Abstract
We establish a new fixed point theorem in the setting of Branciari metric spaces. The obtained result is an extension of the recent fixed point theorem established in Jleli and Samet (J. Inequal. Appl. 2014:38, 2014).
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1 Introduction
The fixed point theorem, generally known as the Banach contraction principle, appeared in an explicit form in Banach’s thesis in 1922 [1], where it was used to establish the existence of a solution to an integral equation. Since then, because of its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. This principle states that if is a complete metric space and is a contraction map (i.e., for all , where is a constant), then T has a unique fixed point.
The Banach contraction principle has been generalized in many ways over the years. In some generalizations, the contractive nature of the map is weakened; see [2–9] and others. In other generalizations, the topology is weakened; see [10–23] and others. In [24], Nadler extended the Banach fixed point theorem from single-valued maps to set-valued contractive maps. Other fixed point results for set-valued maps can be found in [25–30] and the references therein.
In 2000, Branciari [11] introduced the concept of generalized metric spaces, where the triangle inequality is replaced by the inequality for all pairwise distinct points . Various fixed point results were established on such spaces; see [10, 13, 17–20, 22, 31–33] and the references therein.
We recall the following definitions introduced in [11].
Definition 1.1 Let X be a non-empty set and be a mapping such that for all and for all distinct points , each of them different from x and y, one has
-
(i)
;
-
(ii)
;
-
(iii)
.
Then is called a generalized metric space (or for short g.m.s).
Definition 1.2 Let be a g.m.s, be a sequence in X and . We say that is convergent to x if and only if as . We denote this by .
Definition 1.3 Let be a g.m.s and be a sequence in X. We say that is Cauchy if and only if as .
Definition 1.4 Let be a g.m.s. We say that is complete if and only if every Cauchy sequence in X converges to some element in X.
The following result was established in [17] (see also [34]).
Lemma 1.1 Let be a g.m.s and be a Cauchy sequence in such that as for some . Then as for all . In particular, does not converge to y if .
We denote by Θ the set of functions satisfying the following conditions:
() θ is non-decreasing;
() for each sequence , if and only if ;
() there exist and such that ;
() θ is continuous.
Recently, Jleli and Samet [35] established the following generalization of the Banach fixed point theorem in the setting of Branciari metric spaces.
Theorem 1.1 (Jleli and Samet [35])
Let be a complete g.m.s and be a given map. Suppose that there exist and such that
Then T has a unique fixed point.
Note that the condition () is not supposed in Theorem 1.1.
The aim of this paper is to extend the result given by Theorem 1.1.
2 Result and proof
Now, we are ready to state and prove our main result.
Theorem 2.1 Let be a complete g.m.s and be a given map. Suppose that there exist and such that
where
Then T has a unique fixed point.
Proof Let be an arbitrary point in X. If for some we have , then will be a fixed point of T. So, without restriction of the generality, we can suppose that for all . Now, from (1), for all , we have
where from (2)
If , then inequality (3) turns into
which implies that
that is a contradiction with . Hence, from (4) we have , and inequality (3) yields
Thus we have
Letting in (5), we obtain
which implies from () that
From condition (), there exist and such that
Suppose that . In this case, let . From the definition of the limit, there exists such that
This implies that
Then
where .
Suppose now that . Let be an arbitrary positive number. From the definition of the limit, there exists such that
This implies that
where .
Thus, in all cases, there exist and such that
Using (5), we obtain
Letting in the above inequality, we obtain
Thus, there exists such that
Now, we shall prove that T has a periodic point. Suppose that it is not the case, then for every such that . Using (1), we obtain
where from (2)
Since θ is non-decreasing, we obtain from (8) and (9)
Let I be the set of such that
If , then there exists such that for every ,
In this case, we obtain from (10)
for all . Letting in the above inequality and using (6), we get
If , we can find a subsequence of , that we denote also by , such that
In this case, we obtain from (10)
for n large enough. Letting in the above inequality, we obtain
Then in all cases, (11) holds. Using (11) and the property (), we obtain
Similarly, from condition (), there exists such that
Let . We consider two cases.
Case 1. If is odd, then writing , , using (7), for all , we obtain
Case 2. If is even, then writing , , using (7) and (12), for all , we obtain
Thus, combining all the cases, we have
From the convergence of the series (since ), we deduce that is a Cauchy sequence. Since is complete, there is such that as . Without restriction of the generality, we can suppose that for all n (or for n large enough). Suppose that , using (1), we get
where
Letting in the above inequality, using () and Lemma 1.1, we obtain
which is a contradiction. Thus we have , which is also a contradiction with the assumption: T does not have a periodic point. Thus T has a periodic point, say z, of period q. Suppose that the set of fixed points of T is empty. Then we have
Using (1), we obtain
which is a contradiction. Thus, the set of fixed points of T is non-empty, that is, T has at least one fixed point. Now, suppose that are two fixed points of T such that . Using (1), we obtain
which is a contradiction. Then we have one and only one fixed point. □
3 Some consequences
We start by deducing the following fixed point result.
Corollary 3.1 Let be a complete g.m.s and be a given map. Suppose that there exists such that
Then T has a unique fixed point.
Proof From (13), we have
Clearly the function defined by belongs to Θ. So, the existence and uniqueness of the fixed point follows from Theorem 2.1. □
The following fixed point result established in [11] is an immediate consequence of Corollary 3.1.
Corollary 3.2 Let be a complete g.m.s and be a given map. Suppose that there exists such that
Then T has a unique fixed point.
The following fixed point result established in [34] is an immediate consequence of Corollary 3.1.
Corollary 3.3 Let be a complete g.m.s and be a given map. Suppose that there exist with such that
Then T has a unique fixed point.
The following fixed point result is also an immediate consequence of Corollary 3.1.
Corollary 3.4 Let be a complete g.m.s and be a given map. Suppose that there exist with such that
Then T has a unique fixed point.
We note that Θ contains a large class of functions. For example, for
we obtain from Theorem 2.1 the following result.
Corollary 3.5 Let be a complete g.m.s and be a given map. Suppose that there exist such that
where is given by (2). Then T has a unique fixed point.
Finally, since a metric space is a g.m.s, from Theorem 2.1 we deduce immediately the following result.
Corollary 3.6 Let be a complete metric space and be a given map. Suppose that there exist and such that
where is given by (2). Then T has a unique fixed point.
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Acknowledgements
The first and third authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No: RG-1435-034.
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Jleli, M., Karapınar, E. & Samet, B. Further generalizations of the Banach contraction principle. J Inequal Appl 2014, 439 (2014). https://doi.org/10.1186/1029-242X-2014-439
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DOI: https://doi.org/10.1186/1029-242X-2014-439