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Fixed points in uniform spaces
Fixed Point Theory and Applications volume 2014, Article number: 134 (2014)
Abstract
We improve Angelov’s fixed point theorems of Φ-contractions and j-nonexpansive maps in uniform spaces and investigate their fixed point sets using the concept of virtual stability. Some interesting examples and an application to the solution of a certain integral equation in locally convex spaces are also given.
1 Introduction
In 1987 [1], Angelov introduced the notion of Φ-contractions on Hausdorff uniform spaces, which simultaneously generalizes the well-known Banach contractions on metric spaces as well as γ-contractions [2] on locally convex spaces, and he proved the existence of their fixed points under various conditions. Later in 1991 [3], he also extended the notion of Φ-contractions to j-nonexpansive maps and gave some conditions to guarantee the existence of their fixed points. However, there is a minor flaw in his proof of Theorem 1 [3] where the surjectivity of the map j is implicitly used without any prior assumption. Additionally, we observe that such a map j can be naturally replaced by a multi-valued map J to obtain a more general, yet interesting, notion of J-nonexpansiveness. Therefore, in this work, we aim to correct and simplify the proof of Theorem 1 [3] as well as extend the notion of j-nonexpansive maps to J-nonexpansive maps and investigate the existence of their fixed points. Then we introduce J-contractions, a special kind of J-nonexpansive maps, that play the similar role as Banach contractions in yielding the uniqueness of fixed points. With the notion of J-contractions, we are able to recover results on Φ-contractions proved in [1] as well as present some new fixed point theorems in which one of them naturally leads to a new existence theorem for the solution of a certain integral equation in locally convex spaces. Finally, we prove that, under a mild condition, J-nonexpansive maps are always virtually stable in the sense of [4] and hence their fixed point sets are retracts of their convergence sets. An example of a virtually stable J-nonexpansive map whose fixed point set is not convex is also given.
2 Fixed point theorems
For any set S, we will use and to denote the set of all nonempty finite subsets of S and the cardinality of S, respectively. Let be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics indexed by A, , and . Interested readers should consult [5] for general topological concepts of uniform spaces, and [6] for the complete development of fixed point theory in uniform spaces that motivates this work. We first give the definition of a J-nonexpansive map as follows:
Definition 2.1 A self-map is said to be J-nonexpansive if for each ,
for any .
Example 2.2 Let , be equipped with the weak topology, and be defined by
for any . Then , where for each .
By Theorem 4.6 in [7], we have
for each , and . Here, .
By letting be defined by , for each , where
for each , it follows that T is J-nonexpansive.
The above definition of a J-nonexpansive map clearly extends the definition of a j-nonexpansive map in [3]. Before giving general existence criteria for fixed points of J-nonexpansive maps, we need the following notations. For each and , we let
and
When there is no ambiguity, we will denote an element of both and simply by . Notice that for each and , the sets and are finite, where denotes the n th coordinate projection .
Lemma 2.3 Every J-nonexpansive map is continuous.
Proof Suppose is J-nonexpansive. Let and be a net in X converging to x. Then for each , we have
Since converges to x, converges to 0 for any , and this proves the continuity of T. □
Theorem 2.4 Let be J-nonexpansive whose is finite for any . Then T has a fixed point in X if and only if there exists such that
-
(i)
the sequence has a convergence subsequence, and
-
(ii)
for each and , .
Proof (⇒): It is obvious by letting be a fixed point of T.
(⇐): Suppose that converges to some . Let and . Then and . We can choose sufficiently large so that and , for all . It follows that
Since α is arbitrary, converges to z. By the continuity of T, we have and hence z is a fixed point of T. □
As a corollary of the previous theorem, we immediately obtain Theorem 1 [3], with a corrected and simplified proof, as follows:
Corollary 2.5 Let be a j-nonexpansive map. If there exists such that
-
(i)
the sequence has a convergence subsequence, and
-
(ii)
for every , ,
then T has a fixed point.
Proof The proof follows directly from the previous theorem by considering the map . Notice that which is finite. □
We will now consider a special kind of J-nonexpansive maps that resemble Banach contractions in yielding the uniqueness of fixed points. Let Φ denote the family of all functions satisfying the following conditions:
(Φ1) ϕ is non-decreasing and continuous from the right, and
(Φ2) for any .
Notice that , and we will call subadditive if for all . Also, for a subfamily of Φ, , and , we let
Definition 2.6 A self-map is said to be a J-contraction if for each , there exists such that
for any , and is subadditive whenever .
Clearly, a Φ-contraction as defined in [1] is a J-contraction and a J-contraction is always J-nonexpansive. A natural example of a J-contraction can be obtained by adding (finitely many) appropriate Φ-contractions as shown in the following example.
Example 2.7 Given two Φ-contractions and as defined [1]. Then there exist , and for each , there exist such that
for any and . If for each , and there is a subadditive so that and for any , then the map is clearly a J-contraction with respect to and for any .
Lemma 2.8 If is a J-contraction. Then we have
for any , and .
Proof Recall that is assumed to be subadditive whenever . Then, for any , and , we clearly have
□
We now obtain some general criteria for the existence of fixed points of J-contractions.
Theorem 2.9 Suppose X is sequentially complete and is a J-contraction whose is finite for any . If T satisfies the following conditions:
-
(i)
for each , there exists such that
for any , , , and
-
(ii)
there exists such that for each , , and , we have
for some ,
then T has a fixed point. Moreover, if for each and , there exists such that
for all and , then the fixed point of T is unique.
Proof For each and , since is non-decreasing, we have
and by letting , it follows that
Also, for a given , since , we have for some . Since is right continuous, we have , and hence . Therefore, . By (1), it follows that . Since α is arbitrary, is a Cauchy sequence and, by sequential completeness, converges to some . Notice also that z must be a fixed point of T by continuity.
Now suppose that for each and , there exists such that for all and . If x, y are fixed points of T, then by Lemma 2.8, we have for each and ,
Since , we must have . □
As a corollary of the previous theorem, we immediately obtain Theorem 1 in [1] as follows.
Corollary 2.10 Suppose X is a bounded and sequentially complete subset of E and is Φ-contraction. If
-
(i)
for each , there exists such that for all and ,
-
(ii)
for each , ,
then there exists a unique fixed point of T.
Proof For each , , and , by letting and , we have , and . Hence, by Theorem 2.9, T has a unique fixed point. □
Theorem 2.11 Suppose X is sequentially complete and is a self-map satisfying: for each and , there exist , a finite set and a map such that
for any .
-
1.
If there exists such that for each there exists so that and
for all and , then T has a fixed point in X.
-
2.
If for each and , there exists such that and
for all and , then T has a unique fixed point in X and, for any , the sequence converges to the fixed point of T.
Proof First notice that T is clearly a J-contraction.
-
1.
For any and , we have
Also, since , is a Cauchy sequence and converges to a fixed point of T by the sequential completeness of X and the continuity of T.
-
2.
For any , and , we have
Also, since , is a Cauchy sequence and converges to a fixed point of T by the sequential completeness of X and the continuity of T.
Now, since for each , and ,
and , we have the uniqueness. □
Corollary 2.12 (Theorem 5 in [1])
Let us suppose
-
(i)
for each and , there exist and such that
for any ,
-
(ii)
there exists such that (), and .
Then T has at least one fixed point in X.
Proof By letting for any and and . Then for each , we have and . By Theorem 2.11(2), T has a fixed point. □
Theorem 2.13 Suppose X is sequentially complete and is a J-contraction whose is finite for each . If, for each , there exists satisfying:
-
(i)
is non-decreasing in t,
-
(ii)
for any , and , and
-
(iii)
there exist and such that for any and ,
then T has a fixed point in X.
Proof Let , , and for any , , , and . Then for any and , we have, by Lemma 2.8,
Since
for any , we have . Then by Theorem 2.11(1), T has a fixed point. □
Corollary 2.14 (Theorem 2 in [1])
Let us suppose
-
(i)
the operator is a Φ-contraction,
-
(ii)
for each there exists a Φ-function such that for all and is non-decreasing,
-
(iii)
there exists an element such that ().
Then T has at least one fixed point in X.
Proof By letting for any and . Then , and, by Theorem 2.13, T has a fixed point. □
Example 2.15 Given a sequentially complete locally convex space X, and two Φ-contractions ; i.e., there exist , and for each , there exist such that
for any and . Suppose further that
-
(i)
and for any ,
-
(ii)
for each , and for some , and
-
(iii)
there exists such that and for any and .
Then is a J-contraction with and . Also, by (i) and (iii), we have and
Hence, H satisfies all conditions in Theorem 2.13, and it has a fixed point in X. Notice that H may not be a Φ-contraction, by choosing and so that for some , and hence Theorem 2 in [1] cannot be applied.
We now end this section by giving an application to the solution of a certain integral equation in locally convex spaces.
Example 2.16 Following terminologies in [8], let X be an -space topologized by the family of seminorms and the space of all continuous functions from into X topologized by the family of seminorms , where for any . Let denote the set of all continuous linear operators on X,
and let be a -semigroup on X such that is locally bounded.
Now, we replace H3 and H5 in [8] by conditions (N1), (N2) and (N3) as follows:
(N1) is an operator such that there exists so that for any , there is such that
for any .
(N2) is continuous and there exist and such that for each ,
for any and ,
(N3) .
Consider the integral equation
whose solution is closely related to the mild solution of the differential equation
where a denotes the infinitesimal generator of .
We now define an operator G on by
for any . Following the proof of Theorem 3 in [8] and for each , , then we can show that, for any , there exists such that
where . It is easy to see that if for each , and , then G is a J-contraction with .
In particular, if we assume further that for each , , such that and . Then for any and , we have
Now, by letting , , , and , we have
-
(i)
for any , , , and ,
-
(ii)
for any and .
Therefore, by Theorem 2.11(2), G has a unique fixed point, so the integral equation (2) has a unique solution.
3 Fixed point sets
In this section, we will show that, under a mild condition, a J-nonexpansive map is always virtually stable. This immediately gives a connection between the fixed point set and the convergence set of a J-nonexpansive map. Recall that a continuous self-map , whose fixed point set is nonempty, on a Hausdorff space X is said to be virtually stable [4] if for each and each neighborhood U of x, there exist a neighborhood V of x and an increasing sequence of positive integers such that for all . When the sequence is independent of the point x and the neighborhood U, we simply call T a uniformly virtually stable map with respect to . For example, a (quasi-) nonexpansive self-map, whose fixed point set is nonempty, on a metric space is always uniformly virtually stable with respect to the sequence of all natural numbers. An important feature of a virtually stable map is the connection between its fixed point set and its convergence set as given in the following theorem.
Theorem 3.1 ([4], Theorem 2.6)
Suppose X is a regular space. If is virtually stable, then is a retract of , where is the (Picard) convergence set of T defined as follows:
As in the previous section, let be a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics indexed by A and . The following theorem gives a general criterion for a self-map on X to be virtually stable.
Theorem 3.2 Let be a self-map whose fixed point set is nonempty, and which satisfies the following conditions:
-
(i)
for each and , there exist a finite set and a map such that
for any ,
-
(ii)
there exists such that and for any and .
Then T is uniformly virtually stable with respect to the sequence of all natural numbers.
Proof Let and let U be a neighborhood of z. We may assume that for some and . For each , let
By (ii), there exists such that and for any and . Let which is clearly a nonempty open subset of X, , and . It follows that
If , then
If , since , we have for each , and hence
Hence, T is uniformly virtually stable with respect to the sequence of all natural numbers. □
Corollary 3.3 Suppose that T is J-nonexpansive with . If there exists such that and for any and , then T is uniformly virtually stable with respect to the sequence of all natural numbers.
Proof By letting and for any and , we have
for any . The result then follows from Theorem 3.2. □
Example 3.4 Let equipped with the weak topology and be defined by
for any . Then , and by Lemma 4.5 and Theorem 4.6 in [7], we have
for each , , and .
By letting be defined by for each , where
for each , it follows that T is J-nonexpansive.
Notice that is a fixed point of T, and for each and , . Then, by Theorem 3.2, T is virtually stable and hence the fixed point set of T is a retract of the convergence set of T. Moreover, the fixed point set is not convex because and are fixed points of T, while the convex combination is not.
References
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Acknowledgements
The authors are grateful to the anonymous referee(s) for their valuable comments and suggestions for improving this manuscript. This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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Chaoha, P., Songsa-ard, S. Fixed points in uniform spaces. Fixed Point Theory Appl 2014, 134 (2014). https://doi.org/10.1186/1687-1812-2014-134
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DOI: https://doi.org/10.1186/1687-1812-2014-134