Abstract
We obtain a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler and Jachymski type. Our result generalizes in various directions the celebrated fixed point theorems of Boyd and Wong, and Matkowski. Some illustrative examples are also given.
MSC:47H10, 54H25, 54E50.
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1 Introduction and preliminaries
In their celebrated paper [1], Kada, Suzuki and Takahashi introduced and studied the notion of a w-distance on a metric space. By using that notion they obtained, among other results, generalizations of the nonconvex minimization theorem of Takahashi [2], of Caristi’s fixed point theorem [3] and of Ekeland’s variational principle [4], as well as a general fixed point theorem that improves fixed point theorems of Subrahmanyam [5], Kannan [6] and Ćirić [7]. This study was continued by Suzuki and Takahashi [8], and by Park [9] who extended several results from [1] to quasi-metric spaces. Park’s approach was successful continued by Al-Homidan, Ansari and Yao [10], who obtained, among other interesting results, quasi-metric versions of Caristi-Kirk’s fixed point theorem and Nadler’s fixed point theorem by using Q-functions (a slight generalization of w-distances). More recently, Latif and Al-Mezel [11], and Marín et al. [12–14] have proved some fixed point theorems both for single-valued and multi-valued mappings in complete quasi-metric spaces and preordered quasi-metric spaces by using Q-functions and w-distances, and generalizing in this way well-known fixed point theorems of Mizoguchi and Takahashi [15], Bianchini and Grandolfi [16], and Boyd and Wong [17], respectively.
In this paper we shall obtain a fixed point theorem for generalized contractions with respect to w-distances on complete quasi-metric spaces from which we deduce w-distance versions of Boyd and Wong’s fixed point theorem [17] and of Matkowski’s fixed point theorem [18]. Our approach uses a kind of functions considered by Jachymski in [[19], Corollary of Theorem 2] and that generalizes the notion of a function of Meir-Keeler type.
In the sequel the letters , ℕ and ω will denote the set of non-negative real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.
By a quasi-metric on a set X we mean a function such that for all :
-
(i)
, and
-
(ii)
.
A quasi-metric space is a pair such that X is a set and d is a quasi-metric on X.
Each quasi-metric d on a set X induces a topology on X which has as a base the family of open balls , where for all and .
Given a quasi-metric d on X, the function defined by for all , is also a quasi-metric on X, and the function defined by for all , is a metric on X.
There exist several different notions of Cauchy sequence and of complete quasi-metric space in the literature (see e.g. [20]). In this paper we shall use the following general notion.
A quasi-metric space is called complete if every Cauchy sequence in the metric space converges with respect to the topology (i.e., there exists such that ).
A w-distance on a quasi-metric space is a function satisfying the following three conditions:
(W1) for all ;
(W2) is lower semicontinuous on for all ;
(W3) for each there exists such that and imply .
Several examples of w-distances on quasi-metric spaces may be found in [9–12].
Note that if d is a metric on X then it is a w-distance on . Unfortunately, this does not hold for quasi-metric spaces, in general. Indeed, in [[12], Lemma 2.2] there was observed the following.
Lemma 1 If q is a w-distance on a quasi-metric space , then for each there exists such that and imply .
It follows from Lemma 1 (see [[12], Proposition 2.3]) that if a quasi-metric d on X is also a w-distance on , then the topologies induced by d and by the metric coincide, so is a metrizable topological space.
2 Results and examples
Meir and Keeler proved in [21] that if f is a self-map of a complete metric space satisfying the condition that for each there is such that, for any , with we have , then f has a unique fixed point and for all .
This well-known result suggests the notion of a Meir-Keeler function:
A function is said to be a Meir-Keeler function if , and satisfies the following condition:
(MK) For each there exists such that
Remark 1 It is obvious that if ϕ is a Meir-Keeler function then for all .
Later on, Jachymski proved in [19] the following interesting result and showed that both Boyd and Wong’s fixed point theorem and Matkowski’s fixed point theorem are easy consequences of it.
Theorem 1 ([[19], Corollary of Theorem 2])
Let f be a self-map of a complete metric space such that for , and for all , where satisfies the condition
(Ja) for each there exists such that for any ,
Then f has a unique fixed point and for all .
Theorem 1 suggests the following notion:
A function is said to be a Jachymski function if and it satisfies condition (Ja) of Theorem 1.
Remark 2 Obviously, each Meir-Keeler function is a Jachymski function. However, the converse does not follow even in the case that for all : Indeed, let defined as for all and otherwise. Clearly ϕ is a Jachymski function such that for all . Finally, for and any we have , so ϕ is not a Meir-Keeler function.
Now we establish the main result of this paper.
Theorem 2 Let f be a self-map of a complete quasi-metric space . If there exist a w-distance q on and a Jachymski function such that for all , and
for all , then f has a unique fixed point . Moreover .
Proof Fix . For each let . Then
for all .
First, we shall prove that is a Cauchy sequence in .
To this end put for all .
If there is such that , then for all by (2) and our assumption that . Therefore whenever by condition (W1), and consequently, by Lemma 1. Thus for all .
Otherwise, we assume, without loss of generality, that for all . Then converges to some . Of course, for all .
If there exists such that
Take such that for all . Therefore , so by condition (2), for all , a contradiction. Consequently .
Now choose an arbitrary . There exists , with , for which conditions (W3) and (Ja) hold. Similarly, for there exists , with for which conditions (W3) and (Ja) also hold, i.e.,
and , imply , and for any , implies .
Since , there exists such that for all .
By using a similar technique to the one given by Jachymski in [[19], Theorem 2] we shall prove, by induction, that for each and each , we have
Indeed, fix . Since , condition (3) follows for .
Assume that (3) holds for some . We shall distinguish two cases.
-
Case 1: . Then we deduce from the induction hypothesis and condition (Ja) that
so by (1), . Therefore
-
Case 2: .
If , we deduce that by (1). So, by (W1),
If , we deduce that , so
Now take with . Then and for some . Hence, by (3),
Now, from Lemma 1 it follows that whenever . We conclude that is a Cauchy sequence in .
Since is complete, there exists such that .
Next we show that : Indeed, choose an arbitrary . We have proved (see (3)) that there is such that for all and . Fix . Since it follows from condition (W2) that, for n sufficiently large,
Hence for all . We deduce that .
From (1) it follows that . So by Lemma 1. Consequently , i.e., is a fixed point of f. Furthermore . In fact, otherwise we have
a contradiction.
Finally, let such that and . If we deduce that
a contradiction. So . Similarly we check that . Since , we deduce from Lemma 1 that , i.e., . We conclude that z is the unique fixed point of f. □
Corollary 1 Let f be a self-map of a complete metric space . If there exist a w-distance q on and a Jachymski function such that for all , and
for all , then f has a unique fixed point . Moreover .
Corollary 2 Let f be a self-map of a complete quasi-metric space . If there exist a w-distance q on and a Meir-Keeler function such that
for all , then f has a unique fixed point . Moreover .
Proof Apply Remarks 1 and 2, and Theorem 2. □
Corollary 3 [13]
Let f be a self-map of a complete quasi-metric space . If there exist a w-distance q on and a right upper semicontinuous function such that , for all , and
for all , then f has a unique fixed point . Moreover .
Proof It suffices to show that ϕ is a Meir-Keeler function. Assume the contrary. Then there exist and a sequence of positive real numbers such that but for all . Since , it follows from right upper semicontinuity of ϕ that eventually, i.e., , a contradiction. We conclude that f has a unique fixed point by Corollary 2. □
Corollary 4 Let f be a self-map of a complete quasi-metric space . If there exist a w-distance q on and a non-decreasing function such that , for all , and
for all , then f has a unique fixed point . Moreover .
Proof Again it suffices to show that ϕ is a Meir-Keeler function. Assume the contrary. Then there exist and a sequence of positive real numbers such that but for all . Since ϕ is non-decreasing we deduce that whenever . Hence whenever , which contradicts the hypothesis that for all . We conclude that f has a unique fixed point by Corollary 2. □
Remark 3 In [22] the authors proved Corollary 2 for the case that is a complete metric space. Note also that Boyd and Wong’s fixed point theorem [17] and Matkowski’s fixed point theorem [18] are special cases of Corollaries 3 and 4, respectively, when is a complete metric space and q is the metric d.
We conclude the paper with some examples that illustrate and validate the obtained results.
The first example shows that condition ‘ for all ’ in Theorem 2 cannot be omitted.
Example 1 Let and let d be the discrete metric on X, i.e., for all and whenever . Let defined as and , and defined as and for all . It is clear that ϕ is a Jachysmki function such that
for all . However, f has no fixed point.
The next is an example where we can apply Theorem 2 for an appropriate w-distance q on a complete quasi-metric space but not for d. Moreover, Corollary 1 cannot be applied for any w-distance on the metric space .
Example 2 Let and let d be the quasi-metric on X defined as
Clearly is complete (observe that is a Cauchy sequence in with ).
Let q be the w-distance on given by for all .
Now define as and for all , and as and where , .
It is routine to check that ϕ is a Jachymski function satisfying for all (in fact, it is a Meir-Keeler function).
Since for all , and for each with , we have
it follows that all conditions of Theorem 2 are satisfied. In fact is the unique fixed point of f.
However, the contraction condition (1) is not satisfied for d. Indeed, for any we have
Finally, note that we cannot apply Corollary 1 because is not complete (observe that is a Cauchy sequence in that does not converge in ).
We conclude with an example where we can apply Corollary 2 but not Corollaries 3 and 4.
Example 3 Let d be the quasi-metric on given by for all . Since is the usual metric on it immediately follows that is complete.
Define as . It is clear that q is a w-distance on .
Now let , defined by if , and otherwise.
Then ϕ is a Meir-Keeler function: Indeed, we first note that . Now, given we distinguish the following cases:
-
(1)
if , we take , and thus, from , it follows ;
-
(2)
if , we take , and thus, from , it follows , whereas ;
-
(3)
if , we take , and thus, from , it follows ;
-
(4)
if , we fix , and thus, from , it follows because and for .
Finally, taking , we obtain for all , because
Therefore, all conditions of Corollary 2 are satisfied. In fact, is the unique fixed point of f.
However, ϕ is not right upper semicontinuous at , so we cannot apply Corollary 3.
Similarly, we cannot apply Corollary 4 because ϕ is not a non-decreasing function.
Observe also that the w-distance q cannot be replaced by the quasi-metric d because for we have
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The authors are grateful to the referees for several useful suggestions. They also thank the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
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Alegre, C., Marín, J. & Romaguera, S. A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory Appl 2014, 40 (2014). https://doi.org/10.1186/1687-1812-2014-40
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DOI: https://doi.org/10.1186/1687-1812-2014-40