Abstract
In a quasi-gauge space with quasi-gauge , using the left (right) -families of generalized quasi-pseudodistances on X (-families on X generalize quasi-gauge ), the left (right) quasi-distances () of Hausdorff type on are defined, , the three kinds of left (right) set-valued contractions of Nadler type are constructed, and, for such contractions, the left (right) -convergence of dynamic processes starting at each point is studied and the existence and localization of periodic and fixed point results are proved. As implications, two kinds of left (right) single-valued contractions of Banach type are defined, and, for such contractions, the left (right) -convergence of Picard iterations starting at each point is studied, and existence, localization, periodic point, fixed point and uniqueness results are established. Appropriate tools and ideas of studying based on -families and also presented examples showed that the results: are new in quasi-gauge, topological, gauge, quasi-uniform and quasi-metric spaces; are new even in uniform and metric spaces; do not require completeness and Hausdorff properties of the spaces , continuity of contractions, closedness of values of set-valued contractions and properties () and (), , ; provide information concerning localizations of periodic and fixed points; and substantially generalize the well-known theorems of Nadler and Banach types.
MSC:54A05, 54C60, 47H09, 37C25, 54H20, 54H25, 54E15.
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1 Introduction
There are in the literature many different versions of the well-known theorems due to Banach [1] and Nadler [2] concerning fixed points for single-valued and set-valued dynamic systems, respectively, in complete metric spaces. Especially, their analogues in more general spaces and concerning nontrivial problems and more complicated situations are important, fascinating and challenging (cf. [3–59]).
Recall that a set-valued dynamic system is defined as a pair , where X is a certain space and T is a set-valued map ; here denotes the family of all nonempty subsets of a space X. In particular, a set-valued dynamic system includes the single-valued dynamic system where T is a single-valued map , i.e., . For , define (m-times) and (an identity map on X).
Let be a set-valued dynamic system. By and we denote the sets of all fixed points and periodic points of T, respectively, i.e., and . A dynamic process or a trajectory starting at or a motion of the system at is a sequence defined by for (see Aubin and Siegel [5], Aubin and Ekeland [60], Aubin and Frankowska [4] and Yuan [58]).
Let be a single-valued dynamic system. For each , a sequence such that is called a Picard iteration starting at of the system .
The notion of Banach’s contraction belongs to the most fundamental mathematical ideas and a classic result of Banach, from 1922, is the milestone in the history of fixed point theory and its applications.
Theorem 1.1 (Banach [1])
Let be a complete metric space. Assume that the single-valued dynamic system is -contraction, i.e.,
Then T has a unique fixed point w in X (i.e., and ) and, for each , the sequence satisfies .
Recall that the Hausdorff metric on the class of all nonempty closed and bounded subsets of the metric space is defined as
where .
In a slightly different direction is the following elegant result of Nadler on set-valued dynamic systems.
Theorem 1.2 (Nadler [2], Theorem 5])
Let be a complete metric space. Assume that the set-valued dynamic system satisfying is -contraction, i.e.,
Then (i.e., there exists such that ).
Remark 1.1 Clearly, and , as metrics, are Hausdorff spaces, and the completeness of implies the completeness of . Observe that in the proofs of Theorems 1.1 and 1.2 the following play an important role: (a) the continuity of d and ; (b) the completeness and the separability of the spaces and ; (c) the continuity of maps and satisfying conditions (1.1) and (1.3), respectively; (d) in Theorem 1.2 the assumption that for each , is closed in X; (e) the properties and , .
By analyzing Theorems 1.1 and 1.2, one may build many examples without properties (a)-(e) and such that the assertions are obtainable and remain valid. These remarks suggest that more subtle investigations and modifications of structures on X, and the concept of distance of Hausdorff defined by (1.2), and the concepts of contractions of Banach and Nadler defined by (1.1) and (1.3) respectively are necessary. The aim of this paper is to provide new modifications of Theorems 1.1 and 1.2 removing the assumptions (a)-(e) mentioned in Remark 1.1 and leaving the assertions such as in Theorems 1.1 and 1.2, even in more general forms.
More precisely, let X be a nonempty set, let the family of quasi-pseudometrics , , be a quasi-gauge on X such that is a quasi-gauge space (in the sense of Dugundji [61] and Reilly [62]), and let the family of generalized quasi-pseudodistances , , be a left (right) -family on X (-families on X generalize quasi-gauge on X). Then, in , using the left (right) -families on X, the left (right) quasi-distances (), , of Hausdorff type on are defined, the three kinds of left (right) set-valued contractions of Nadler type are constructed, and, for such contractions, the left (right) -convergence of dynamic processes starting at each point is studied and the existence and localization of periodic and fixed point results are proved. As implications, two kinds of left (right) single-valued contractions of Banach type are defined, the left (right) -convergence of Picard iterations starting at each point is studied, and existence, localization, periodic point, fixed point and uniqueness results for such contractions are established.
The left (right) set-valued and single-valued contractions are studied here on X, on and on , where () are left (right) -balls centered in of radius .
Moreover, in our investigations, we assume additionally that these left (right) contractions are left (right) -admissible or left (right) partially -admissible. Also, the cases when these left (right) contractions are left (right) -quasi-closed maps are described.
Appropriate tools and ideas of studying based on asymmetric structures determined by -families and also presented examples showed that the results: are new in quasi-gauge, topological, gauge, quasi-uniform and quasi-metric spaces; are new even in uniform and metric spaces; do not require completeness and Hausdorff properties of the spaces , continuity of contractions, closedness of values of set-valued contractions and properties () and (), , ; provide information concerning localizations of periodic and fixed points; and substantially generalize the well-known theorems of Nadler and Banach types.
2 Quasi-gauge spaces
Before proceeding further, let us record the following.
Definition 2.1 Let X be a nonempty set.
-
(A)
A quasi-pseudometric on X is a map such that: (a) ; and (b) . For given quasi-pseudometric p on X, a pair is called quasi-pseudometric space. A quasi-pseudometric space is called Hausdorff if
(2.1) -
(B)
Each family of quasi-pseudometrics , , is called a quasi-gauge on X.
-
(C)
Let the family be a quasi-gauge on X. The topology having as a subbase the family of all balls , , , , is called the topology induced by on X.
-
(D)
(Dugundji [61], Reilly [34, 62]) A topological space such that there is a quasi-gauge on X with is called a quasi-gauge space and is denoted by .
-
(E)
A quasi-gauge space is called Hausdorff if the quasi-gauge has the property:
(2.2) -
(F)
Let the family be a quasi-gauge on X, and let be a quasi-gauge space. If , where , then is a quasi-gauge space and is called the conjugate of .
Remark 2.1 Each quasi-uniform space and each topological space is a quasi-gauge space (Reilly [62], Theorems 4.2 and 2.6]). The quasi-gauge spaces are the greatest general spaces with asymmetric structures.
3 Left (right) -families
Historically, the first work on the distances in metric spaces was done by Tataru [45]. Next, the concepts of w-distances, τ-functions and τ-distances in these spaces, which generalize Tataru distances and metrics d, were introduced by Kada et al. [23], Lin and Du [27] and Suzuki [37], respectively. Distances in uniform spaces had first been formulated by Vályi [46]. From rich literature it follows that the above distances provide useful and powerful tools for investigating problems of fixed point theory. Using these ideas, more general and various distances have been demonstrated in [47–57].
For a different purpose, in quasi-gauge spaces with quasi-gauges on X, we recall the left (right) -families of generalized quasi-pseudodistances on X (left (right) -families generalize quasi-gauges ).
Definition 3.1 ([57], Section 3])
Let be a quasi-gauge space.
-
(A)
The family of maps , , is said to be a left (right) -family of generalized quasi-pseudodistances on X (left (right) -family on X, for short) if the following two conditions hold:
(1) ; and
(2) for any sequences and in X satisfying
(3.1)(3.2)and
(3.3)(3.4)the following holds:
(3.5)(3.6) -
(B)
Define
In the following remark, we list some basic properties of left (right) -families in .
Remark 3.1 Let be a quasi-gauge space. The following hold:
-
(a)
.
-
(b)
Let or . If , then for each , is quasi-pseudometric.
-
(c)
There are examples of and such that the maps , , are not quasi-pseudometrics. Indeed, in Example 4.1 below, if , then .
-
(d)
([57], Proposition 3.1]) If is a Hausdorff quasi-gauge space and or , then .
4 Left (right) -balls
In this section we define and characterize the left (right) -balls in .
Definition 4.1 Let be a quasi-gauge space, and let the family of maps , , be a left (right) -family on X. We define the left (right) -ball centered in of radius by
Remark 4.1 Notice, however, that there exist a quasi-gauge space , a left (right) -family on X, and such that (). This follows from Example 4.1 below.
Example 4.1 Let X contain at least two different points, let the family of quasi-pseudometrics , , be a quasi-gauge on X, and let be a quasi-gauge space.
Let the set containing at least two different points be arbitrary and fixed, and let satisfy , where .
Let the family , , , be defined by the formula:
Then .
Indeed, we see that condition (1) does not hold only if there exist some and such that , , and . However, then we conclude that there exists such that and , which is impossible. Therefore, , i.e., condition (1) holds.
Now suppose that the sequences and in X satisfy (3.1) and (3.3). Then, in particular, (3.3) yields
By (4.2) and (4.1), denoting , we conclude that
From (4.3), the definition of and (4.2), we get
The result is that the sequences and satisfy (3.5). Therefore, is a left -family.
Analogously, we prove that if and in X satisfy (3.2) and (3.4), then also (3.6) holds, therefore is a right -family.
5 Left (right) -convergences and left (right) -sequential completeness
Now, using left (right) -families, we define the following natural concept of left (right) -completeness in .
Definition 5.1 Let be a quasi-gauge space, and let be a left (right) -family on X.
-
(A)
We say that a sequence in X is left (right) -Cauchy sequence in X if
-
(B)
Let and let be a sequence in X. We say that is left (right) -convergent to u if (), where
-
(C)
We say that a sequence in X is left (right) -convergent in X if (), where
-
(D)
If every left (right) -Cauchy sequence in X is left (right) -convergent in X (i.e., ()), then is called a left (right) -sequentially complete quasi-gauge space.
Remark 5.1 Let be a quasi-gauge space.
-
(a)
It is clear that if is left (right) -convergent in X, then
for each subsequence of .
-
(b)
There exist examples of quasi-gauge spaces and left (right) -families on X, , such that are left (right) -sequentially complete, but not left (right) -sequentially complete.
6 Left (right) -closed sets
Definition 6.1 Let be a quasi-gauge space, and let the family of maps , , be a left (right) -family on X.
-
(A)
We say that a set is a left (right) -closed in X if (), where (), the left (right) -closure in X, denotes the set of all for which there exists a sequence in Y which left (right) -converges to x.
-
(B)
Define (); that is, () denotes the class of all nonempty left (right) -closed subsets of X.
Remark 6.1 If is a left (right) -sequentially complete quasi-gauge space and a set (), then is a left (right) -sequentially complete quasi-gauge space.
7 Left (right) -admissible and left (right) partially -admissible set-valued maps
The following terminologies will be much used in the sequel.
Definition 7.1 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and let .
-
(A)
We say that a set-valued map is left (right) -admissible in a point if for each sequence satisfying the properties and
there exists such that
We say that a set-valued map is left (right) -admissible in Y if is left (right) -admissible in each point .
-
(B)
We say that a set-valued map is left (right) partially -admissible in a point if for each sequence satisfying the properties and
there exists such that
We say that a set-valued map is left (right) partially -admissible in Y if is left (right) partially -admissible in each point .
Remark 7.1 Let be a quasi-gauge space, and let the family of maps , , be a left (right) -family on X.
-
(a)
If is a left (right) -sequentially complete quasi-gauge space, then a set-valued dynamic system , , is left (right) -admissible on X.
-
(b)
If is a left (right) -sequentially complete quasi-gauge space and is symmetric, i.e., , then is left (right) partially -admissible on X.
-
(c)
It is evident that each left (right) partially -admissible on X a set-valued dynamic system is left (right) -admissible on X but the converse not necessarily holds.
8 Left (right) -quasi-closed maps
We can define the following generalizations of continuity.
Definition 8.1 Let be a quasi-gauge space, let be a set-valued dynamic system, , and let . The map is said to be a left (right) -quasi-closed map on X if for every sequence in , left (right) -converging in X (thus ()) and having subsequences and satisfying
the following property holds: there exists () such that ().
Definition 8.2 Let be a quasi-gauge space, let Y be a nonempty subset of X, and let be a set-valued map. The map T is said to be a left (right) -quasi-closed map on Y if for every sequence in , left (right) -converging in X (thus ()) and having subsequences and satisfying
the following property holds: there exists () such that ().
9 Left (right) quasi-distances of Hausdorff type and three kinds of set-valued left (right) contractions of Nadler type
In not necessarily Hausdorff quasi-gauge spaces, we define the left (right) Hausdorff quasi-distances (Definition 9.1(A)) and the set-valued left (right) contractions of Nadler type (Definition 9.1(B)).
Definition 9.1 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, let , and let
-
(A)
Define on the left (right) quasi-distance () of Hausdorff type, where , (, ) are defined as follows:
(A.1)
,
and
if ;
(A.2)
,
and
if .
-
(B)
Let and let . We say that a set-valued map is left (right) -contraction on Y (-contraction on Y) if:
(B.1) if ;
(B.2) if .
Remark 9.1 Let be a quasi-gauge space, and let the family of maps , , be a left (right) -family on X.
-
(a)
Generally, () are not symmetric, i.e., () not necessarily hold. Moreover, () not necessarily hold; see Remarks 12.1 and 12.3.
-
(b)
Each -contraction on Y (-contraction on Y), , is -contraction on Y (-contraction on Y) but the converse not necessarily holds.
10 Convergence, existence, fixed point, periodic point and localization results for left (right) set-valued contractions of Nadler type
We have the following theorem.
Theorem 10.1 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and suppose that . Assume, moreover, that and a set-valued dynamic system , , satisfy the following:
-
(i)
T is -contraction on X (T is -contraction on X); and
-
(ii)
For every and for every , there exists such that
(10.1)(10.2)
-
(A)
If is left (right) -admissible in a point , then there exist a dynamic process of the system starting at , a point and such that () and is left (right) -convergent to w.
If, moreover, is left (right) partially -admissible in a point , then the point w above satisfies ().
-
(B)
If is left (right) -admissible in a point and if, for some , is left (right) -quasi-closed on X, then and there exist a dynamic process of the system starting at , a point and such that () and is left (right) -convergent to w.
If, moreover, is left (right) partially -admissible in a point , then the point w above satisfies ().
Proof We prove Theorem 10.1 only in the case when is a left -family on X, is left -admissible on X or left partially -admissible on X, and is left -quasi-closed map on X, respectively. We omit the proof in the case of ‘right’, which is based on an analogous technique.
Part 1. Assume that is left -admissible in a point .
By (9.1) and the fact that , , we choose
such that
Put
In view of (10.3) and (10.4), this implies , and we apply (10.1) to find such that
We see from (10.5) and (10.6) that
Observe that (10.7) implies .
Put now
Then, in view of (10.7), we get , and applying again (10.1) we find such that
Also note that
Indeed, from (10.9), (9.1), Definition 9.1 and (10.8), we get
Thus (10.10) holds. Further, by (1), (10.7) and (10.10), we observe that
Hence .
Proceeding as before, using Definition 9.1 and property (10.1), we get that there exists a sequence in X satisfying
For calculational purposes, upon letting , where
we observe that ,
and
We see from (10.13) that .
Let now . Using (1) and (10.12), we get
This means that
and
Now, since is left -admissible on X, by Definition 6.1(A), properties (10.14) and (10.15) imply that there exists such that
Next, defining and for , by (10.15) and (10.16) we see that conditions (3.1) and (3.3) hold for the sequences and in X. Consequently, by () we get (3.5) which implies that
and so, in particular, we see that .
Additionally, by () and (10.13), we note that
Part 2. Assume that is left -admissible in a point and, for some , is left -quasi-closed on X.
By Part 1, and since by (10.11), for , thus defining we see that , , the sequences and satisfy and, as subsequences of , are left -converging to each point of the set . Moreover, by Remark 5.1(a), and . By above, since is left -quasi-closed, we conclude that .
Part 3. Assume that is left partially -admissible in a point .
Using Part 1, (10.14) and (10.15), by Definition 6.1(B), we have that there exists such that
The consequence of (10.17) and (10.18) is .
Part 4. The result now follows at once from Parts 1-3. □
Theorem 10.1 and its proof immediately yields the following theorem.
Theorem 10.2 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and suppose that . Assume, moreover, that , , and a set-valued map
satisfy:
-
(i)
T is -contraction on (T is -contraction on );
-
(ii)
(); and
-
(iii)
for every () and for every , there exists such that
-
(A)
If T is left (right) partially -admissible in , then there exist a dynamic process of the system starting at and a point such that () and is left (right) -convergent to w.
-
(B)
If T is left (right) partially -admissible in and if T is left (right) -quasi-closed on (on ), then and there exist a dynamic process of the system starting at and a point such that (), is left (right) -convergent to w and ().
11 Convergence, existence, periodic point, fixed point, localization and uniqueness results for single-valued left (right) contractions of Banach type
In this section we indicate how to extend the results of the preceding section to single-valued maps.
Definition 11.1 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and let .
-
(A)
Define on X the left (right) distance () as follows:
(A.1) and if ;
(A.2) and if .
-
(B)
Let and let . We say that a single-valued map is -contraction on Y (-contraction on Y) if:
(B.1) if ;
(B.2) if .
As a consequence of Definition 11.1 and Theorems 10.1 and 10.2, we have the following results.
Theorem 11.1 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and suppose that . Let and let a single-valued dynamic system , , be -contraction on X (-contraction on X).
-
(A)
If is left (right) -admissible in a point , then there exist a point and such that the sequence is left (right) -convergent to w and ().
If, moreover, is left (right) partially -admissible in a point , then the point w above satisfies .
-
(B)
If is left (right) -admissible in a point and if, for some , is left (right) -quasi-closed on X, then and there exist a point and such that the sequence is left (right) -convergent to w, () and
(11.1)
If, moreover, is left (right) partially -admissible in a point , then the point w above satisfies ().
-
(C)
If is left (right) -admissible in a point , if, for some , is left (right) -quasi-closed on X and if is a Hausdorff space, then there exists a point such that , the sequence is left (right) -convergent to w, () and
(11.2)
If, moreover, is left (right) partially -admissible in a point , then the point w above satisfies ().
Proof We prove only (11.1) and (11.2) and only in the case when is a left -family on X, is left -admissible in or left partially -admissible in , and is left -closed map on X, respectively. We omit the proof in the case of ‘right’, which is based on an analogous technique.
Part 1. Property (11.1) holds.
Indeed, first suppose that . Of course, , and, for , by Definition 11.1, 0 < = ⩽ ⩽ ≤ ≤ ≤ ⋯ ≤ < , which is impossible.
Suppose now that . Then, by Definition 11.1, using the fact that , we get, for , 0 < = ≤ ≤ ≤ = 0, which is impossible.
Therefore, (11.1) holds.
Part 2. Property (11.2) holds.
If is a Hausdorff space, then Remark 3.1(d) and property (11.1) imply and . Therefore, and
Suppose now that and . Then, by Remark 3.1(d), . Of course, for , we then have {[ > 0 ∧ = ≤ ≤ < ] ∨ [ > 0 ∧ = ≤ ≤ < ]}, which is impossible. This gives that is a singleton.
Therefore, (11.2) holds. □
Theorem 11.2 Let be a quasi-gauge space, let the family of maps , , be a left (right) -family on X, and suppose that . Let , , and a single-valued map () be such that
and ().
-
(A)
If T is left (right) partially -admissible in , then there exists a point such that the sequence is left (right) -convergent to w, () and ().
-
(B)
If T is left (right) partially -admissible in and if T is left (right) -quasi-closed on (), then , and there exists a point such that the sequence is left (right) -convergent to w, (), and
-
(C)
If T is left (right) partially -admissible in , if T is left (right) -quasi-closed on () and if is a Hausdorff space, then there exists such that , the sequence is left (right) -convergent to w, (), () and
12 Examples illustrating Theorem 10.1 and comparison of Theorem 10.1 with Theorem 1.2 of Nadler
Example 12.1 Let , , be a metric space with a metric of the form , , and let . Let be of the form
Let and let , where J is of the form
(I.1) is a quasi-gauge space and is left and right -sequentially complete.
(I.2) The property holds.
(I.3) J is symmetric and . See Example 4.1.
(I.4) The set-valued dynamic system is a -contraction on X, i.e., , where
Indeed, denoting , we see that this follows from (I.1)-(I.3) and from Cases I.4.1-I.4.6 below.
Case I.4.1. If , then and . Hence, and, consequently, .
Case I.4.2. If and , then , , , , , since , and since implies
Therefore, .
Case I.4.3. If and , then , , , and . We calculate: (a) If , then and, consequently, ; (b) If , then and, consequently, ; (c) Inequality is a consequence of (a) and (b).
Case I.4.4. If and , then , , , and . We calculate: (a) If , then
and, consequently, ; (b) If , then ; (c) The consequence of (a) and (b) is that .
Case I.4.5. If , then , , , for and , and .
Case I.4.6. If , then , , , for and , and .
(I.5) Property (10.1) holds, i.e., . Indeed, this follows from Cases I.5.1-I.5.4 below.
Case I.5.1. If and , then, since , . Hence, .
Case I.5.2. If and , then and .
Case I.5.3. If and , then and .
Case I.5.4. If and , then and .
(I.6) The set-valued dynamic system is partially -admissible in X. In fact, observing that for , it remains to verify that if and are such that and , then . One way to check this is as follows: We see that and, in view of (12.2), implies . Moreover, . From this information we deduce that .
(I.7) The set-valued dynamic system is a left and right -quasi-closed map in X. Indeed, let be a left (thus also right) -converging sequence in X. Since , thus . Then we remark that the following two cases hold.
Case I.7.1. If , then and if and are subsequences of satisfying , then we get . Moreover, .
Case I.7.2. Suppose now that . Then and if and are subsequences of satisfying , then we get , which is impossible. Let us observe, additionally, that then also .
(I.8) All the assumptions of Theorem 10.1 are satisfied. This follows from (I.1)-(I.7).
We conclude that and one shows the following.
Claim I.8.1. If and are arbitrary and fixed, then defining for we get that , and .
Claim I.8.2. If and are arbitrary and fixed, then defining for we get that , and .
Claim I.8.3. If and are arbitrary and fixed, then defining for we get that , and .
Claim I.8.4. If and are arbitrary and fixed, then defining for we get that , and .
Remark 12.1 Let . By (12.2), .
Example 12.2 Let be a complete metric space where and is of the form , , and let be defined by
We see that, for each , where . Moreover, . However, for each , condition (1.3) for does not hold. We argue by contradiction and suppose that
Consider then the case when and . Then we deduce the following: (i) For , we have and ; (ii) For , we have and ; (iii) Consequently,
which is absurd.
Remark 12.2 Observe that and defined in Examples 12.1 and 12.2 are identical. However, Example 12.1 shows that we may apply Theorem 10.1 with defined by (12.2) and satisfying , and Example 12.2 shows that we do not apply Theorem 1.2 of Nadler since (1.3) does not hold.
Example 12.3 Let and let where p is a quasi-pseudometric on X defined by
Let and let where J is of the form
Define by
(III.1) is not symmetric. In fact, by (12.3), and .
(III.2) is a quasi-gauge space and . See Example 4.1.
(III.3) The property holds. This follows from (12.3) and Definitions 6.1 and 5.1(C).
(III.4) The set-valued dynamic system is a -contraction on X, i.e., , where
Indeed, denoting , we see that this follows from (III.1)-(III.3) and from Cases III.4.1-III.4.3 below.
Case III.4.1. If , then , and, by (12.3), . Thus .
Case III.4.2. If and , then , , , and . Hence, by (12.3), implies
On the other hand, implies . Therefore, .
Case III.4.3. If and , then , , , and . Consequently, by (12.3), implies . Next, by (12.3), implies . Therefore, .
Case III.4.4. If , then , and . Thus, .
(III.5) Property (10.1) holds, i.e., . Indeed, this follows from Cases III.5.1-III.5.3 below.
Case III.5.1. If and , then , and .
Case III.5.2. If and , then , and .
Case III.5.3. If and , then and .
(III.6) The set-valued dynamic system is left -admissible in X. We verify that if and are arbitrary and fixed and such that
and
then
In fact, first note that
Next we see that (12.7) is equivalent to and so, in particular in view of (12.9), (12.3) and (12.4), this implies
Now in view of (12.9), (12.10), (12.3) and (12.4), we conclude that and hence, since and , we must have where , and this implies (12.8). Therefore is left -admissible in X.
(III.7) The set-valued dynamic system is a left -quasi-closed map in X. Indeed, let be a left -converging sequence in X. Since , thus . In other words, and thus, by (12.4) and (12.3), we obtain or, equivalently, . Of course, then also . We remark that the considerations above show that if and are arbitrary and fixed subsequences of satisfying , then .
(III.8) For defined by (12.3) and (12.4), all the assumptions of Theorem 10.1 in the case of left are satisfied. This follows from (III.1)-(III.7).
We conclude that , and we claim that if , , and are arbitrary and fixed, and , then the sequence is a dynamic process of T starting at and left -converging to each point and these points w satisfy .
Remark 12.3 Let a quasi-gauge space and -family be as in Example 12.3.
-
(a)
From Cases III.4.2 and III.4.3 it follows that for and .
-
(b)
Observe that if .
Example 12.4 Let a quasi-pseudometric space , , and a set-valued dynamic system be as in Example 12.3. Observe that does not hold; here , . In fact, we argue by contradiction and suppose that the above condition holds. Then we remark that for and , we have , and . Thus, for , we get . This shows that , which is absurd.
Remark 12.4 Observe that and defined in Examples 12.3 and 12.4 are identical, note that we may apply Theorem 10.1 with defined by (12.3) and (12.4) and satisfying and note, however, that we do not apply Theorem 10.1 with . Thus the existence of a -family such that is essential.
Example 12.5 Let and let be a metric space where is of the form
Defining the set
we consider two maps , , defined by:
Observe that
Remark 12.5 We claim that for and for , , defined in Example 12.5, we do not use Theorem 1.2 of Nadler. Indeed, we note that is complete, (12.15) holds, and , whereas, for each and for each , is not closed in .
Example 12.6 Let . Define by
here A is defined by (12.12). Let , , be as in (12.13) and (12.14).
(VI.1) The map p defined by (12.16) is quasi-pseudometric on X and , , is a quasi-gauge space. See [57], (VIII.1), p.23].
(VI.2) The space is a left -sequentially complete. See [57], (IX.2), p.24].
(VI.3) For , has the property . Indeed, let , let be an arbitrary and fixed point of X, and let be an arbitrary and fixed sequence in the set which is left -convergent to each point of a nonempty set . Thus we see that if and, by (12.16), we conclude that .
As a consequence we have the following cases.
Case VI.3.1. If , then and, by (12.16), and ;
Case VI.3.2. If , then and, by (12.16), ; i.e., .
(VI.4) For , the set-valued dynamic system is left partially -admissible in X. In fact, observing that and for , it remains to verify that if and are such that
or
and
then
here we remark that by (12.16) property (12.19) shows that
One way to check this is as follows: If , then by (12.16), (12.17) and (12.21), and . If , then by (12.16), (12.18) and (12.21), we have . By symmetry of p, this shows that (12.20) holds.
(VI.5) Let . For each , the set-valued dynamic system is a -contraction on X. Indeed, we see that, for each ,
Therefore, .
(VI.6) For , the set-valued dynamic system is left -quasi-closed on X. Indeed, let . Let be an arbitrary and fixed sequence in , left -convergent to each point of a nonempty set and having subsequences and satisfying . Of course, and . Let now be arbitrary and fixed. Since , thus, by (12.16),
As a consequence, we have the following cases.
Case VI.6.1. If , then by (12.13), (12.16) and (12.21), . Hence, , and . This gives, by Section 8, that is left -quasi-closed on X.
Case VI.6.2. If , then by (12.14), (12.16) and (12.21), . Hence, and . This gives, by Section 8, that is left -quasi-closed on X.
(VI.7) For each , property (10.1) holds, i.e.,
Indeed, this follows from Cases VI.7.1 and VI.7.2 below.
Case VI.7.1. Let .
If , then , and .
If , then , and ;
Case VI.7.2. Let .
If , then , and .
If , then , and .
(VI.8) All the assumptions of Theorem 10.1(B) when are satisfied. This is proved in (VI.1)-(VI.7).
We conclude that and , and one shows the following.
Claim VI.8.1. Let . If and , , satisfies , then the sequence is left -convergent to w and . Moreover, and whenever , and, for each , and whenever ;
Claim VI.8.2. Let . If and , , satisfies , then the sequence is left -convergent to w and . Moreover, and whenever , and, for each , and whenever .
Remark 12.6 Let , let , , be defined by (12.13) and (12.14), let d be of the form (12.11) and let p be of the form (12.16). We point out some facts concerning Examples 12.5 and 12.6.
-
(a)
is not Hausdorff. Indeed, if and , then , and we notice that evidently (2.1) does not hold.
-
(b)
The basic idea of Example 12.6 is as follows. First, in Example 12.5, in the metric space , we showed that it is not possible to use the metric structure on X determined by d despite the fact that (12.15) and thus also (1.3) holds for , . Next, by a suitable choice of the not Hausdorff structure on X, defined by , we proved that for and , , the assumptions of Theorem 10.1(B) when hold and we can then apply this theorem.
Example 12.7 Let and let A and be defined by (12.12) and (12.16), respectively. For , , , we define the map
(VII.1) We claim that, for each , is a -contraction and -contraction on X. In view of (12.16), this follows from the fact that . Indeed, by (12.23), implies . Hence, by (12.16) we have that
(VII.2) The set-valued dynamic system is left and right partially -admissible in X. In fact, since for , thus if and are arbitrary and fixed and such that , then and (thus also ). Consequently, by (12.16), . By Definition 7.1(B), is left and right partially -admissible in X.
(VII.3) The set-valued dynamic system ,
is left and right -quasi-closed on X. Indeed, let be an arbitrary and fixed sequence in , left -convergent to each point w of a nonempty set and having subsequences and satisfying . It follows from (12.16) that then and . Analogously, we prove that is right -quasi-closed on X.
(VII.4) Properties (10.1) and (10.2) hold. Indeed, this follows from Cases VII.4.1-VII.4.3 below.
Case VII.4.1. If and , then , , and ;
Case VII.4.2. If and , then , , and .
Case VII.4.3. Finally, and follow from the fact that is symmetric.
(VII.5) All the assumptions of Theorem 10.1(B) when are satisfied. This is proved in (VII.1)-(VII.4).
We conclude that and , and one shows the following.
Claim VII.5.1. If , and , , then , , the sequence is left and right -convergent to w, and, for each , and ; here
Claim VII.5.2. If , and , , then , , the sequence is left -convergent to w, and ; here .
Remark 12.7 We point out some facts concerning Example 12.7.
-
(a)
is not Hausdorff; see Remark 12.6(a).
-
(b)
Noting that is symmetric, we see that holds.
-
(c)
The property does not hold. In fact, it is not hard to see that, for , and for , .
-
(d)
and .
References
Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Nadler SB: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
Alemany S, Romaguera AS: On right K -sequentially complete quasi-metric spaces. Acta Math. Hung. 1997, 75: 267–278. 10.1023/A:1006559507624
Aubin JP, Frankowska H: Set-Valued Analysis. Birkhäuser, Boston; 1990.
Aubin JP, Siegel J: Fixed points and stationary points of dissipative multivalued maps. Proc. Am. Math. Soc. 1980, 78: 391–398. 10.1090/S0002-9939-1980-0553382-1
Aydi H, Abbas M, Vetro C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 2012, 159: 3234–3242. 10.1016/j.topol.2012.06.012
Benavides TD, Ramírez PL: Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions. J. Math. Anal. Appl. 2004, 291: 100–108. 10.1016/j.jmaa.2003.10.019
de Blasi FS, Myjak J, Reich S, Zaslawski AJ: Generic existence and approximation of fixed points for nonexpansive set-valued maps. Set-Valued Var. Anal. 2009, 17: 97–112. 10.1007/s11228-009-0104-5
Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.
Caristi J, Kirk WA Lecture Notes in Math. 490. In Geometric Fixed Point Theory and Inwardness Conditions. Springer, Berlin; 1975:74–83.
Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116
Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5–11. 10.1007/BF02771543
Ekeland I: Remarques sur les problémes variationnels. I. C. R. Acad. Sci. Paris Sér. A-B 1972, 275: 1057–1059.
Ekeland I: On the variational principle. J. Math. Anal. Appl. 1974, 47: 324–353. 10.1016/0022-247X(74)90025-0
Ekeland I: Nonconvex minimization problems. Bull. Am. Math. Soc. 1979, 1: 443–474. 10.1090/S0273-0979-1979-14595-6
Espínola R, Petruşel A: Existence and data dependence of fixed points for multivalued operators on gauge spaces. J. Math. Anal. Appl. 2005, 309: 420–432. 10.1016/j.jmaa.2004.07.006
Espínola R, Kirk WA: Set-valued contractions and fixed points. Nonlinear Anal. 2003, 54: 485–494. 10.1016/S0362-546X(03)00107-X
Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 2006, 317: 103–112. 10.1016/j.jmaa.2005.12.004
Frigon M: Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. Fixed Point Theory Appl. 2010., 2010: Article ID 183217
Frigon M: Fixed point results for generalized contractions in gauge spaces with applications. Proc. Am. Math. Soc. 2000, 128: 2957–2965. 10.1090/S0002-9939-00-05838-X
Frigon M: Fixed point results for multivalued contractions on gauge spaces. Ser. Math. Anal. Appl. 4. In Set-Valued Mappings with Applications in Nonlinear Analysis. Taylor & Francis, London; 2002:175–181.
Frigon M: Fixed point and continuation results for contractions in metric and gauge spaces. Banach Cent. Publ. 2007, 77: 89–114.
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44: 381–391.
Kirk WA: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 2003, 277: 645–650. 10.1016/S0022-247X(02)00612-1
Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064
Leader S, Hoyle SL: Contractive fixed points. Fundam. Math. 1975, 87: 93–108.
Lin L-J, Du W-S: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 2006, 323: 360–370. 10.1016/j.jmaa.2005.10.005
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X
Nadler SB: Periodic points of multi-valued ε -contractive maps. Topol. Methods Nonlinear Anal. 2003, 22: 399–409.
Pathak HK, Shahzad N: Fixed point results for set-valued contractions by altering distances in complete metric spaces. Nonlinear Anal. 2009, 70: 2634–2641. 10.1016/j.na.2008.03.050
Petruşel G: Fixed point results for multivalued contractions on ordered gauge spaces. Cent. Eur. J. Math. 2009, 7: 520–528. 10.2478/s11533-009-0027-2
Quantina K, Kamran T: Nadler’s type principle with hight order of convergence. Nonlinear Anal. 2008, 69: 4106–4120. 10.1016/j.na.2007.10.041
Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.
Reilly IL: A generalized contraction principle. Bull. Aust. Math. Soc. 1974, 10: 349–363.
Reilly IL, Subrahmanyam PV, Vamanamurthy MK: Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte Math. 1982, 93: 127–140. 10.1007/BF01301400
Smithson RE: Fixed points for contractive multifunctions. Proc. Am. Math. Soc. 1971, 27: 192–194. 10.1090/S0002-9939-1971-0267564-4
Suzuki T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 2001, 253: 440–458. 10.1006/jmaa.2000.7151
Suzuki T: Several fixed point theorems concerning τ -distance. Fixed Point Theory Appl. 2004, 2004: 195–209.
Suzuki T: Several fixed point theorems in complete metric spaces. Yokohama Math. J. 1997, 44: 61–72.
Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl. 2008, 340: 752–755. 10.1016/j.jmaa.2007.08.022
Suzuki T: Kannan mappings are contractive mappings in some sense. Comment. Math. 2005, 45: 43–56.
Tarafdar E, Yuan GX-Z: Set-valued contraction mapping principle. Appl. Math. Lett. 1995, 8: 79–81.
Takahashi W: Existence theorems generalizing fixed point theorems for multivalued mappings. Pitman Res. Notes Math. Ser. 252. In Fixed Point Theory and Applications. Edited by: Baillon JB, Théra M. Longman Sci. Tech., Harlow; 1991:397–406. (Marseille, 1989)
Tarafdar E: An approach to fixed-point theorems on uniform space. Trans. Am. Math. Soc. 1974, 191: 209–225.
Tataru D: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 1992, 163: 345–392. 10.1016/0022-247X(92)90256-D
Vályi I: A general maximality principle and a fixed point theorem in uniform spaces. Period. Math. Hung. 1985, 16: 127–134. 10.1007/BF01857592
Włodarczyk K, Plebaniak R: Maximality principle and general results of Ekeland and Caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl. 2010., 2010: Article ID 175453
Włodarczyk K, Plebaniak R: Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 864536
Włodarczyk K, Plebaniak R, Obczyński C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 2010, 72: 794–805. 10.1016/j.na.2009.07.024
Włodarczyk K, Plebaniak R: Quasi-gauge spaces with generalized quasi-pseudodistances and periodic points of dissipative set-valued dynamic systems. Fixed Point Theory Appl. 2011., 2011: Article ID 712706
Włodarczyk K, Plebaniak R: Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances. J. Math. Anal. Appl. 2012, 387: 533–541. 10.1016/j.jmaa.2011.09.006
Włodarczyk K, Plebaniak R: Generalized uniform spaces, uniformly locally contractive set-valued dynamic systems and fixed points. Fixed Point Theory Appl. 2012., 2012: Article ID 104
Włodarczyk K, Plebaniak R: Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances. Topol. Appl. 2012, 159: 3504–3512. 10.1016/j.topol.2012.08.013
Włodarczyk K, Plebaniak R: Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl. 2012., 2012: Article ID 176
Włodarczyk K, Plebaniak R: Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points. Fixed Point Theory Appl. 2013., 2013: Article ID 128
Włodarczyk K, Plebaniak R: Contractions of Banach, Tarafdar, Meir-Keeler, Ćirić-Jachymski-Matkowski and Suzuki types and fixed points in uniform spaces with generalized pseudodistances. J. Math. Anal. Appl. 2013, 404: 338–350. 10.1016/j.jmaa.2013.03.030
Włodarczyk K, Plebaniak R: New completeness and periodic points of discontinuous contractions of Banach-type in quasi-gauge spaces without Hausdorff property. Fixed Point Theory Appl. 2013., 2013: Article ID 289
Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis. Dekker, New York; 1999.
Zhong C-H, Zhu J, Zhao P-H: An extension of multi-valued contraction mappings and fixed points. Proc. Am. Math. Soc. 1999, 128: 2439–2444.
Aubin JP, Ekeland JI: Applied Nonlinear Analysis. Wiley, New York; 1984.
Dugundji J: Topology. Allyn & Bacon, Boston; 1966.
Reilly IL: Quasi-gauge spaces. J. Lond. Math. Soc. 1973, 6: 481–487.
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Włodarczyk, K. Hausdorff quasi-distances, periodic and fixed points for Nadler type set-valued contractions in quasi-gauge spaces. Fixed Point Theory Appl 2014, 239 (2014). https://doi.org/10.1186/1687-1812-2014-239
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DOI: https://doi.org/10.1186/1687-1812-2014-239