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Abstract and Applied Analysis
Volume 2015, Article ID 201236, 16 pages
http://dx.doi.org/10.1155/2015/201236
Research Article

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

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received 19 February 2015; Revised 17 May 2015; Accepted 18 May 2015

Academic Editor: Poom Kumam

Copyright © 2015 Kazimierz Włodarczyk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, [14]).

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 andthen 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 andthen (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 35). 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 1012) 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 ifA 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 ifAn 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 ifA 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, [2830] 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 satisfyingthe 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 whereIf where, for each , the distance is defined bythen is left and right family generated by .

Proof. Indeed, we see that condition (1) does not hold only if there exist some and such thatThen (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) impliesDenoting , 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 . Ifthen a family is said to be left -quasi-distance on .
Ifthen we say that is a left -quasi-contraction on .
(B) Let . Ifthen a family is said to be right -quasi-distance on .
Ifthen 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 choosesuch that Put In view of (28) and (29) this implies and we apply (26) to find such thatWe see from (30) and (31) thatPut now Then, in view of (32), we get and applying again (26) we find such thatObserve that Indeed, from (34), Definition 23(A), and using (33), in the event that or or , we get Thus (35) holds.
Next defineThen, in view of (35), . Applying (26) in this situation, we conclude that there exists such thatWe seek to show thatBy (38), Definition 23(A), and using (37), in the event that or or , it follows thatThus (39) holds.
Proceeding as before, using Definition 23(A), we get that there exists a sequence in satisfyingand for calculational purposes, upon letting wherewe observe that , Let now . Using (1), we getHence, 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 thatNext, 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 thatand 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 whereOne says that is left -quasi-contraction on if (B) If , then one defines the right -quasi-distance on by whereOne 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