1) α is continuous,
(α 2) \alpha (t)=0 if and only if t=0.
We denote by Γ the set of functions \beta :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) satisfying the following hypotheses:
(β 1) β is lower semicontinuous,
(β 2) \beta (t)=0 if and only if t=0.
2 Main results
Definition 3 ([14])
Let X be a nonempty set and let T,F:X\to X. The mappings T, F are said to be weakly compatible if they commute at their coincidence points, that is, if Tx=Fx for some x\in X implies that TFx=FTx.
Lemma 1 Let \{{a}_{n}\} be a sequence of nonnegative real numbers. If
for all n\in \mathbb{N}, where \psi \in \mathrm{\Psi}, \alpha \in \mathrm{\Phi}, \beta \in \mathrm{\Gamma} and
then the following hold:

(i)
{a}_{n+1}\le {a}_{n} if {a}_{n}>0,

(ii)
{a}_{n}\to 0 as n\to +\mathrm{\infty}.
Proof (i) Let, if possible, {a}_{n}<{a}_{n+1} for some n\in \mathbb{N}. Then, using the monotone property of ψ and (2.1), we have
which implies that {a}_{n}=0 by (2.2), a contradiction with {a}_{n}>0. Therefore, for all n\in \mathbb{N},
(ii) By (i) the sequence \{{a}_{n}\} is nonincreasing, hence there is a\ge 0 such that {a}_{n}\to a as n\to +\mathrm{\infty}. Letting n\to +\mathrm{\infty} in (2.1), using the lower semicontinuity of β and the continuities of ψ and α, we obtain \psi (a)\le \alpha (a)\beta (a), which by (2.2) implies that a=0. □
Theorem 1 Let (X,d) be a Hausdorff and complete g.m.s. and let T,F:X\to X be selfmappings such that TX\subseteq FX, and FX is a closed subspace of X, and that the following condition holds:
for all x,y\in X, where \psi \in \mathrm{\Psi}, \alpha \in \mathrm{\Phi}, \beta \in \mathrm{\Gamma} 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 {x}_{0} be an arbitrary point in X. Since TX\subseteq FX, we can define the sequence \{{x}_{n}\} in X by
Substituting x={x}_{n} and y={x}_{n+j} for every j\in \mathbb{N} in (2.3), using (2.4), we have
By (ii) of Lemma 1, we obtain that
Next we prove that \{T{x}_{n}\} is a g.m.s. Cauchy sequence. Suppose that \{T{x}_{n}\} is not a g.m.s. Cauchy sequence. Then there exists \epsilon >0, for which we can find subsequences \{T{x}_{{m}_{k}}\} and \{T{x}_{{n}_{k}}\} of \{T{x}_{n}\} with {n}_{k}>{m}_{k}>k such that
Further, corresponding to {m}_{k}, we can choose {n}_{k} in such a way that it is the smallest integer with {n}_{k}>{m}_{k} satisfying (2.6). Then
Now, using (2.6), (2.7) and the rectangular inequality, we have
Letting k\to +\mathrm{\infty} in the above inequality, using (2.5) with j=1,2, we obtain
Again, the rectangular inequality gives us
Taking k\to +\mathrm{\infty} in the above inequalities and using (2.5) and (2.8), we get
Substituting x={x}_{{n}_{k}} and y={x}_{{m}_{k}} in (2.3), we have
Letting k\to +\mathrm{\infty} in (2.10) and using the lower semicontinuity of β and the continuities of ψ and α, we obtain
which implies that \epsilon =0 by (2.2), a contradiction with \epsilon >0. It then follows that \{T{x}_{n}\} is a g.m.s. Cauchy sequence, and hence \{T{x}_{n}\} is convergent in the complete g.m.s. (X,d). Since FX is closed and by (2.4), T{x}_{n}=F{x}_{n+1} for all n\ge 0, we have that there exists w\in FX for which
We can find y in X such that Fy=w. From (2.3), we get
On taking limit as n\to +\mathrm{\infty} and using (2.11), we have
which implies that \psi (d(w,Ty))=0, and Ty=w. 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 TFy=FTy, and
If y=w, then y is a common fixed point. If y\ne w, then by (2.3) we have
From (2.2), Fy=Fw. Then, by (2.12) and (2.13), we have w=Fw=Tw. Consequently, w is the unique common fixed point of T and F. □
Denote by Λ the set of functions \gamma :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) satisfying the following hypotheses:
({h}_{1}) γ is a Lebesgueintegrable mapping on each compact of [0,+\mathrm{\infty}).
({h}_{2}) For every \epsilon >0, we have
We have the following result.
Theorem 2 Let (X,d) be a Hausdorff and complete g.m.s. and let T,F:X\to X be selfmappings such that TX\subseteq FX, and FX is a closed subspace of X, and that the following condition holds:
for all x,y\in X, where {\gamma}_{1},{\gamma}_{2},{\gamma}_{3}\in \mathrm{\Lambda} 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 \psi (t)={\int}_{0}^{t}{\gamma}_{1}(s)\phantom{\rule{0.2em}{0ex}}ds, \alpha (t)={\int}_{0}^{t}{\gamma}_{2}(s)\phantom{\rule{0.2em}{0ex}}ds and \beta (t)={\int}_{0}^{t}{\gamma}_{3}(s)\phantom{\rule{0.2em}{0ex}}ds. □
Taking {\gamma}_{3}(s)=(1k){\gamma}_{2}(s) for k\in [0,1) in Theorem 2, we obtain the following result.
Corollary 1 Let (X,d) be a Hausdorff and complete g.m.s. and let T,F:X\to X be selfmappings such that TX\subseteq FX, and FX is a closed subspace of X, and that the following condition holds:
for all x,y\in X, where {\gamma}_{1},{\gamma}_{2}\in \mathrm{\Lambda} and k\in [0,1) 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 (\psi ,\alpha ,\beta )weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl 2013, 131 (2013). https://doi.org/10.1186/168718122013131
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DOI: https://doi.org/10.1186/168718122013131
Keywords
 fixed point
 generalized metric
 weakly contractive condition
 contraction of integral type
 common fixed points