Abstract
In quasi-pseudometric spaces (not necessarily Hausdorff), the concepts of the left quasi-closed maps (generalizing continuous maps) and generalized quasi-pseudodistances (generalizing in metric spaces: metrics, Tataru distances, w-distances of Kada et al., τ-distances of Suzuki and τ-functions of Lin and Du) are introduced, the asymmetric structures on X determined by J (generalizing the asymmetric structure on X determined by quasi-pseudometric p) are described and the contractions with respect to J (generalizing Banach and Rus contractions) are defined. Moreover, if are left sequentially complete (in the sense of Reilly, Subrahmanyam and Vamanamurthy), then, for these contractions such that is left quasi-closed for some , the global minimum of the map is studied and theorems concerning the existence of global optimal approximate solutions of the equation are established. The results are new in quasi-pseudometric and quasi-metric spaces and even in metric spaces. Examples showing the difference between our results and the well-known ones are provided. In the literature the fixed and periodic points in not Hausdorff spaces were not studied.
MSC:41A65, 47H09, 47H10, 37C25, 54E15.
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1 Introduction
Let X be a space and let . By and we denote the sets of all fixed points and periodic points of , respectively, i.e., and .
Motivated by results from the large literature concerning the global optimal approximate solution theorems, in which the spaces X are metric, the maps are contractions of Banach or Rus types and optimal approximate solutions belong to the set , our main interest of this paper is the following.
Question 1.1 Let be a quasi-pseudometric. Are there generalized pseudodistance generalizing p (generalizing asymmetric structure on X determined by a quasi-pseudometric p), the map satisfying the condition and such that the map attains its global minimum at the approximate solution w of the equation and w satisfies the equation but: (i) the space is not Hausdorff; (ii) T does not satisfy the condition ; (iii) T is not continuous in ; (iv) ?
Our aim in this paper is to answer the question affirmatively.
The fixed point theory is currently a very active field. In this theory, the notion of contractivity introduced by Banach belongs to the most fundamental mathematical ideas and the following theorem concerning the existence of global optimal approximate solution has important generalizations and applications.
Theorem 1.1 (Banach [1], Caccioppoli [2])
Let be a complete metric space. If satisfies the contractive condition
then T has a unique fixed point w in X and .
This shows that the map is continuous and this map attains its global minimum at the approximate unique solution of the equation ; we note that here d and T are continuous, the space is Hausdorff and d determines the symmetric structure on X.
It is important to observe that the map satisfying (1.1) satisfies (Rus [3])
and the converse is not true. In general, the maps satisfying (1.2) are not continuous.
Banach’s global optimal approximate solution theorem has inspired a large body of work over the last 40 years.
Rus [3] proved that the conclusions of Theorem 1.1 hold without the uniqueness assertion but under a slightly weaker contractive assumption; for other results in this direction, we refer to Subrahmanyam [4], Kasahara [5] and Hicks and Rhoades [6].
Theorem 1.2 (Rus [3])
Let be a complete metric space and let . If T is a continuous map satisfying contractive condition (1.2), then and .
The concepts of the asymmetric structures became established and investigated in mathematics and its applications. For details, see, e.g., [7] in normed spaces; [8–11] in metric spaces; [12] in uniform spaces; and [13–22] in quasi-gauge, cone uniform and uniform spaces.
There are many different generalizations of Theorems 1.1 and 1.2 in the literature where the distances are more general than d. In particular, we give some references where various contractions of Rus type are naturally defined. We refer to the works [8] and [10, 23–26] where in complete metric spaces the distance d in condition (1.2) is replaced by w-distances and τ-distances, respectively. The authors [13] introduced and examined, in not necessarily sequentially complete uniform spaces, contractions of Rus type with respect to the families of generalized pseudodistances.
In this paper, in quasi-pseudometric spaces (not necessarily Hausdorff), the concepts of the left quasi-closed maps (generalizing continuous maps) and generalized quasi-pseudodistances (generalizing the quasi-pseudometrics p in quasi-pseudometric spaces , and generalizing in metric spaces : metrics d, distances of Tataru [11], w-distances of Kada et al. [8], τ-distances of Suzuki [10] and τ-functions of Lin and Du [9]) are introduced. Next, the asymmetric structures on X determined by J (generalizing asymmetric structure on X determined by quasi-pseudometrics p) are described and the contractions with respect to J (generalizing Banach and Rus contractions) are defined. Moreover, if are left sequentially complete (in the sense of Reilly et al. [27]), then, for these contractions , assuming that is left quasi-closed for some , the global minimum of the map is studied and theorems concerning the existence of global optimal approximate solutions of the equation are established.
In addition, the provided examples illustrating generalized quasi-pseudodistances and our theorems, describe the techniques which enable one to compute the periodic and fixed points as well as to give precise information about the difference between our results and the well-known ones. The techniques require considerably more machinery from fixed point theory, asymmetric structures and iterative approximation.
Note that quasi-pseudometric spaces generalize quasi-metric and metric spaces and the studies of asymmetric structures in quasi-pseudometric and quasi-metric spaces and their applications to problems in theoretical computer science are important.
2 Definitions, notations and statement of results
Let X be a nonempty set.
-
(i)
A map is called quasi-pseudometric on X if it satisfies the two conditions:
(2.1)and
(2.2)For given quasi-pseudometric p on X, a pair is called quasi-pseudometric space.
-
(ii)
A quasi-pseudometric space is called Hausdorff if
-
(iii)
A map is called quasi-metric on X if it satisfies (2.1), (2.2) and
(2.3)For given quasi-metric p on X, a pair is called quasi-metric space.
-
(iv)
A map is called metric on X if it satisfies (2.1)-(2.3) and . For given metric p on X, a pair is called metric space.
In order to investigate the new contractivity in quasi-pseudometric spaces, we need to introduce the concept of ‘generalized quasi-pseudodistances’.
Definition 2.2 Let be a quasi-pseudometric space. The map is said to be a generalized quasi-pseudodistance on X if the following two conditions hold:
-
(J1)
; and
-
(J2)
For any sequences and in X satisfying
(2.4)and
(2.5)the following holds:
(2.6)
Remark 2.1 Let be a quasi-pseudometric space and let be a class defined as follows: . Then:
-
(a)
since ;
-
(b)
, see Examples 4.2 and 4.7;
-
(c)
Each quasi-pseudometric is a generalized quasi-pseudodistance, but converse is not true (see Section 4).
One can prove the following proposition.
Proposition 2.1 Let be a Hausdorff quasi-pseudometric space and let . Then
Proof Assume that there are , , such that . Then since, by using (J1), it follows that . Defining the sequences and in X by and or and for , and observing that , we see that (2.4) and (2.5) for these sequences hold. Then, by (J2), (2.6) holds, so it is . But this is a contradiction since is Hausdorff and thus implies or . □
Recall the following definition.
Let be a quasi-pseudometric space.
-
(i)
We say that a sequence is left Cauchy sequence in X if
-
(ii)
We say that a sequence is left convergent in X if
(which we write as ).
-
(iii)
If every left Cauchy sequence in X is left convergent to some point in X, then is called left sequentially complete quasi-pseudometric space.
Using this we can define the following natural generalization of continuity.
Definition 2.4 Let be a left sequentially complete quasi-pseudometric space, let and let . The map is called left quasi-closed in X if every sequence in , left converging to each point of the set and having subsequences and satisfying , has the property .
By and we denote the sets of all fixed points and periodic points of , respectively, i.e., and .
Motivated by papers [8, 10, 23–26], we raise a question.
Question 2.1 In quasi-pseudometric spaces (and thus also in quasi-metric and in metric spaces), is it possible to find an effective construction of a condition of Rus type with respect to generalized quasi-pseudodistances and techniques for obtaining periodic and fixed point theorems for left quasi-closed maps satisfying this condition?
The purpose of this paper is to answer this question in the affirmative. The first result in this direction is the following.
Theorem 2.1 Assume that is a left sequentially complete quasi-pseudometric space, the map is a generalized quasi-pseudodistance on X and the map satisfies
(S1) .
The following statements hold.
-
(A)
For each there exists a nonempty set such that the sequence is left convergent to each point ; i.e., (A1) For each , .
-
(B)
If: (b1) is left quasi-closed in X for some , then: (B1) ; (B2) For each there exists such that (i.e., ); and (B3) For each , .
-
(C)
If: (c1) is a Hausdorff space; and (c2) for some , then: (C1) ; and (C2) .
A version of Definition 2.2 in metric spaces is as follows.
Definition 2.5 Let be a metric space. The map , is said to be a generalized pseudodistance on X if the following two conditions hold: (J1) ; and (J2) For any sequences and in X satisfying and , the following holds .
The following is a metric analog of Theorem 2.1.
Theorem 2.2 Assume that is a complete metric space, the map is a generalized pseudodistance on X and the map satisfies
(S2) .
The following statements hold:
-
(A)
For any there exists such that .
-
(B)
If: (b1) is continuous in X for some , then, for any , there exists such that: (B1) ; (B2) ; and (B3) .
Remark 2.2 If we assume that and , then a special case of Theorem 2.2 gives Theorem 1.2.
3 Proof of Theorem 2.1
Proof In the sequel, for each , a sequence is defined by for ; we see that and .
-
(A)
The proof will be divided into four steps.
Step 1. We show that
Indeed, if is arbitrary and fixed, and , then, by (J1) and (S1),
Step 2. We show that
Indeed, by (3.1), we get . This implies (3.2).
Step 3. For each the sequence is a left Cauchy sequence on X.
Indeed, let be arbitrary and fixed. Then, by (3.2), we have
Hence, if is arbitrary and fixed and if we define a sequence as for , then we get
Now, from (3.2), (3.3) and (J2) of Definition 2.2, we conclude that
The consequence of (3.4) and definition of is
here is arbitrary and fixed. This gives
Now, let be arbitrary and fixed. From (3.5) we get that
We see that if m and n satisfy , then for some . Therefore by (3.6),
so it is
Thus the sequence is left Cauchy sequence on X.
Step 4. For each there exists a nonempty set such that the sequence is left convergent to each point .
Indeed, let be arbitrary and fixed. By Step 3, the sequence is left Cauchy on X. Hence, since is left sequentially complete quasi-pseudometric space, there exists a nonempty subset of X, such that the sequence is left convergent to each point .
(B) We have
Clearly, by Step 4, for each , the sequences and , as subsequences of , are also left convergent to each point of ; more precisely, is left convergent to each point of , is left convergent to each point of , and . Additionally, . Since is left quasi-closed, by (3.7) and Definition 2.4, we obtain that .
We show that (B3) holds. Indeed, assume that is arbitrary and fixed.
We see that
Otherwise, . Hence, by (J1) and (S1), since , we get which is impossible. Therefore, (3.8) holds.
Next, we see that
Otherwise, . Hence, by (J1), (S1) and (3.8), since and , we get which is impossible. Therefore, (3.9) holds.
-
(C)
From (3.8), (3.9) and the fact that is Hausdorff, using Proposition 2.1, we get , i.e., .
Finally, by (J1), (3.8) and (3.9), we get . □
4 Examples and comparisons
In this section we present some examples illustrating the concepts introduced so far.
The following two examples illustrate the concept of a quasi-pseudometric space and generalized quasi-pseudodistances, respectively.
Example 4.1 Let be a nonempty set and let be defined by the formula
(I.1) The map p is quasi-pseudometric on X, and is quasi-pseudometric space (see Reilly et al. [27]).
(I.2) is Hausdorff. Indeed, let , . Then, by (4.1), implies and implies . By Definition 2.1(ii), is Hausdorff.
Example 4.2 Let be a quasi-pseudometric space. Let the set , containing at least two different points, be arbitrary and fixed and let satisfy , where . Let be defined by the formula
(II.1) The map J is a generalized quasi-pseudodistance on X (see [14]).
Now, we present the examples illustrating Theorems 2.1 and 2.2.
Example 4.3 Let be a Hausdorff quasi-pseudometric space, where and let be as in Example 4.1. Let and let
by (II.1), J is a generalized quasi-pseudodistance on X. Let be a map given by the formula
(III.1) is a left sequentially complete quasi-pseudometric space. Indeed, if is a left Cauchy sequence on X, then there exists such that , i.e., . Thus is a left sequentially complete quasi-pseudometric space.
(III.2) The map T satisfies condition (S1) for . Indeed, if is arbitrary and fixed, then the following four cases hold.
Case 1. Fixing , by (4.4), we obtain . Hence, by (4.3) and (4.1), we have . This gives that condition (S1) holds.
Case 2. Fix an arbitrary . By (4.4), . Then, we obtain from (4.3) and (4.1) that and thus condition (S1) holds.
Case 3. Let be fixed. Note that and, by (4.4), , , . By (4.3) and (4.1), we also have and . This implies . Therefore, for , condition (S1) holds.
Case 4. Let . By (4.4), . Hence, by (4.2) and (4.1), . Therefore, for , condition (S1) holds.
(III.3) is left quasi-closed on X. Indeed, we have
and . Let be an arbitrary and fixed sequence in , left convergent to each point of a nonempty set and having subsequences and satisfying . Clearly, and . Hence, by (4.5), and , which gives the following.
Case 1. If and are such that , then also . Consequently, by Definition 2.3(ii) and Example 4.1, .
Case 2. If and are such that
or
then also
or
respectively. Consequently, .
Case 3. If and are such that , then also . Consequently, .
Of course, since , therefore in Cases 1-3. Finally, we see that in Cases 1-3. By Definition 2.4, is left quasi-closed in X.
(III.4) All the assumptions and assertions of Theorem 2.1 hold. It is straightforward to verify that , , and .
We note that the existence of a generalized quasi-pseudodistance such that is essential.
Example 4.4 Let and T be such as in Example 4.3.
(IV.1) T does not satisfy condition (S1) for . In fact, if holds and , then , and, by Example 4.1 and formulae (4.1) and (4.5), . This is absurd.
Next, we notice that the assumption that is left quasi-closed on X for some is essential.
Example 4.5 Let be such as in Example 4.3. Let be of the form
(V.1) T satisfies (S1) for and for each . Indeed, we have the following.
Case 1. Fixing , by (4.6), we obtain , . Therefore, by Example 4.1, . This implies that condition (S1) holds.
Case 2. Fix an arbitrary . Then, by (4.6), , and, by Example 4.1, . Thus (S1) holds.
Case 3. Let be fixed. By (4.6), and . However, . Hence, by Example 4.1, . This also gives (S1) for all .
(V.2) For each , the map is not left quasi-closed in X. Indeed, if is arbitrary and fixed, then, fixing , we get by (4.6) that and that a sequence satisfies and is left converging to each point of the set .
Let now be a sequence of the form and let be a sequence of the form ; of course, , where and . Then .
Now we see that . This means that the map is not left quasi-closed in X.
(V.3) In summary:
-
(a)
is a left sequentially complete quasi-pseudometric space (see (III.1)).
-
(b)
T satisfies (S1) for and for each .
-
(c)
We calculate that
(4.7)
and thus, for , Theorem 2.1(A) holds.
-
(d)
For each , the map is not a left quasi-closed in X and thus the assumption (b1) in Theorem 2.1(B) does not hold. Since thus assertion (B1) holds. Fixing , by (4.7), we get that the sequence is not left convergent to 3 and and thus, for , the assertion (B2) of Theorem 2.1 does not hold.
-
(e)
is Hausdorff (see (I.2)), and . This means that, for , Theorem 2.1(C) holds.
We compare Theorem 2.1 and [32].
Example 4.6 Let and T be such as in Example 4.3.
(VI.1) T is not a generalized contraction of Reilly type [32]. Indeed, suppose that . Obviously, this inequality holds for and and since, by (4.4), and , thus, by (4.1), we get . This is absurd.
At the end of this paper, in Examples 4.7 and 4.8, we illustrate Theorem 2.1 when is not Hausdorff.
Example 4.7 Let , let and let be of the form
(VII.1) The map p is quasi-pseudometric on X. Indeed, from (4.8), we have that for each and thus condition (2.1) holds.
Now, it is worth noticing that condition (2.2) does not hold only if there exists such that . This inequality is equivalent to , where
and
Conditions (4.10) and (4.11) imply or and or , respectively. We consider the following four cases:
Case 1. If and , then which, by (4.8), implies . By (4.9) this is absurd.
Case 2. If and , then . Hence, by (4.8), . By (4.9) this is absurd.
Case 3. If and , then . Hence, by (4.8), . By (4.9) this is absurd.
Case 4. If and , then . Hence, by (4.8), . By (4.9) this is absurd.
Thus, condition (2.2) holds.
We proved that p is quasi-pseudometric on X and is the quasi-pseudometric space.
(VII.2) is not Hausdorff. Indeed, for and , we have and . Hence, by (4.8), we obtain . This, by Definition 2.1(ii), means that is not Hausdorff.
Example 4.8 Let , let p be the same as in Example 4.7 and let be given by the formula
(VIII.1) The space is a not Hausdorff space. See (VII.2).
(VIII.2) The space is a left sequentially complete. Indeed, let be a left Cauchy sequence in X. By (4.8), not losing generality, we may assume that
Now, we have the following two cases:
Case 1. Let . By (4.8), in particular, since , we have that . This, by Definition 2.3(iii), means that is left convergent in X (we have that );
Case 2. Let . Then we have the following two subcases:
Subcase 2(a) Let . Then, by (4.8), we get and this implies ;
Subcase 2(b) Let . Then, by (4.8), since and , . However, since and , this, by (4.13), implies when and when . This is absurd.
We proved that if (4.13) holds, then . By Definition 2.3(ii), the sequence is left convergent in X.
(VIII.3) For the assumption (S1) of Theorem 2.1 holds (more precisely, the map T satisfies condition (S1) for and for each ). This follows from the fact that, by (4.8), for each .
(VIII.4) The map T is not left quasi-closed on X. Indeed, let a sequence in be of the form
Since thus, by (4.8), and . Hence . Moreover, its subsequences and satisfy . Clearly, . However, there does not exist such that .
(VIII.5) The map is left quasi-closed on X. Indeed, we have
and let be an arbitrary and fixed sequence in , left convergent to each point of a nonempty set and having subsequences and satisfying . Thus, , and . Hence, by (4.8), we conclude that
This gives . Next, we see that . By Definition 2.4, is left quasi-closed on X.
(VIII.6) For , the statements (A) and (B) of Theorem 2.1 hold. This follows from (VIII.1)-(VIII.5). From the above, it follows:
and
Moreover, by (4.8), since , thus, by (4.12), we get , so (B3) holds.
(IX.7) For , the statement (C) of Theorem 2.1 does not hold. We have: the assumption (c1) does not hold; for the assumption (c2) holds; ; properties (C1) and (C2) do not hold since .
Remark 4.1 (a) We see that in Example 4.3: (i) The map T is not left quasi-closed in X and is left quasi-closed in X; (ii) The map T satisfies condition (S1) for J defined by (4.3) and for ; (iii) When then, for each , the map T does not satisfy condition (S1) (see Example 4.4); (iv) Assumptions of Theorem 2.1 are satisfied; (v) In complete metric spaces, the assumptions of Banach [1], Rus [3], Subrahmanyam [4], Kada et al. [[8], Corollary 2] and Suzuki [[10], Theorem 1] theorems are not satisfied.
(b) In metric spaces , the generalized pseudodistances J (see Definition 2.5) generalize: metrics p, distances of Tataru [11], w-distances of Kada et al. [8], τ-distances of Suzuki [23] and τ-functions of Lin, Du [9]; for details, see [15, 16].
(c) It is important to observe that we provide the conditions guaranteeing the existence of fixed points and periodic points of the maps , and in our studies we determine the optimal global minima of the maps , .
(d) It is worth noticing that in the literature the fixed and periodic points of contractions in not Hausdorff spaces were not studied.
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Włodarczyk, K., Plebaniak, R. Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points. Fixed Point Theory Appl 2013, 128 (2013). https://doi.org/10.1186/1687-1812-2013-128
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DOI: https://doi.org/10.1186/1687-1812-2013-128