A new generalization of the Banach contraction principle
© Jleli and Samet; licensee BioMed Central Ltd. 2013
Received: 2 August 2013
Accepted: 26 December 2013
Published: 24 January 2014
We present a new generalization of the Banach contraction principle in the setting of Branciari metric spaces.
KeywordsBanach contraction generalized metric fixed point
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 .
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).
, 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.
Then T has a unique fixed point.
Let . We consider two cases.
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.
Then T has a unique fixed point.
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.
for some . We consider two cases.
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.
we obtain from Theorem 2.1 the following result.
Then T has a unique fixed point.
This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.
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