We present a new generalization of the Banach contraction principle in the setting of Branciari metric spaces.
The fixed-point theorem, generally known as the Banach contraction principle, appeared in explicit form in Banach’s thesis in 1922 , 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 , 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 references therein.
In 2000, Branciari  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] and references therein.
In this paper, we introduce a new type of contractive maps and we establish a new fixed-point theorem for such maps on the setting of generalized metric spaces.
2 Main results
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 .
Before we prove the main results, we recall the following definitions introduced in .
Definition 2.1 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
Then is called a generalized metric space (or for short g.m.s.).
Definition 2.2 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 2.3 Let be a g.m.s. and be a sequence in X. We say that is Cauchy if and only if as .
Definition 2.4 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  (Lemma 1.10).
Lemma 2.1 Letbe a g.m.s., be a Cauchy sequence in, and. Suppose that there exists a positive integer N such that
, for all ;
and x are distinct points in X, for all ;
and y are distinct points in X, for all ;
Then we have.
We observe easily that if one of the conditions (ii) or (iii) is not satisfied, then the result of Lemma 2.1 is still valid.
Now, we are ready to state and prove our main result.
Theorem 2.1 Letbe a complete g.m.s. andbe a given map. Suppose that there existandsuch that
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
Thus, we have
Letting in (2), 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
Suppose now that . Let be an arbitrary positive number. From the definition of the limit, there exists such that
This implies that
Thus, in all cases, there exist and such that
Using (2), 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
Letting in the above inequality and using (), we obtain
Similarly, from condition (), there exists such that
Let . We consider two cases.
Case 1. If is odd, then writing , , using (4), for all , we obtain
Case 2. If is even, then writing , , using (4) and (6), 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 . On the other hand, observe that T is continuous, indeed, if , then we have from (1)
which implies from () that
From this observation, for all , we have
Letting in the above inequality, we get . From Lemma 2.1, we obtain , which is 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. □
Since a metric space is a g.m.s., from Theorem 2.1, we deduce immediately the following result.
Corollary 2.1 Letbe a complete metric space andbe a given map. Suppose that there existandsuch that
Then T has a unique fixed point.
Observe that the Banach contraction principle follows immediately from Corollary 2.1. Indeed, if T is a Banach contraction, i.e., there exists such that
then we have
Clearly the function defined by belongs to Θ. So, the existence and uniqueness of the fixed point follows from Corollary 2.1. In the following example (inspired by ), we show that Corollary 2.1 is a real generalization of the Banach contraction principle.
Example Let X be the set defined by
We endow X with the metric d given by for all . It is not difficult to show that is a complete metric space. Let be the map defined by
Clearly, the Banach contraction is not satisfied. In fact, we can check easily that
Now, consider the function defined by
It is not difficult to show that . We shall prove that T satisfies the condition (1), that is,
for some . The above condition is equivalent to
So, we have to check that
for some . We consider two cases.
Case 1. and . In this case, we have
Case 2. . In this case, we have
Thus, the inequality (7) is satisfied with . Theorem 2.1 (or Corollary 2.1) implies that T has a unique fixed point. In this example is the unique fixed point of T.
Note that Θ contains a large class of functions. For example, for
we obtain from Theorem 2.1 the following result.
Corollary 2.2 Letbe a complete g.m.s. andbe a given map. Suppose that there existsuch that
Then T has a unique fixed point.
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This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final version.
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Jleli, M., Samet, B. A new generalization of the Banach contraction principle. J Inequal Appl 2014, 38 (2014). https://doi.org/10.1186/1029-242X-2014-38