#### 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 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, [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 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 . If