Abstract
In quasi-gauge spaces (in the sense of Dugundji and Reilly), we introduce the concept of the left (right) -family of generalized quasi-pseudodistances, and we use this -family to define the new kind of left (right) -sequential completeness, which extends (among others) the usual -sequential completeness. We use this -family to construct more general contractions than those of Banach and Rus, and for such contractions (which are not necessarily continuous), we establish the conditions guaranteeing the existence of periodic points (when is not Hausdorff), fixed points (when is Hausdorff), and iterative approximation of these points. The results are new in quasi-gauge, topological and quasi-uniform spaces and, in particular, generalize the well-known theorems of Banach and Rus types in the matter of fixed points. Various examples illustrating ideas, methods of investigations, definitions and results, and fundamental differences between our results and the well-known ones are given.
MSC:54H25, 54A05, 47J25, 47H09, 54E15.
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1 Introduction
Let X be a nonempty set. If , then, for each , we define a sequence starting with as follows , where (m-times), and is an identity map on X.
By and , we denote the sets of all fixed points and periodic points of , respectively, i.e., and .
The famous theorem of Banach-Caccioppoli [1, 2] states the following.
Theorem 1.1 If is a complete metric space with metric d, then the map satisfying the condition
has a unique fixed point w in X (i.e., ) and .
Another is a theorem of Rus [3] (see also [4, 5] and [6]), which states the following.
Theorem 1.2 If is a complete metric space with metric d, then a continuous map satisfying the condition
has the properties x and .
It is clear that the map T satisfying (1.1) is continuous and satisfies (1.2), and in the assertion of Theorem 1.2, the uniqueness such as in the assertion of Theorem 1.1 does not necessarily hold.
These results are basic facts in the metric fixed point theory and their applications, and in the last four decades, the question concerning important generalizations of [1, 2] and [3] has received considerable attention from various researchers, and some very interesting results have been obtained in several hundred papers and several books. It is not our purpose to give a complete list of related papers and books here.
In important and various directions, there are elegant results discovered by [7–13], in which more general and natural settings, by using asymmetric structures in considerable spaces, are studied; in [7–9] a complete metric space in results of [1–3] is replaced by a left (right) -sequentially complete quasi-gauge space , and in construction of contractive conditions of (1.1) and (1.2) types, the quasi-gauge is used, whereas [10] and [11–13] provide substantial and inspiring tools for investigations in complete metric spaces the existence of fixed points of maps which are the contractions of [1–3] types with respect to w-distances and τ-distances, respectively.
Note that quasi-gauge , w-distances and τ-distances generate asymmetric structures and generalize metric d, and that the studies of asymmetric structures and their applications in theoretical computer science are important.
Our main interest of this paper is the following.
Question 1.1 For which not necessarily Hausdorff and not necessarily complete spaces or not necessarily sequentially complete spaces and for which new families of distances on these spaces, there exist symmetric or asymmetric structures determined by these new families of distances which are more general than those determined by quasi-gauges , w-distances, τ-distances or metrics d, and for which not necessarily continuous contractions of the Banach or Rus types with respect to these new families of distances the assertions such as in the results of [1, 2] or [3], respectively, hold (and not only for fixed points but also for periodic points)?
In this paper, in the quasi-gauge spaces (see Definition 2.1), to answer this question affirmatively, we introduce the concepts of the left (right) -families of generalized quasi-pseudodistances (see Definition 3.1), and we show how these left (right) -families can be used, in a natural way, to define the left (right) -sequential completeness (see Definition 3.2) which generalize (among others) the usual left (right) -sequential completeness, to construct the not necessarily continuous contractions of Banach and Rus types (see conditions (H1) and (H2)), and assuming additionally that is a left (right) -quasi-closed map in X for some (see Definition 3.3), to obtain the new periodic and fixed point theorems (see Theorems 4.1 and 4.2) which, in particular, generalize Banach and Rus results in the matter of fixed points. The results are new in quasi-gauge, topological and quasi-uniform spaces (see Remarks 2.1, 3.1, 3.2 and 6.1). Various examples illustrating ideas, methods of investigations, definitions and results, and fundamental differences between our results and the well-known ones are given (see Section 6).
2 Quasi-gauge spaces
The following terminologies will be much used.
Definition 2.1 Let X be a nonempty set.
-
(i)
A quasi-pseudometric on X is a map such that
-
() ; and
-
() .
For given quasi-pseudometric p on X, a pair is called quasi-pseudometric space. A quasi-pseudometric space is called Hausdorff if .
-
-
(ii)
Each family of quasi-pseudometrics , , is called a quasi-gauge on X (-index set).
-
(iii)
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.
-
(iv)
(Dugundji [24], Reilly [7, 25]) A topological space such that there is a quasi-gauge on X with is called a quasi-gauge space and is denoted by .
-
(v)
A quasi-gauge space is called Hausdorff if a quasi-gauge has the property
Remark 2.1 Each quasi-uniform space and each topological space is a quasi-gauge space (Reilly [[7], Theorems 4.2 and 2.6]).
3 Left (right) -families, left (right) -sequential completeness and left (right) -quasi-closed maps in quasi-gauge spaces with generalized quasi-pseudodistances
We next record the definitions of left (right) -families, left (right) -sequential completeness and left (right) -quasi-closed maps needed in the next sections.
Definition 3.1 Let be a quasi-gauge space. 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:
-
() ; and
-
() for any sequences and in X satisfying
(3.1)(3.2)and
(3.3)(3.4)the following holds
(3.5)(3.6)
Remark 3.1 If is a quasi-gauge space, then , where
and , where
One can prove the following proposition.
Proposition 3.1 Let be a Hausdorff quasi-gauge space, and let be a left (right) -family on X. Then
Proof Assume that is a left -family, and that there are , , such that . Then , by using property () in Definition 3.1, it follows that
Defining the sequences and in X by and or by and for , observing that , and using property () of Definition 3.1 for these sequences, we see that (3.1) and (3.3) hold, and, therefore, (3.5) is satisfied, which gives . But this is a contradiction, since is Hausdorff, and thus, . When is a right -family, then the proof is based on the analogous technique. □
The necessity of defining the various concepts of completeness in quasi-gauge spaces became apparent with the investigation of asymmetric structures in these spaces. General results of this sort were progressively shown in a series of papers, and important ideas are to be found in [7–9, 24–27], which also contain many examples.
Now, using left (right) -families, we define the following new natural concept of completeness.
Definition 3.2 Let be a quasi-gauge space, and let be a left (right) -family on X.
-
(i)
We say that a sequence in X is left (right) -Cauchy sequence in X if
-
(ii)
Let , and let be a sequence in X. We say is left (right) -convergent to u if
where
-
(iii)
We say that a sequence in X is left (right) -convergent in X if
where
-
(iv)
If every left (right) -Cauchy sequence in X is left (right) -convergent in X
then is called a left (right) -sequentially complete quasi-gauge space.
Remark 3.2 (a) It is clear that if is left (right) -convergent in X, then
for each subsequence of (see Example 3.1).
(b) There exist examples of quasi-gauge spaces and left (right) -family on X, such that is left (right) -sequentially complete, but not left (right) -sequentially complete (see Section 6).
Example 3.1 Let , and let , where
Let
If , then
If , then
Also, using Definition 3.2, we can define the following generalization of continuity.
Definition 3.3 Let be a quasi-gauge space, let , and let . The map is said to be a left (right) -quasi-closed map if every sequence in , left (right) -converging in X
and having subsequences and satisfying has the property
4 Main results
Using the above, we can now state the main results of this paper.
Theorem 4.1 Let be a quasi-gauge space. Let the family be a left (right) -family on X such that is left (right) -sequentially complete. Let a map satisfy
(H1) .
The following statements hold:
-
(A)
For each the sequence is left (right) -convergent in X; i.e.,
-
(a1) ().
-
-
(B)
Assume that
Then
-
(B1)
is left (right) -quasi-closed on X for some .
-
(b1) ;
-
(b2) (); and
-
(b3) .
-
-
(B1)
-
(C)
Assume that Then
-
(C1)
is a Hausdorff space; and
-
(C2)
there exists such that .
-
(c1) for some ;
(c2) (); and
(c3) .
-
-
(C1)
Theorem 4.2 Let be a quasi-gauge space. Let the family be a left (right) -family on X such that is left (right) -sequentially complete. Let a map satisfy
(H2) .
The following statements hold:
-
(D)
For each the sequence is left (right) -convergent in X; i.e.,
-
(d1) ().
-
-
(E)
Assume that Then
-
(E1)
is left (right) -quasi-closed on X for some .
-
(e1) ;
-
(e2) (); and
-
(e3) .
-
-
(E1)
-
(F)
Assume that Then
-
(F1)
is a Hausdorff space; and
-
(F2)
there exists such that .
-
(f1) ;
-
(f2) (); and
-
(f3) .
-
-
(F1)
Remark 4.1 (i) It is worth noticing that each map T satisfying (H1) satisfies (H2).
(ii) If condition (B1) or (E1) holds, then condition (C2) or (F2) holds, respectively.
(iii) Since in the results of [1, 2] and [4], the spaces are Hausdorff and complete, and the maps are continuous, therefore, Theorems 4.1 and 4.2 are new generalizations of [1, 2] and [3], respectively; more precisely, the assertions are identical, but assumptions are weaker.
(iv) The statements (C) and (F) say that each periodic point is a fixed point when is Hausdorff; for illustrations, see Examples 6.1-6.7.
(v) The situations when is not Hausdorff and the periodic points exist but they are not fixed points are described in Examples 6.8 and 6.9.
5 Proofs
We prove Theorems 4.1 and 4.2 in the case when is left -family and a quasi-gauge space is left -sequentially complete; we omit the proof when is a right -family and is right -sequentially complete, which is based on the analogous technique.
Proof of Theorem 4.2 (D) The assertion (d1) holds. The proof will be broken into four steps.
Step D.I. The following holds:
Indeed, if and are arbitrary and fixed, and , then by (1) and (H2), we get that
Step D.II. We show that
Indeed, by Step D.I, we get
This implies a required condition.
Step D.III. The following holds:
Indeed, it is a consequence of Step D.II.
Step D.IV. For each , .
Indeed, let be arbitrary and fixed. By (5.1) and Definition 3.2(i), the sequence is left -Cauchy on X. Hence, since is a left -sequentially complete quasi-gauge space, we get that is left -convergent in X, i.e., there exists, by Definition 3.2(ii)-(iv), a nonempty set , such that for all , we have
However, is left -family. Therefore, from (5.1) and (5.2), fixing , defining and and using Definition 3.1 for these sequences, we conclude that
i.e., . Clearly, this means that .
We proved that the assertion (d1) holds.
-
(E)
The assertions of (e1)-(e3) hold.
The proof will be broken into three steps.
Step E.I. We show that (e1) holds. Indeed, let be arbitrary and fixed. By (D), , and since
thus, defining , we see that
the sequences
and
satisfy
and, as subsequences of , are left -converges to each point of . Moreover, by Remark 3.2(a),
By above, since is left -quasi-closed for some , we conclude that
Consequently, (e1) holds.
Step E.II. We show that (e2) holds. Assertion (e2) follows from assertion (d1) and Step E.I.
Step E.III. We show that (e3) holds. Assume that is arbitrary and fixed.
First, we see that
otherwise, and using this and (1), we get
and
which is impossible.
Next, we show that
otherwise, and, since and , then by (H2), and since , we have that
which is impossible.
The above show that
This means that (e3) holds.
-
(F)
The assertions of (f1)-(f3) hold.
The proof will be broken into three steps.
Step F.I. We show that (f1) holds. By (F1) and Proposition 3.1, condition (e3) implies that if , then , i.e., . Thus, (f1) holds.
Step F.II. We show that (f2) holds. We see that (e2) and (f1) gives (f2).
Step F.III. We show that (f3) holds. By (1), using (e3) and (f1), we get
i.e., (f3) holds.
The proof of Theorem 4.2 is complete. □
Proof of Theorem 4.1 By Remark 4.1(i) and Theorem 4.2, it is enough to prove (c1). With this aim, first notice that if and , then (H1) gives
However, since , by Proposition 3.1,
This gives
which is absurd. Therefore, is a singleton. Consequently, (c1) holds.
By (c1), we see that (f2) and (f3) gives (c2) and (c3), respectively.
The proof of Theorem 4.1 is complete. □
6 Examples and comparisons of our results with [1, 3, 7–9, 11–13] results
Definitions and results are illustrated with simple examples making clear their general nature.
First, in Examples 6.1-6.7, we consider the situation when is Hausdorff.
Example 6.1 Let , and
The map is quasi-pseudometric on X and is the quasi-gauge space.
Example 6.2 Let , and , where p is defined in Example 6.1.
(II.1) We show that is not a left -sequentially complete quasi-gauge space.
Indeed, let . By (6.1),
Thus, this sequence is left -Cauchy. However, this sequence in not left -convergent in X. Otherwise, supposing that for some we may assume, not losing generality, that
Then, the following two cases hold:
Case 1. If , then, by (6.1), since , we have
which is impossible;
Case 2. If , then for some and, using (6.1), we see that
and taking the limit interior as , we find , which, by (6.2), is impossible.
We conclude that is not a left -sequentially complete.
Example 6.3 Let be a quasi-pseudometric space, where and p is a quasi-pseudometric on X. Let the set , containing at least two different points, be arbitrary and fixed, and let satisfy , where
Define by
(III.1) The family is left -family on X.
Indeed, it is worth noticing that condition (1) does not hold only if there exist some satisfying
This inequality is equivalent to
where , and . However, by (6.3), we get the following.
Case 1. gives that there exists such that ;
Case 2. gives ;
Case 3. gives .
This is impossible. Therefore, , i.e., condition () holds.
To prove that (2) holds, we assume that the sequences and in X satisfy (3.1) and (3.3). Then, in particular, (3.3) yields
By (6.4) and (6.3), since , we conclude that
From (6.5), (6.3) and (6.4), we get
Therefore, the sequences and satisfy (3.5). Consequently, property (2) holds. Thus, is left -family.
(III.2) The family is right -family on X.
We omit the proof since it is based on the analogous technique as in (III.1).
Example 6.4 Let , and , where p is such as in Example 6.1. Let , and let be given by the formula
(IV.1) is a left -family on X.
This follows from (III.1).
(IV.2) is not a left -sequentially complete quasi-gauge space.
This follows from (II.1).
(IV.3) is a left -sequentially complete quasi-gauge space.
Indeed, let be a left -Cauchy sequence; not losing generality, we may assume that
Then, by (6.7), (6.6) and (6.1), we get
and
We consider the following two cases.
Case 1. Let . This together with (6.8)-(6.10) shows that or or and, therefore, the sequence is left -convergent to the point or or , respectively;
Case 2. Let . We note that then
Otherwise, , and let . By definition of S, and , which, by (6.6) and (6.1), gives
and this, by (6.7), is impossible. Thus, (6.11) holds. Now, since is complete, is closed in ℝ, by (6.9), and is Cauchy with respect to (indeed, by (6.8), we get that
holds), thus, there exists such that
Next, by (6.11) and (6.12),
which, by (6.6) and (6.1), implies that
and we conclude that is left -convergent to u.
This means that is left -sequentially complete.
Theorem 4.2 is quite general, and does not require left -sequential completeness; in Example 6.5, T satisfies (H2) for some , and , T satisfies (H2) for and , and is left -sequentially complete but not left -sequentially complete.
Example 6.5 Let X, , E and J be as in Example 6.4, and let be given by
(V.1) is left -sequentially complete for .
This follows from (IV.3).
(V.2) We claim that T satisfies condition (H2) for , and J defined in (6.6).
To establish this, we see that
and consider the following five cases.
Case 1. If , then, by (6.13) and (6.14), and . Therefore, , and . Hence, by (6.6) and (6.1),
Case 2. If , then, by (6.13) and (6.14), and . Therefore, , and . Hence, by (6.6) and (6.1),
Case 3. If , then, by (6.13) and (6.14), , , and . Therefore, and . Hence, by (6.6) and (6.1), we get
Case 4. If , then and . Hence, by (6.6) and (6.1), we obtain
Case 5. If then . Moreover, . Hence, by (6.6) and (6.1),
Consequently, the map T satisfies (H2) for and defined by (6.6).
(V.3) T is left -quasi-closed on X.
Indeed, let be arbitrary and fixed sequence in , left -convergent to each point of a nonempty set , and having subsequences and satisfying .
Let be arbitrary and fixed. Then, by (6.1), (6.13) and Definition 3.2, we conclude that
Consequently,
This gives .
We see that . Otherwise, there exists , and then
However, this gives, in particular, the following
and hence, we get that
which is impossible since for , and when .
We proved that
By Definition 3.3, T is left -quasi-closed on X.
(V.4) All assumptions and all assertions of Theorem 4.2 hold for .
This follows from (V.1)-(V.3). We get
(V.5) T satisfies condition (H2) for and .
Indeed, the following three cases hold.
Case 1. Let . Then , , and, by (6.1),
and
Case 2. If , then , and, by (6.1),
and
Case 3. Let . Then and, by (6.1), .
(V.6) is not a left -sequentially complete.
This follows from (II.1).
(V.7) Assumptions of Theorem 4.2 for do not hold.
This follows from (V.6).
Now, we notice that the existence of -family such that is essential; in Example 6.6, T satisfies (H2) for some and does not satisfy (H2) for , and is left -sequentially complete but not left -sequentially complete.
Example 6.6 Let X, , E and be as in Example 6.4. Define by
(VI.1) For , is -sequentially complete.
This follows from (IV.3).
(VI.2) T satisfies (H2) for and for defined in (6.6).
Indeed, we get
and using (6.15) and (6.16), we consider the following four cases.
Case 1. If , then and, by (6.15) and (6.16), , , . Consequently, by (6.6) and (6.1),
Case 2. If , then, by (6.15) and (6.16), , , , and . Thus, by (6.6) and (6.1), we obtain that
Case 3. If , then and . Hence, since , by (6.6) and (6.1), we obtain that
Case 4. If , then . Hence, by (6.6) and (6.1),
Consequently, for and defined in (6.6) and (6.1), the map T satisfies condition (H2).
(VI.3) T is left -quasi-closed on X.
Indeed, let be arbitrary and fixed sequence in , left -convergent to each point of a nonempty set and having subsequences and satisfying .
Let be arbitrary and fixed. Then, by (6.1), (6.13) and Definition 3.2, we conclude that
Consequently,
This gives .
We see that . Otherwise, there exists , and then
Of course, since , we have or . Hence, in particular, we obtain that
or
Consequently,
or
Hence, when and additionally,
or
for , and , which is absurd.
We proved that
and
By Definition 3.3, T is left -quasi-closed on X.
(VI.4) All assumptions and all assertions of Theorem 4.2 for hold.
This follows from (VI.1)-(VI.3). We obtained that
(VI.5) T does not satisfy (H2) for .
Indeed, assuming that
and putting in this inequality, we get
which is absurd.
(VI.6) is not a left -sequentially complete.
This follows from (II.1).
(VI.7) Assumptions of Theorem 4.2 for do not hold.
This follows from (VI.5) and (VI.6).
(VI.8) Assumptions of Theorem 4.1 for do not hold.
Indeed, it follows from (VI.5) that T does not satisfy (H1) for . Additionally, (VI.6) holds.
Now, we show that the uniqueness in Theorem 4.2 does not necessarily hold; in Example 6.7, T satisfies (H2) for some and does not satisfy (H2) for , and is not a singleton.
Example 6.7 Let , and , where p is defined in Example 3.1, i.e.,
and
Let , and let
(VII.1) The map p is quasi-pseudometric on X, and is a quasi-gauge.
See Reilly et al. [[8], Example 1].
(VII.2) Condition (F1) holds; i.e., is Hausdorff.
Indeed, let , . Then, by (6.17), implies that , and implies that . By Definition 2.1(v), is Hausdorff.
(VII.3) is a left -family on X.
This follows from (III.1).
(VII.4) is left -sequentially complete.
To establish this, let be an arbitrary and fixed left -Cauchy sequence on X. Then, by Definition 3.2(i),
which, by (6.19), gives
This means , and using now the facts that also and , by (6.19) and (6.17), we obtain
i.e., . We claim that is left -sequentially complete.
(VII.5) T satisfies condition (H2) for and J defined by (6.19) and (6.17).
Indeed, first we see that
and we consider the following seven cases.
Case 1. If then, by (6.18) and (6.20), , so, by (6.19) and (6.17),
Case 2. If , then, by (6.18) and (6.20), , so by (6.19) and (6.17),
Case 3. If , then and, by (6.18) and (6.20), , , so by (6.19) and (6.17),
Case 4. If , then, by (6.18) and (6.20), , so by (6.19) and (6.17), . Hence,
Case 5. If , then, by (6.18) and (6.20), , . Since , by (6.19) and (6.17), . Therefore,
Case 6. If , then, by (6.18) and (6.20), and . Since , by (6.19) and (6.17), . But and, by (6.19) and (6.17), . Therefore,
Case 7. If , then, by (6.18) and (6.20), and . Since , by (6.19) and (6.17), . But and, by (6.19) and (6.17), . Therefore,
Consequently, for and defined in (6.19) and (6.17), the map T satisfies condition (H2).
(VII.6) Condition (E1) holds.
Indeed, we prove that is left -quasi-closed on X. With this aim, we see that, by (6.18) and (6.20),
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 (6.21), we obtain and , which gives the following.
Case 1. If and are such that , then also . Consequently,
Case 2. If and are such that or , then, by (6.21), also or . Consequently,
Of course, since , therefore, . Finally, we see that in Cases 1 and 2. By Definition 3.3, is left -quasi-closed on X.
(VII.7) Statements (D)-(F) of Theorem 4.2 hold.
This follows from (VII.1)-(VII.7). We get
(VII.8) T does not satisfy (H2) for .
To establish this, let
Since , by (6.17), we get
which is absurd.
Finally, in Examples 6.8 and 6.9, we consider the situation when is not Hausdorff.
Example 6.8 Let , let , and let where is of the form
(VIII.1) The map p is quasi-pseudometric on X and is the quasi-gauge space.
Indeed, from (6.22), we have that for each , and thus, condition () holds.
Now, it is worth noticing that condition () does not hold only if there exists such that . This inequality is equivalent to , where
and
Conditions (6.24) and (6.25) imply that or and or , respectively. We consider the following four cases.
Case 1. If and , then which, by (6.22), implies that . By (6.23), this is absurd;
Case 2. If and , then . Hence, by (6.22), . By (6.23), this is absurd;
Case 3. If and , then . Hence, by (6.22), . By (6.23), this is absurd;
Case 4. If and , then . Hence, by (6.22), . By (6.23), this is absurd.
Thus, condition () holds.
We proved that p is quasi-pseudometric on X, and is the quasi-gauge space.
(VIII.2) The quasi-gauge space is not Hausdorff.
Indeed, for and we have and . Hence, by (6.22), we obtain . This, by Definition 2.1(v), means that is not Hausdorff.
Example 6.9 Let , let , where p is defined as in Example 6.8, and let be given by the formula
(IX.1) The pair is a not a Hausdorff quasi-gauge space.
This is a consequence of (VIII.1) and (VIII.2).
(IX.2) The space is a left -sequentially complete.
Indeed, let be a left -Cauchy sequence in X. By (6.22), not losing generality, we may assume that
Now, we have the following two cases.
Case 1. Let . By (6.22), in particular, we have that . This gives, , i.e., ;
Case 2. Let . Then we have the following two subcases: Subcase 2.1. If , then, by (6.22), we get , and this implies that , i.e., ; Subcase 2.2. If , then, by (6.22), . However, since and , this, by (6.27), implies that . This is absurd.
We proved that if (6.27) holds, then
By Definition 3.2(ii), the sequence is left -convergent in X.
(IX.3) For , assumption (H2) of Theorem 4.2 holds (more precisely, the map T satisfies condition (H2) for and for each ).
This follows from the fact that, by (6.22), for each .
(IX.4) The map T is not left -quasi-closed on X.
Indeed, let a sequence in be of the form
Since , thus, by (6.22), and . Hence, . Moreover, its subsequences and satisfy . Clearly,
However, there does not exist such that .
(IX.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 (6.22), we conclude that
This gives
Next, we see that
By Definition 3.3, is left -quasi-closed on X.
(IX.6) For , statements (D) and (E) of Theorem 4.2 hold.
This follows from (IX.1)-(IX.5). From the above it follows:
Moreover, by (6.22), since , thus, by (6.22), we get
so (e3) holds.
(IX.7) For , statement (F) of Theorem 4.2 does not hold.
We have: assumption (F1) does not hold; for , assumption (F2) holds; ; properties (f1)-(f3) do not hold since .
Remark 6.1 (a) If is a metric space, then the generalized quasi-pseudodistances J of -families on X generalize: metrics d, distances of Tataru [28], w-distances of Kada et al. [10], τ-distances of Suzuki [11] and τ-functions of Lin and Du [29]. Moreover, in uniform spaces, the -families on these spaces generalize distances of Vályi [30]. For details, see [14].
(b) In metric spaces, beautiful generalizations of Rus’ and Subrahmanyam’s results [3, 4] are established by Kada et al. [[10], Corollary 2] for w-distances and Suzuki [[11], Theorem 1] for τ-distances. Interesting conclusions of Theorem 1.2 are given by Suzuki [12].
(c) Reilly [7] and Subrahmanyam and Reilly [9] proved extensions of Banach’s theorem for continuous maps in quasi-gauge spaces.
(d) In all results mentioned above, the restrictive assumptions about metric spaces or quasi-gauge spaces, which must be Hausdorff and complete or sequentially complete, respectively, or maps are continuous, are essential. Further, the mentioned results do not concern periodic points of the considered maps.
(e) We see that in Examples 6.5-6.7 and 6.9, the assumptions of Theorem 4.2 are satisfied, but assumptions of Banach’s [1], Rus’ [3], Subrahmanyam and Reilly’s [[9], Section 3], Reilly’s [25], Reilly-Subrahmanyam-Vamanamurthy’s [[8], Theorem 9] and Suzuki’s [[11], Theorem 1] theorems are not.
(f) Let us finally mention that properties of Definitions 3.1-3.3 and Theorems 4.1 and 4.2 concerning ‘right’ were omitted in our presentation; we may provide them by constructing appropriate examples (without assuming that T is continuous, without completeness of spaces in a usual sense and without separability of spaces) and applying analogous technique as above.
(g) Finally, it remains to note that the results of this paper are new in quasi-gauge, topological and quasi-uniform spaces.
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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, 289 (2013). https://doi.org/10.1186/1687-1812-2013-289
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DOI: https://doi.org/10.1186/1687-1812-2013-289