Abstract
In this paper, we shall investigate the existence and uniqueness of a fixed point of α-ψ-contractive mappings in the context of uniform spaces. We shall also prove some common fixed point theorems by introducing the notion of α-admissible pairs. We shall construct some examples to support our novel results.
MSC:46T99, 47H10, 54H25.
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1 Introduction
One of the interesting metric fixed point results was given by Samet et al. [1] by introducing the notions of α-admissible and α-ψ-contractive type mappings. They reported results via these new notions, and they extended and unified most of the related existing metric fixed point results in the literature. In particular, the authors [2] showed that fixed point results via cyclic contractions are consequences of their related results. Naturally, many authors have started to investigate the existence and uniqueness of a fixed point theorem via admissible mappings and variations of the concept of α-ψ-contractive type mappings, for reference see [1–19]. The notion of cyclic contraction was introduced by Kirk et al. [20]. The main advantage of the cyclic contraction is that the given mapping does not need to be continuous. It has been appreciated by several authors; see e.g. [21–26] and related references therein.
In this paper, we shall consider the characterization of the notions of α-ψ-contractive and α-admissible mappings in the context uniform spaces. Further, we shall prove some fixed point theorems by using these concepts. We shall also use α-admissible pairs to investigate the existence and uniqueness of a common fixed point in the setting of uniform spaces. We shall also establish some examples to illustrate the main results.
For the sake of completeness, we shall recollect some basic definitions and fundamental results. Let X be a nonempty set. A nonempty family, ϑ, of subsets of is called a uniform structure of X it satisfies the following properties:
-
(i)
if G is in ϑ, then G contains the diagonal ;
-
(ii)
if G is in ϑ and H is a subset of which contains G, then H is in ϑ;
-
(iii)
if G and H are in ϑ, then is in ϑ;
-
(iv)
if G is in ϑ, then there exists H in ϑ, such that, whenever and are in H, then is in G;
-
(v)
if G is in ϑ, then is also in ϑ.
The pair is called a uniform space and the element of ϑ is called entourage or neighborhood or surrounding. The pair is called a quasiuniform space (see e.g. [27, 28]) if property (v) is omitted.
Let be the diagonal of a nonempty set X. For , we shall use the following setting in the sequel:
and
For a subset , a pair of points x and y are said to be V-close if and . Moreover, a sequence in X is called a Cauchy sequence for ϑ, if, for any , there exists such that and are V-close for . For , there is a unique topology on X generated by where .
A sequence in X is convergent to x for ϑ, denoted by , if, for any , there exists such that for every . A uniform space is called Hausdorff if the intersection of all the is equal to Δ of X, that is, if for all implies . If then we shall say that a subset is symmetrical. Throughout the paper, we shall assume that each is symmetrical. For more details, see e.g. [27, 29–32].
Now, we shall recall the notions of A-distance and E-distance.
Let be a uniform space. A function is said to be an A-distance if, for any , there exists such that if and for some , then .
Let be a uniform space. A function is said to be an E-distance if
-
(i)
p is an A-distance,
-
(ii)
, .
Let be a uniform space and let d be a metric on X. It is evident that is a uniform space where is a set of all subsets of containing a ‘band’ for some . Moreover, if , then d is an E-distance on .
Let be a Hausdorff uniform space and p be an A-distance on X. Let and be sequences in X and , be sequences in converging to 0. Then, for , the following results hold:
-
(a)
If and for all , then . In particular, if and , then .
-
(b)
If and for all , then converges to z.
-
(c)
If for all with , then is a Cauchy sequence in .
Let p be an A-distance. A sequence in a uniform space with an A-distance is said to be a p-Cauchy if, for every , there exists such that for all .
Let be a uniform space and p be an A-distance on X.
-
(i)
X is S-complete if, for every p-Cauchy sequence , there exists x in X with .
-
(ii)
X is p-Cauchy complete if, for every p-Cauchy sequence , there exists x in X with with respect to .
-
(iii)
is p-continuous if implies .
Remark 1.6 Let be a Hausdorff uniform space which is S-complete. If a sequence be a p-Cauchy sequence, then we have . Regarding Lemma 1.4(b), we derive with respect to the topology , and hence S-completeness implies p-Cauchy completeness.
Definition 1.7 [20]
Let X be a nonempty set, m a positive integer and a mapping. is said to be a cyclic representation of X with respect to T if
-
(i)
, are nonempty sets;
-
(ii)
.
2 Main results
Let Ψ be the family of functions satisfying the following conditions:
() ψ is nondecreasing;
() for all , where is the n th iterate of ψ.
These functions are known in the literature as (c)-comparison functions. It is easily proved that if ψ is a (c)-comparison function, then for any .
Definition 2.1 [1]
Let and . We shall say that T is α-admissible if, for all , we have
We shall characterize the notion of α-ψ-contractive mapping, introduced by Samet et al. [1], in the context of uniform space as follows.
Definition 2.2 Let be a uniform space such that p is an E-distance on X and be a given mapping. We shall say that T is an α-ψ-contractive mapping if there exist two functions and such that
Theorem 2.3 Let be a S-complete Hausdorff uniform space such that p be an E-distance on X. Let be an α-ψ-contractive mapping satisfying the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that and ;
-
(iii)
T is p-continuous.
Then T has a fixed point .
Proof By hypothesis (ii) of the theorem we have such that . Define the sequence in X by for all . If for some , then is a fixed point of T. So, we can assume that for all n. Since T is α-admissible, we have
Inductively, we have
From (2.1) and (2.2), it follows that, for all , we have
Iteratively, we derive
Since p is an E-distance, for , we have
To show that is a p-Cauchy sequence, consider
Thus from (2.4) we have
Since , there exists such that . Thus by (2.5) we have
Since p is not symmetrical, by repeating the same argument we have
Hence the sequence is a p-Cauchy in the S-complete space X. Thus, there exists such that , which implies . Since T is p-continuous, we have , which implies that . Hence we have and . Thus by Lemma 1.4(a) we have . □
In the following theorem, we omit the p-continuity by replacing a suitable condition on the obtained iterative sequence.
Theorem 2.4 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X. Let be an α-ψ-contractive mapping satisfying the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that and ;
-
(iii)
for any sequence in X with as and for each , then for each .
Then T has a fixed point .
Proof By following the proof of Theorem 2.3, we know that is a p-Cauchy in the S-complete space X. Thus, there exists such that , which implies . By using (2.1) and assumption (iii), we get
Letting in above inequality, we shall have . Hence we have and . Thus by Lemma 1.4(a) we have . □
Example 2.5 Let be endowed with the usual metric d. Define . It is easy to see that is a uniform space. Define by
and by
and for all . One can easily see that T is α-ψ-contractive and α-admissible mapping. Also for we have . Moreover, for any sequence in X with as and for each we have for each . Therefore by Theorem 2.4, T has a fixed point.
In the sequel, we shall investigate the uniqueness of a fixed point. For this purpose, we shall introduce the following condition.
-
(H)
For all , there exists such that and .
Here, denotes the set of fixed points of T.
The following theorem guarantees the uniqueness of a fixed point.
Theorem 2.6 Adding the condition (H) to the hypothesis of Theorem 2.3 (respectively, Theorem 2.4), we obtain the uniqueness of fixed point of T.
Proof Suppose, on the contrary, that is another fixed point of T. From (H), there exists such that
Owing to the fact that T is α-admissible, from (2.10), we have
We define the sequence in X by for all and . From (2.11) and (2.1), we have
for all . This implies that
Letting in the above inequality, we obtain
Similarly,
From (2.13) and (2.14) together with Lemma 1.4(a), it follows that . Thus we have proved that u is the unique fixed point of T. □
Definition 2.7 [9]
A pair of two self-mappings is said to be α-admissible, if, for any with , we have and .
Definition 2.8 Let be a uniform space. A pair of two self-mappings is said to be an α-ψ-contractive pair if
for any , where .
Theorem 2.9 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that the pair of is an α-ψ-contractive pair satisfying the following conditions:
-
(i)
is α-admissible;
-
(ii)
there exists such that and ;
-
(iii)
for any sequence in X with as and for each , then for each .
Then T and S have a common fixed point.
Proof By hypothesis (ii) of the theorem, we have such that and . Since is an α-admissible pair, we can construct a sequence such that
From (2.15) for all , we have
Hence, we conclude that
Similarly, we find that
Hence, we derive
Thus from (2.16) and (2.17), and by induction, we get
We shall show that is a p-Cauchy sequence, Since p is an E-distance, for , we have
Now, we shall consider
Thus, from (2.19) we have
Since , there exists such that . Thus, by (2.20) we have
Since p is not symmetrical, by repeating the same argument we have
Hence the sequence is p-Cauchy in the S-complete space X. Thus, there exists such that , which implies . By using (2.15) and assumption (iii), we get
Letting in (2.23), we have . Hence we have and . Thus by Lemma 1.4(a) we have . Analogously, one can derive . Therefore . □
Remark 2.10 Note that Theorem 2.9 is valid if one replaces condition (ii) with
(ii)′ there exists such that and .
We shall get the following result by letting in Theorem 2.9.
Corollary 2.11 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that a mapping is satisfying the condition
for any , where . Also suppose that the following conditions are satisfied:
-
(i)
T is α-admissible;
-
(ii)
there exists such that and ;
-
(iii)
for any sequence in X with as and for each , then for each .
Then T has a fixed point.
Example 2.12 Let is a dislocated metric space where and . Define , where . It is easy to see that is a uniform space. Define by
and by
and by
and for all . One can easily see that is an α-ψ-contractive and α-admissible pair. Also for we have . Moreover, for any sequence in X with as and for each , we have for each . Therefore by Theorem 2.9, T and S have a common fixed point.
To investigate the uniqueness of a common fixed point, we shall introduce the following condition.
-
(I)
For each , we have , where is the set of all common fixed points of T and S.
Theorem 2.13 Adding the condition (I) to the hypothesis of Theorem 2.9, we obtain the uniqueness of the common fixed point of T and S.
Proof On the contrary suppose that are two distinct common fixed points of T and S. From (I) and (2.15) we have
which is impossible for . Consequently, we have . Analogously, one can show that . Thus we have , which is a contradiction to our assumption. Hence T and S have a unique common fixed point. □
3 Consequences
3.1 Standard contractions on uniform space
Taking in Theorem 2.6, for all , we shall obtain immediately the following fixed point theorems.
Corollary 3.1 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X and be a given mapping. Suppose that there exists a function such that
for all . Then T has a unique fixed point.
By substituting , where , in Corollary 3.1, we shall get the following.
Corollary 3.2 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that is a given mapping satisfying
for all . Then T has a unique fixed point.
Taking in Theorem 2.13 for all , we shall obtain immediately the following common fixed point theorem.
Corollary 3.3 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X and are given mappings. Suppose that there exists a function such that
for all . Then T and S have a unique common fixed point.
Corollary 3.4 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X and be a given mapping. Suppose that there exists a function such that
for all . Then T has a unique fixed point.
3.2 Cyclic contraction on uniform space
Corollary 3.5 Let be a S-complete Hausdorff uniform space such that p is an E-distance on X and , are nonempty closed subsets of X with respect to the topological space . Let be a mapping, where . Suppose that the following conditions hold:
-
(i)
and ;
-
(ii)
there exists a function such that
Then T has a unique fixed point that belongs to .
Proof Since and are closed subsets of X, is an S-complete Hausdorff uniform space. Define the mapping by
From (ii) and the definition of α, we can write
for all . Thus T is an α-ψ-contractive mapping.
Let such that . If , from (i), , which implies that . If , from (i), , which implies that . Thus in all cases, we have . This implies that T is α-admissible.
Also, from (i), for any , we have , which implies that .
Now, let be a sequence in X such that for all n and as . This implies from the definition of α that
Since is a closed subset of X with respect to the topological space , we get
which implies that . Thus we can easily get from the definition of α that for all n.
Finally, let . From (i), this implies that . So, for any , we have and . Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem 2.6 are satisfied, and we deduce that T has a unique fixed point that belongs to (from (i)). □
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Ali, M.U., Kamran, T. & Karapınar, E. Fixed point of α-ψ-contractive type mappings in uniform spaces. Fixed Point Theory Appl 2014, 150 (2014). https://doi.org/10.1186/1687-1812-2014-150
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DOI: https://doi.org/10.1186/1687-1812-2014-150