Quasi-Triangular Spaces, Pompeiu-Hausdorff Quasi-Distances, and Periodic and Fixed Point Theorems of Banach and Nadler Types

Abstract

Let , -index set. A quasi-triangular space is a set with family satisfying . For any , a left (right) family generated by is defined to be , where and furthermore the property    holds whenever two sequences and in satisfy and    and . In , using the left (right) families generated by ( is a special case of ), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on ; for each type we construct of left (right) set-valued quasi-contraction , and we prove the convergence, existence, and periodic point theorem for such quasi-contractions. We also construct two types of left (right) single-valued quasi-contractions and we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions. () generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric, -metric, partial metric, partial -metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).

1. Introduction

The set-valued dynamic system is defined as a pair , where is a certain space and is a set-valued map ; here denotes the family of all nonempty subsets of the space . For , we define (-times) and (an identity map on ). By and we denote the sets of all fixed points and periodic points of , respectively; that is, and for some . A dynamic process or a trajectory starting at or a motion of the system at is a sequence defined by for (see, [1–4]).

Recall that a single-valued dynamic system is defined as a pair , where is a certain space and is a single-valued map ; that is, . By and we denote the sets of all fixed points and periodic points of , respectively; that is, and for some . For each , a sequence is called a Picard iteration starting at of the system .

Let be a (nonempty) set. A distance on is a map . The set , together with distances on , is called distance spaces.

The following distance spaces are important for several reasons.

Definition 1. Let be a (nonempty) set, and let .
(A) is called metric if (i) , (ii) , and (iii) .
(B) (See [5])   is called ultra metric if (i) , (ii) , and (iii) .
(C) (See [6, 7]) is called -metric with parameter if (i) , (ii) , and (iii) .
(D) (See [8]) is called partial metric if (i) , (ii) , (iii) , and (iv) .
(E) (See [9]) is called partial -metric with parameter if (i) , (ii) , (iii) , and (iv) .
(F) (See [10]) is called quasi-metric if (i) and (ii) .
(G) is called ultra quasi-metric if (i) and (ii) .
(H) The distance is called pseudometric (or the gauge) on if (i) , (ii) , and (iii) .
(I) The distance is called quasi-pseudometric (or the quasi-gauge) on if (i) and (ii) .
(J) (See [11]) The distance is called ultra quasi-pseudometric (or the ultra quasi-gauge) on if (i) and (ii) .

Definition 2 (see [12]). Let be a (nonempty) set, and let be an index set.
(A) Each family of pseudometrics , is called gauge on . The gauge on is called separating if .
(B) Let the family be separating gauge on . The topology having as a subbase the family of all balls with , , and is called topology induced by on ; the topology is Hausdorff.
(C) A topological space such that there is a separating gauge on with is called a gauge space and is denoted by .

Definition 3 (see [13]). Let be a (nonempty) set, and let be an index set.
(A) Each family of quasi-pseudometrics , is called quasi-gauge on .
(B) Let the family be quasi-gauge on . The topology having as a subbase of the family of all balls with , and is called topology induced by on .
(C) A topological space such that there is a quasi-gauge on with is called quasi-gauge space and is denoted by .

Remark 4 (see [13, Theorems 4.2 and 2.6]). Each quasi-uniform space and each topological space is the quasi-gauge space.

There is a growing literature concerning set-valued and single-valued dynamic systems in the above defined distance spaces. These studies contain also various extensions of the Banach [14] and Nadler [15, 16] theorems. Of course, there is a huge literature on this topic. For some such spaces and theorems in these spaces, see, for example, M. M. Deza and E. Deza [17], Kirk and Shahzad [18], and references therein.

Recall that the first convergence, existence, approximation, uniqueness, and fixed point result concerning single-valued contractions in complete metric spaces were obtained by Banach in 1922 [14].

Theorem 5 (see [14]). Let be a complete metric space. If and then the following are true: (i) has a unique fixed point in (i.e., there exists such that and ; and (ii) for each , the sequence converges to .

The Pompeiu-Hausdorff metric on the class of all nonempty closed and bounded subsets of the metric space is defined as follows: where for each and , . Using Pompeiu-Hausdorff metric new contractions were received by Nadler in 1967 and 1969 [15, 16] as a tool to study the existence of fixed points of set-valued maps in complete metric spaces.

Theorem 6 (see [15], [16, Theorem ]). Let be a complete metric space. If and then (i.e., there exists such that ).

Markin [19, 20] gave a slighty defferent version of Theorem 6.

Our primary interest is to construct new very general distance spaces, deliver new contractive set-valued and single-valued dynamic systems in these distance spaces, present the new global methods for studying of these dynamic systems in these spaces, and prove new convergence, approximation, existence, uniqueness, periodic point, and fixed point theorems for such dynamic systems.

The goal of the present paper is to introduce and describe the quasi-triangular spaces (Section 2) and more general quasi-triangular spaces with left (right) families generated by (Sections 3–5). Moreover, we use new methods and adopt ideas of Pompeiu and Hausdorff (Section 7) (see [21] for an excellent introduction to these ideas), to establish in these spaces some versions of Banach and Nadler theorems (Sections 8 and 9). Here studied dynamic systems are left (right) -admissible or left (right) -closed (Section 6). Examples are provided (Sections 10–12) and concluding remarks are given (Section 13).

2. Quasi-Triangular Spaces

It is worth noticing that the distance spaces , introduced and described below, are not necessarily topological or Hausdorff or sequentially complete.

Definition 7. Let be a (nonempty) set, let be an index set, and let .
(A) One says that a family of distances is a quasi-triangular family on if A quasi-triangular space is a set together with the quasi-triangular family on .
(B) Let be the quasi-triangular space. One says that is separating if (C) If is an quasi-triangular space and , then . One says that the quasi-triangular space , , is the conjugation of .

Remark 8. In the spaces , in general, the distances , do not vanish on the diagonal; they are asymmetric and do not satisfy triangle inequality (i.e., the properties or or do not necessarily hold); see Section 10.

Definition 9. Let be a (nonempty) set, let be an index set, and let .
(A) One says that a family of distances on is a ultra quasi-triangular family if An ultra quasi-triangular space is a set together with the ultra quasi-triangular family on .
(B) One says that a family of distances on is a partial quasi-triangular family if A partial quasi-triangular space is a set together with the partial quasi-triangular family on .

Remark 10. It is worth noticing that quasi-triangular spaces generalize ultra quasi-triangular and partial quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric, -metric, partial metric, partial -metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).

3. Left (Right) Families Generated by in Quasi-Triangular Spaces

In the metric spaces there are several types of distances (determined by ) which generalize metrics . First these distances were introduced by Tataru [22]. More general concepts of distances in metric spaces which generalize , of this sort, are given by Kada et al. [23] (-distances), Lin and Du [24] (-functions), Suzuki [25] (-distances), and Ume [26] (-distance). Distances in uniform spaces were given by Vályi [27]. In the appearing literature, these distances and their generalizations in other spaces provide efficient tools to study various problems of fixed point theory; see, for example, [28–30] and references therein. In this paper we also generalize these ideas.

Let be the quasi-triangular family on . It is natural to define the notions of left (right) families generated by which provide new structures on .

Definition 11. Let be the quasi-triangular space.
(A) The family of distances , is said to be a left (right) family generated by if; and furthermore. For any sequences and in satisfying  the following holds
(B)    is the set of all left (right) families on generated by .

Remark 12. From Definition 11 if follows that . Moreover, there are families and such that the distances , do not vanish on the diagonal, are asymmetric, and are quasi-triangular and thus are not metric, ultra metric, quasi-metric, ultra quasi-metric, -metric, partial metric, partial -metric, pseudometric (gauge), quasi-pseudometric (quasi-gauge), and ultra quasi-pseudometric (ultra quasi-gauge).

4. Relations between and

Remark 13. The following result shows that Definition 11 is correct and that and .

Theorem 14. Let be the quasi-triangular space. Let be a set containing at least two different points and let where If where, for each , the distance is defined by then is left and right family generated by .

Proof. Indeed, we see that condition (1) does not hold only if there exist some and such that Then (15) implies and the following Cases 1–4 hold.
Case 1. If , then and, by (16) and (15), . Therefore, by (14), . This is impossible.
Case 2. If and , then (16) and (15) give whenever or whenever . This is impossible.
Case 3. If and , then (16) and (15) give whenever or whenever . This is impossible.
Case 4. If and , then (16) and (15) give for . This is impossible.
Therefore, ; that is, the condition (1) holds.
Assume now that the sequences and in satisfy (8) and (10). Then (12) holds. Indeed, (10) implies Denoting , we see, by (17) and (15), that . Then, in view of Definition 11(A), (15), and (17), this implies . Hence we obtain that the sequences and satisfy (12). Thus we see that is left family generated by .
In a similar way, we show that (13) holds if and in satisfy (9) and (11). Therefore, is right family generated by . We proved that holds.

The following is interesting in respect to its use.

Theorem 15. Let be the quasi-triangular space, and let be the left (right) family generated by . If is separating on (i.e., (5) holds), then is separating on ; that is, holds.

Proof. We begin by supposing that , , and . Then (1) implies or, equivalently, and . Assuming that and , , we conclude that and . Therefore, it is not hard to see that (8)–(11) hold and, by (), the above considerations lead to the following conclusion: or, equivalently, . However, is separating. A contradiction. Therefore, is separating.

5. Left (Right) -Convergences and Left (Right) -Sequentially Completeness

Definition 16. Let be the quasi-triangular space, and let be the left (right) family generated by .
(A) One says that a sequence is left (right) -Cauchy sequence if   .
(B) Let and let . One says that the sequence is left (right) -convergent to if    where (C) One says that a sequence is left (right) -convergent in if   .
(D) If every left (right) -Cauchy sequence is left (right) -convergent in (i.e.,   , then is called left (right) -sequentially complete.

Remark 17. The structures on determined by left (right) families generated by are more general than the structure on determined by ; see Remark 34.

Remark 18. Let be the quasi-triangular space. It is clear that if is left (right) -convergent in , then    for each subsequence of .

Definition 19. One says that is left (right) Hausdorff if for each left (right) -convergent in sequence the set    is a singleton.

6. Left (Right) -Admissible and Left (Right) -Closed Set-Valued Maps

The following terminologies will be much used in the sequel.

Definition 20. Let be the quasi-triangular space, and let be the left (right) family generated by . Let be the set-valued dynamic system, .
(A) Given , One says that is left (right) -admissible in if, for each dynamic processes starting at , ,    whenever (B) One says that is left (right) -admissible on , if is left (right) -admissible in each point .

Remark 21. Let be the quasi-triangular space and let be the left (right) family generated by . Let be the set-valued dynamic system on . If is left (right) -sequentially complete, then is left (right) -admissible on but the converse not necessarily holds.

We can define also the following generalization of continuity.

Definition 22. Let be the quasi-triangular space. Let be the set-valued dynamic system, , and let . The set-valued dynamic system is said to be a left (right) -closed on if for every sequence in , left (right) -converging in (thus    and having subsequences and satisfying , the following property holds: there exists    such that   .

7. Left (Right) Pompeiu-Hausdorff Quasi-Distances and Left (Right) Set-Valued Quasi-Contractions

In this section, in the quasi-triangular spaces , using left (right) families generated by , we define three types of left (right) Pompeiu-Hausdorff quasi-distances on , and for each type a left (right) set-valued quasi-contraction is constructed.

Definition 23. Let be the quasi-triangular space, and let be the left (right) family generated by . Let , let be a set-valued dynamic system, , and let . Let (A) Let . If then a family is said to be left -quasi-distance on .
If then we say that is a left -quasi-contraction on .
(B) Let . If then a family is said to be right -quasi-distance on .
If then we say that is a right -quasi-contraction on .

Remark 24. Observe that and extend (2). Quasi-contractions (23) and (25) extend (3).

Remark 25. Each -quasi-contraction -quasi-contraction), , is -quasi-contraction -quasi-contraction) but the converse does not necessarily hold.

8. Convergence, Existence, Approximation, and Periodic Point Theorem of Nadler Type for Left (Right) Set-Valued Quasi-Contractions

The following result extends Theorem 6 to spaces .

Theorem 26. Let be the quasi-triangular space, and let be the set-valued dynamic system, . Let , and let .
Assume that there exist a left (right) family generated by and a point with the following properties.
(A1) is left -quasi-contraction (right -quasi-contraction) on .
(A2) is left (right) -admissible in .
(A3) For every and for every there exists such that Then the following hold.
(B1) There exist a dynamic process of the system starting at , , and a point such that is left (right) -convergent to .
(B2) If the set-valued dynamic system is left (right) -closed on for some , then and there exist a dynamic process of the system starting at , , and a point such that is left (right) -convergent to .

Proof. We prove only the case when is a left family generated by , is left -admissible in a point , and is left -closed on . The case of “right” will be omitted, since the reasoning is based on the analogous technique.
Part 1. Assume that (A1)–(A3) hold.
By (21) and the fact that , we choose such that Put In view of (28) and (29) this implies and we apply (26) to find such that We see from (30) and (31) that Put now Then, in view of (32), we get and applying again (26) we find such that Observe that Indeed, from (34), Definition 23(A), and using (33), in the event that or or , we get Thus (35) holds.
Next define Then, in view of (35), . Applying (26) in this situation, we conclude that there exists such that We seek to show that By (38), Definition 23(A), and using (37), in the event that or or , it follows that Thus (39) holds.
Proceeding as before, using Definition 23(A), we get that there exists a sequence in satisfying and for calculational purposes, upon letting where we observe that , Let now . Using (1), we get Hence, by (44), for each , This and (41) mean that and since implies , Now, since is left -admissible in , by Definition 20(A), properties (47) and (48) imply that there exists such that Next, defining and for , by (48) and (49) we see that conditions (8) and (10) hold for the sequences and in . Consequently, by (), we get (12) which implies that and so in particular we see that .
Part 2. Assume that (A1)–(A3) hold and that, for some , is left -closed on .
By Part 1, and since, by (47), 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 18, and . By the above and by Definition 22, since is left -closed, we conclude that there exist such that .
Part 3. The result now follows at once from Parts 1 and 2.

9. Theorem of Banach Type in Quasi-Triangular Spaces

In this section, in the quasi-triangular spaces , using left (right) families generated by , we construct two types of left (right) single-valued quasi-contractions , and convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions is also proved.

The following Definition 27 can be stated as a single-valued version of Definition 23.

Definition 27. Let be the quasi-triangular space, and let be the left (right) family generated by . Let be the single-valued dynamic system, let , and let .
(A) If , then we define the left -quasi-distance on by where One says that is left -quasi-contraction on if (B) If , then one defines the right -quasi-distance on by where One says that is right -quasi-contraction on if

Remark 28. Observe that (52) and (54) extend (1).

The following terminologies will be much used in the sequel.

Definition 29. Let be the quasi-triangular space, and let be the left (right) family generated by . Let be the single-valued dynamic system, .
(A) Given , One says that is left (right) -admissible in if, for the sequence ,    whenever (B) We say that is left (right) -admissible on , if is left (right) -admissible in each point .

Remark 30. Let be the quasi-triangular space, and let be the left (right) family generated by . Let be the single-valued dynamic system on . If is left (right) -sequentially complete, then is left (right) -admissible on .

We can define the following generalization of continuity.

Definition 31. Let be the quasi-triangular space. Let be the single-valued dynamic system, , and let . The single-valued dynamic system is said to be a left (right) -closed on if for each sequence in , left (right) -converging in (thus and having subsequences and satisfying ; the following property holds: there exists such that .

The following result extends Theorem 5 to spaces .

Theorem 32. Let be the quasi-triangular space, and let be the single-valued dynamic system, . Let , and let .
Assume that there exist a left (right) family generated by and a point with the following properties.
(A1) is left -quasi-contraction (right -quasi-contraction) on .
(A2) is left (right) -admissible in a point .
Then the following hold.
(B1) There exists a point such that the sequence starting at is left (right) -convergent to .
(B2) If the single-valued dynamic system is left (right) -closed on for some , then , there exists a point such that the sequence starting at is left (right) -convergent to , and (B3) If the family is separating on and if the single-valued dynamic system is left (right) -closed on for some , then there exists a point such that the sequence starting at is left (right) - convergent to , and

Proof. By Theorem 26, we prove only (56)–(58) and only in the case of “left.” We omit the proof in the case of “right,” which is based on the analogous technique.
Part 1. Property (56) holds. Suppose that . Of course, , and, for , by Definition 27(A), which is impossible. Therefore, Suppose now that . Then, by Definition 27(A) and property (60), using the fact that , we get, for , that which is impossible. Therefore, We see that (56) is a consequence of (60) and (62).
Part 2. Properties (57) and (58) hold. We first observe that in other words, . In fact, if and , then, since the family is separating on , we get that . In view of Theorem 15 this implies . However, by property (56), this is impossible.
Next we see that . In fact, by Definition 11(A) and property (56), we conclude that .
Finally, suppose that and . Then, since the family is separating on , we get . By applying Theorem 15, this implies . Consequently, for , by Definition 27(A), we conclude that which is impossible. This gives that is a singleton.
Thus (57) and (58) hold.

10. Examples of Spaces

Example 1. Let , and let be of the form ()  , , is the quasi-triangular space. In fact, Inequality (66) is a consequence of Cases 1–6.
Case 1. If and , then and . This gives .
Case 2. If , and , then and where, for , and .
Case 3. and .
Case 4. If and , then and . This gives .
Case 5. If , and , then .
Case 6. If , then .
() is asymmetric. Indeed, we have that . Therefore, condition does not hold.
() does not vanish on the diagonal. Indeed, if , then . Therefore, the condition does not hold.
() For the constant sequence of the form the sets and are not singletons. Indeed, by (65), Remark 12, and Definition 16(B), we have that and .

Example 2. Let be a set (nonempty), , , , , and let be of the form () A pair , , is the quasi-triangular space. Indeed, formula (67) yields . Otherwise, . It is clear that then , , and . From this we see that , , and . This is impossible.
() does not vanish on the diagonal. Indeed, if , then . Therefore, the condition does not hold.
() is symmetric. This follows from (67).
() We observe that for each sequence . We conclude this from (67).

Example 3. Let and let be of the form () , , is the quasi-triangular space. In fact, holds. This is a consequence of Cases 1–3.
Case 1. If , then , , and, consequently, .
Case 2. If and , then and where is a minimum point of the map .
Case 3. If , then and, consequently, .
() is asymmetric. Indeed, we have that . Therefore, condition does not hold.
() vanishes on the diagonal. In fact, by (68), it is clear that .
() We observe that and for sequence . We conclude this from (68).

Example 4. Let and let where is of the form Let and let and where is of the form () is not symmetric. In fact, by (69)–(71), and .
() . See Theorem 14.

Remark 33. By Examples 1–4 it follows that the distances defined by (65) and (67)–(69) and defined by (70) and (71) are not metrics, ultra metrics, quasi-metrics, ultra quasi-metrics, -metrics, partial metrics, partial -metrics, pseudometrics (gauges), quasi-pseudometrics (quasi-gauges), and ultra quasi-pseudometrics (ultra quasi-gauges).

11. Examples Illustrating Theorem 26

Example 1. Let , let be arbitrary and fixed, and, for , let Define the set-valued dynamic system by Let and let be of the form   , where , is the quasi-triangular space, and is the left and right family generated by . This is a consequence of Definitions 7 and 11, Example 1, and Theorem 14; we see that .
   is a -quasi-contraction on ; that is, where Indeed, we see that this follows from (73)–(76) and from Cases 1–4 below.
Case 1. If , then and . Thus .
Case 2. If and , then , , and and . Thus . On the other hand, which gives . Therefore, whenever . This gives whenever .
Case 3. If and , then and . Hence we obtain . Therefore, .
Case 4. If , then . Therefore .
Property (26) holds; that is, . Indeed, this follows from Cases 1–4 below.
Case 1. If and , then , , and .
Case 2. If and , then , , and .
Case 3. If and , then , , and .
Case 4. If and , then , , and .
Case 5. If and , then , , and .
   is left and right -admissible in each point . In fact, if and are such that and   , then and, consequently, by (72),   . Hence, by (75) and (76), we get   .
   is a left and right -closed on . Indeed, let be a left (right) -converging sequence in (thus    and having subsequences and satisfying . Then , and    and .
All assumptions of Theorem 26 are satisfied. This follows from (1)–(5) in Example 1.
We conclude that and we have shown the following.
Claim A. and for each and for each dynamic process of the system .
Claim B. and for each and for each dynamic process of the system .

Example 2. Let , , and be such as in Example 1.
For each , condition , where , does not hold. Suppose that . Letting and , it can be shown that , , , , and . Therefore , which is absurd.

Remark 34. We make the following remarks about Examples 1 and 2 and Theorem 26: (a) By Example 1, we observe that we may apply Theorem 26 for set-valued dynamic systems in the left and right quasi-triangular space with left and right family generated by where . (b) By Example 2, we note, however, that we do not apply Theorem 26 in the quasi-triangular space when . (c) From (a) and (b) it follows that, in Theorem 26, the existence of left (right) families generated by and such that are essential.

Example 3. Let , , and Let be of the form and let . Define the set-valued dynamic system by    is quasi-triangular space. See Example 2, Section 11.
   is a -quasi-contraction on ; that is, . Indeed, if , then, by (77)–(79), and .
Property (16) holds; that is, . Indeed, this follows from Cases 1–3 below.
Case 1. Let and be arbitrary and fixed. If , then, by (78), Therefore, .
Case 2. Let and let be arbitrary and fixed. If , then, by (78),   . Therefore, .
Case 3. Let and be arbitrary and fixed. If , then, by (78), Therefore, .
   is left and right -admissible in . Assuming that is arbitrary and fixed we prove that if the dynamic process of starting at is such that , then . Indeed, if , then, by (79), and, by (78), we immediately get .
Set-valued dynamic system is a left and right -closed on . Indeed, if is a left or right -converging sequence in and having subsequences and satisfying , then, by (77)–(79), we have that , , and .
For , , , and defined by (77)–(79), all assumptions of Theorem 26 are satisfied. This follows from (1)–(5) in Example 3.

We conclude that and we claim that if , , and are arbitrary and fixed, and , then sequence is a dynamic process of starting at and left and right -converging to each point of . We observe also that .

Example 4. Let and let where is of the form Define the set-valued dynamic system by Let and let and where is of the form () is not symmetric. In fact, by (82), (84), and (85), and .
() . See Theorem 14.
() is a -contraction on ; that is, where and Indeed, we see that this follows from (1), (2) in Example 4, and from Cases 1–4 below.
Case 1. Let . Then , , and . If , then we have and and if , then we have and By (86),   .
Case 2. If , then and . Therefore, .
Case 3. If and , then , , , , and . We see that since if , then also and . Next, we see that since if , then and Thus .
Case 4. If and , then , , , , , and since, for , and since for . Thus .

(4) Property (26) holds; that is, . Indeed, this follows from Cases 1–3 below.

Case 1. Let and be arbitrary and fixed. If is such that , then and since Then we see that implies . From this we conclude that if , then .

Case 2. Let . Assume that is arbitrary and fixed. Then , and, for each , .

Case 3. Let and be arbitrary and fixed. If is such that , then and, analogously as in Case 1, we get . Therefore, .

() is left -admissible in . Assuming that is arbitrary and fixed we prove that if the dynamic process of starting at is such that , then . We consider the following cases.

Case 1. If , then and and using (82) we immediately get .

Case 2. If , then , , and and using (82) we also immediately get .

This shows that for each and for each dynamic process of the system ; we see that here property of is not required.

() Set-valued dynamic system is a left -quasi-closed on . Indeed, if is a left -converging sequence in and having subsequences and satisfying , then, by (83), we have that . Therefore, in particular, and .

()  For , , and defined by (82)–(85), all assumptions of Theorem 26 in the case of “left” are satisfied. This follows from (1)–(6) in Example 4.

We conclude that and we claim that and that for each and for each dynamic process of the system . We observe also that .

12. Example Illustrating Theorem 32

Example 1. Let , , and be as in Example 3. Define the single-valued dynamic system by    is a -quasi-contraction on ; that is, and . Indeed, we see that if , then and, by (77) and (78), .
   is left and right -admissible in . Assume that is arbitrary and fixed, satisfies , and . Then, by (92) and (78), we have . This gives .
Single-valued dynamic system is a left and right -closed on . Indeed, if is a left -converging sequence in and having subsequences and satisfying , then, by (77), (78), and (92), we have that . In particular, and .
Property (56) holds. Indeed, we have since , , and .
   is not separating on . Indeed, if , then .
For , , and defined by (77), (78), and (79) parts (B1) and (B2) of Theorem 32 hold but part (B3) of Theorem 32 does not hold. This follows from (1)–(5) in Example 1.

13. Concluding Remarks

Remark 2. In Theorems 5 and 6 the following play an important role: (i) Distances and , as metrics, satisfy conditions (A) of Definition 1 on and , respectively. (ii) and , as metric spaces, are topological and Hausdorff spaces and the completeness of implies completeness of . (iii) The continuity of and on and , respectively; (iv) The continuity of maps and (as consequences of contractive properties defined in (1) and (3), resp.); (v) In Theorem 6 the assumption that, for each , .

Remark 3. Conclusions in Theorems 5 and 6 concern only fixed points but not periodic points; this is a consequence of separability of spaces and and also continuity of .

Remark 4. In Theorems 26 and 32, properties concening the spaces and maps such as mentioned above generally need not hold, since spaces with left (right) families generated by are very general, which is an obstruction to use Nadler’s and Banach’s reasoning. Theorems 26 and 32 show how to rectify this situation and are obtained without restrictively required assumptions and with conclusions more profound as in the well known results of this sort existing in the literature.

Conflict of Interests

The author declares that he has no conflict of interests regarding the publication of this paper.