Fixed Point Theorems for Ćirić Type -Contractions in Generating Spaces of Quasi-Metric Family

Abstract

In this paper, the notions of Ćirić type (I) -contractions and Ćirić type (II) -contractions in generating spaces of quasi-metric family are introduced and new fixed point theorems for such two contractions are established. We give examples to illustrate main results.

1. Introduction and Preliminaries

The Banach contraction principle forms the basis of metric fixed point theory. Because its importance, many authors generalized this contraction principle by generalizing the certain contraction conditions.

Especially, Ćirić [1] proved a result on nonunique fixed points as follows:

If a map , where is a metric space, satisfies

for all , where , then has a fixed point whenever is -orbitally complete.

Also, Ćirić [2] obtained the following result:

If a map satisfies the following condition

for all , where , then has a unique fixed point provided that is -orbitally complete.

Recently, Khojasteh et al. [3] presented the notion of -contraction by using a simulation function and obtained the following result:

If a map is -contraction with respect to a simulation funtion , that is,

for all , then has a unique fixed point when is complete.

They unified the some existing metric fixed point results. Afterwards, many authors (for example, [4–8]) obtained generalizations of the result of [3].

Also, a lot of authors generalized the Banach contraction principle by introducing the concepts of generalized metrics, for example, Branciari metric, b-metric, quasi-metric, semi-metric, G-metric, cone metric, fuzzy metric, Menger Probabilistic metric.

In particular, Chang et al. [9, 10] introduced the concept of a generating space of a quasi-metric family and gave some examples and properties of generating space of a quasi-metric family, and they obtained some fixed point and minimization results in such spaces.

In this paper, we introduce the concepts of Ćirić type (I) -contraction maps and Ćirić type (II) -contraction maps in generating spaces of quasi-metric family, and we establish new fixed point theorems for such two contraction maps.

Let be a nonempty set and be a family of mappings .

Then is called a generating space of quasi-metric family [9, 10] if the following are satisfied: (QM1) if and only if ; (QM2) ; (QM3) such that  (QM4) is nonincreasing and left continuous in . It follows from (QM3) and (QM4) that such that 

Example 1. Let be a metric space, and let .
Then is a generating space of quasi-metric family.

Example 2. Let , and let
Then is a generating space of quasi-metric family.

Example 3. Let , and let be a map such thatLet
Then is a generating space of quasi-metric family.
Note that is not a metric on . In fact, the following inequality is not satisfied:Let be a generating space of quasi-metric family, be a sequence, and .
Then we say that(1) converges to (denoted by ) if and only if ;(2) is a Cauchy sequence if and only if such that (3) is complete if and only if every Cauchy sequence in is convergent;(4) is continuous at if and only if such that , whenever ;(5) is continuous if and only if it is continuous at each point in .

Remark 4. Every generating space of quasi-metric family is a Hausdorff space in the topology induced by the family of neighborhoods:where (see [11]).
Thus, every convergent sequence in generating spaces of quasi-metric family has a unique limit.
Note that every convergent sequence is a Cauchy sequence.
Also, note that is continuous at if and only ifLet be a generating space of quasi-metric family, and let be a mapping.
Then we say that
(1) is -orbitally continuous if and only if(2) is - orbitally complete if and only if every Cauchy sequence of the form is convergent in .

Remark 5. Let be a generating space of quasi-metric family, and let be a mapping.(1)if is continuous, then it is -orbitally continuous;(2)if complete, then it is -orbitally complete.

Remark 6. A generating space of quasi-metric family induce a fuzzy metric space and Menger probabilistic metric space (see [12] and [13], respectively).
Let be the family of all mappings such that () ; () ; () for any sequence  We say that is a simulation function [9].
Note that for all , .

Example 7. Let , be functions defined as follows:(1), where are continuous functions such that (2), where are continuous finctions with respect to each variables such that (3), where is a lower semi-continuous function with (4), where is a function such that (5), where is an upper semicontinuous function such that (6), where (7), where is a function such that (8)Then .
For more examples of simulation functions, we can find in [8, 9, 11, 12].

Lemma 8. Let be a generating space of quasi-metric family. Suppose that is not a Cauchy sequence.
Then there exists an for which we can find two subsequences and of such that is the smallest index with Further ifthen we have(1);(2);(3);(4);

Proof. If is not a Cauchy sequence, then from definition (17) holds.
Suppose that (18) is satisfied.
Then from (17) we have that for each By taking in above inequalityHence (1) is proved.
We show that (2) holds.
It follows from (QM3) and (QM4) that for such that ,Letting in Equations (21 and 22), and using Equations (18 and 19) we obtainIn particular, we haveThus proof of Equation (2) is complete.
In similar with the proof of Equation (2), we haveand soWe now show that Equation (4) holds.
For such that andBy taking in Equations (17) and (29), and using Equations (18), (23) and (25) we obtainand soHence (4) is proved.

Lemma 9. Let be a generating space of quasi-metric family. Suppose that Then we have

Proof. Sincewe havewhere and .
Hence we haveBy taking , we obtain

2. Fixed Point Theorems

2.1. Nonunique Fixed Point Results

Let be a generating space of quasi-metric family.

A mapping is called Ćirić type (I) -contraction with respect to if

for all , where

Now, we prove our first fixed point result.

Theorem 10. Let be a generating space of quasi-metric family, and let be a Ćirić type (I)-contraction mapping with respect to . If is -orbitally complete, then has a fixed point.

Proof. Let be any fixed point. Define a sequence by .
If for some , then is a fixed point of , and the proof is finished.
Assume that .
We infer that It follows from (41) and (42) that Consequently, we obtain that Since is a decreasing sequence bounded from below by , there exists such thatWe now show that
On the contrary, assume thatLet and .
ThenFrom condition we havewhich is a contradiction. Hence we have , and henceWe now show that is a Cauchy sequence.
Suppose that is not a Cauchy sequence.
By Lemma 8, there exist and two subsequences of satisfying (17).
Since (49) holds, from Lemma 8 we have thatLetThenIt follows from thatwhich is a contradiction.
Thus is a Cauchy sequence. It follows from -orbitally completness of that there exists such thatFrom Lemma 9Thus it follows from (40) thatwhere HenceLetting in above inequality, and using Equations (49), (54) and (55), we haveThus

Example 11. Let with .
Then is a generating space of quasi-metric family.
Define a mapping as follows:Let , whereThen .
We have thatThus is a Ćirić type (I) -contraction with respect to .
All conditions of Theorem 10 are satisfied. Hence, from Theorem 10 has a fixed point, .

Corollary 12. Let be a metric space, and let be a mapping such thatfor all , where If is -orbitally complete, then has a fixed point.

2.2. Unique Fixed Point Results

Let be a generating space of quasi-metric family.

A mapping is called Ćirić type (II) -contraction with respect to if

where and is nondecreasing with respect to the second variable.

Theorem 13. Let be a generating space of quasi-metric family, and let be a Ćirić type (II) -contraction with respect to If is -orbitally complete, then has a unique fixed point.

Proof. Firstly, we show the uniqueness of fixed point whenever it exists.
Suppose that has two fixed points, say such that .
Then from (54) we havewhich is a contradiction. Thus has a unique fixed point if it exists.
Secondly, we show the existence of fixed point.
As in proof of Theorem 10, let us define a sequence by , where is any fixed point, such thatWe infer thatHenceIt follows from 65 thatwhich impliesThus there exists such thatWe show that
Assume that
LetThen .
It follows from (54 and 55) thatwhich is a contradiction. Hence , and henceWe now show that is a Cauchy sequence.
Suppose that is not a Cauchy sequence.
By Lemma 1.1, there exist and two subsequences of satisfying (17).
Since (73) holds, it follows from Lemma 1.1 thatLet and .
ThenThus from (3) we havewhich is a contradiction. Thus is a Cauchy sequence.
Hence there exists such thatLet and
If , then from Lemma 9 we haveHencewhich is a contradiction.
ThusHence .

Corollary 14. Let be a generating space of quasi-metric family, and let be a mapping such thatwhere and is nondecreasing with respect to the second variable.
If is -orbitally complete, then has a unique fixed point.

Corollary 15. Let be a generating space of quasi-metric family, and let be a mapping such thatwhere is nondecreasing with respect to the second variable.
If is -orbitally complete, then has a unique fixed point.
By taking in Corollary 2.5, we have the following result.

Corollary 16. (Banach) Let be a generating space of quasi-metric family, and let be a mapping such thatIf is -orbitally complete, then has a unique fixed point.

Corollary 17. Let be a metric space, and let be a mapping such thatwhere and is nondecreasing with respect to the second variable.
If is -orbitally complete, then has a unique fixed point.

Example 18. Let , and let

Let be a map defined as

Obviously, is complete, and hence it is -orbitally complete.

We now show that is a Ćirić type (II) -contraction with respect to with .

Case 1: Let .

Then we have

Case 2: Let .

Thus is a Ćrić type (II) -contraction with respect to . All conditions of Theorem 13 are satisfied, and has a unique fixed point theorem

However, Bananch contraction principle in the setting of generating spaces of quasi-metric family, i.e. Corollary 16 is not satisfied.

In fact, if for

then we have that

which implies

By taking , we have which is a contradiction.

Thus Theorem 13 is a generalization of the Banach contraction principle in generating space of quasi-metric family.

3. Consequence

By applying simulation functions of Example 3 to Theorem 10 and Theorem 13, we have some fixed point results.

Especially, taking for in Theorem 10 and Theorem 13, we have the following results.

Corollary 19. Let be a generating space of quasi-metric family, and let be a mapping such thatIf is -orbitally complete, then has a fixed point.

Corollary 20. Let be a metric space, and let be a mapping such thatIf is -orbitally complete, then has a fixed point.

Corollary 21. Let be a gessnerating space of quasi-metric family, and let be a mapping such thatIf is -orbitally complete, then has a unique fixed point.

Corollary 22. Let be a metric space, and let be a mapping such thatIf is -orbitally complete, then has a unique fixed point.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conficts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by Hanseo University.