Abstract
We present the concept of -contractive multivalued mappings in -metric spaces and prove some fixed point results for these mappings in this study. Our results expand and refine some of the literature’s findings in fixed point theory.
1. Introduction and Preliminaries
Fixed point theory is one of the most active research areas of mathematics. The findings are used to solve problems in a wide variety of disciplines, including transportation theory, economics, and biomathematics. The development of functional analysis methods assisted in the broadening of the well-known Banach contraction theory (see [1–21] and references therein). One of them is the extension of the principle of contraction to cover applications and multivalued applications.
Samet et al. [20] presented the definition of -admissible mapping in complete metric spaces and partially ordered metric spaces in 2012 and proved several fixed point theorems under the generalized contraction in both. Following that, Hasanzade et al. [22] introduced the idea of -admissible mapping, which is the multivalued edition of -admissible single-valued mapping established in [20], and they provided a fixed point result for multivalued mappings in complete metric spaces satisfying many generalized contractive conditions.
In 2013, Mohammadi et al. [16] developed the notion of an -admissible mapping to the class of -admissible mappings.
In 2014, Jleli and Samet [11] expanded the contractive condition by considering the following function :
Definition 1 ([11]). Let denote the set of all functions fulfilling the following criteria:
(Θ1): is nondecreasing;
(Θ2): from every sequence , if and only if ;
(Θ3): there exists and such that ;
(Θ4): every is continuous.
Definition 2 ([11]). Let be a metric space. A mapping is called a -contraction if fulfills and there exists a constant such that for all ,
Several researchers extended (1) in various ways and proved fixed point theorems for a single and multivalued contractive mappings (see [7, 12, 17, 18, 23]).
Any Banach contraction is a -contraction, although this is not so for the inverse ([7]).
Definition 3 ([24, 25]). Let be a given real number and be a nonempty set. A function is a -metric if the following conditions are fulfilled for all
(bM1): if and only if ;
(bM2): ;
(bM3): .
The pair is called a -metric space.
It is worth noting that -metric space is a broader category than metric spaces.
The following notations are used in this paper:
, and is the set of all self mapping on such that is a continuous monotone and nondecreasing, and
Definition 4 ([26]). Let be a -metric space. , and consider a sequence in . Then, (i) is -convergent if there exists such that as In this case, we write . (ii) is -Cauchy sequence if as (iii) is closed if and only if for each sequence of points in with as , we have .
Remark 5 ([26]). The following statements hold in a -metric space . (1)A -convergent sequence has a unique limit(2)Each -convergent sequence is a -Cauchy sequence(3)In general, a -metric is not continuous(4) does not generally induce a topology on .
Consider be a -metric space and the class of nonempty closed subsets of On , consider be the generalized Pompeiu-Hausdorff -metric (see [27]); i.e., for all , where
For , we put we note that
Definition 6 ([26, 28]). Let and be two -metric spaces.
(D1): The space is -complete if every -Cauchy sequence in -converges.
(D2): A function is -continuous at a point if it is -sequentially continuous at that is, whenever is -convergent to , is -convergent to .
Definition 7 ([16, 22]). Let be a nonempty set, be a given mapping, and the class of all nonempty subsets of . (1)If the following condition applies, a mapping is called -admissible: for , , where (2)For each and with , we have ; for all , a mapping is said to be -admissible
Hussain et al. [8] in metric spaces used the concept of -complete. We expand and apply it in -metric spaces here.
Definition 8. Let be a -metric space and be a mapping. The -metric space is said to be -complete if and only if every -Cauchy sequence in with , for all , -converges in .
In 2015, Kutbi and Sintunavarat ([13]) introduced the notion of the -continuity for multivalued mappings in metric spaces. We expand and apply it in -metric spaces here.
Definition 9. Let be a -metric space and and be two given mappings. We say is an -continuous multivalued mapping on if (i)for each sequence with as , and(ii), for all , we have as
Remark 10. (1)It is easy to see that -admissibility implies -admissibility. But the converse may not be true as shown in example 15 of [15](2)If is -complete -metric space, then is also -complete -metric space. But the converse is not true (see Example 1)(3)Note that the -continuity implies the -continuity. In general, the converse is not true (see in Example 2)
Example 1. Let for all , and be a closed subset of Define by the following: Clearly, is not a -complete -metric space, but it is an -complete -metric space.
Example 2. Let and for all let and defined by the following: Clearly, is not a -continuous multivalued mapping on Indeed, if in defined by for all , we see that but ; however, is a -continuous multivalued mapping on . Indeed, if as with for all , then for all and so as .
Lemma 11 [29]. Let be a -metric space, and let . If and , then there exists such that .
Lemma 12 [1]. Let be a -metric space, and let and . Then, we have where denotes the closure of .
Lemma 13 ([30]). Let be a sequence in a -metric space with such that for some , and each . Then, is a -Cauchy sequence in .
Lemma 14 [31]. If is a -metric space with and and are -convergent to , respectively, we have In particular, if , then we have . Moreover, for each , we have
We will prove some new results of the fixed point in -complete -metric spaces, we will add a new kind of contraction for multivalued mappings called -contraction multivalued mappings, and we give some examples to illustrate the main results of this paper.
2. Results
Definition 15. Let be a -metric space. A mapping is called weak -contractive multivalued mapping if there exist and constant such that for all .
Example 3. Let , and take the -metric for all . Define , , , and by the following:
and . Note that for all , one has
Now, for all such that , we have
Then,
And thus,
Thus, is weak -contractive multivalued mapping, with and .
The following is our primary result:
Theorem 16. Let be a -complete -metric space and is a weak -contractive multivalued mapping. Assume that the following conditions hold: (i) is a -admissible(ii)There exists such that and (iii)f is -continuousThen, has a fixed point. Moreover, if and are fixed points of such that then .
Proof. If or then is a fixed point of Assume that and
Since is weak -contractive multivalued mapping, we obtain
with
Assume that , and from (15), we get
which is a contradiction; it follows that , and (15) becomes
By Lemma 11, there exists such that , and by the monotonicity of we obtain
Since is a -admissible and we have .
If , this ends the proof; if , we have .
From (10), we have
As above, we obtain
and then, there exists such that
We will assume that there exists the sequence in such that , using an inductive method.
And
for all . This shows that and so .
Prove that is a -Cauchy sequence.
For , we have
and using the monotonicity of , we shall have
for all , where .
By Lemma 13, we conclude that is a -Cauchy sequence.
For , we can use similar arguments as in the proof of Theorem 2.1 of [9] to prove that there exists and such that
for all . Then, for , we have
Since , converges. Therefore, as It follows that is a Cauchy sequence in ; since is -complete -metric space, there exists such that
Since is -continuous, we have
and hence,
which implies that . And since is closed, we obtain
Assume that has two fixed points and such and .
We have
where
Then,
which is a contradiction since ; thus, , which ends the proof.☐
Corollary 17. Let be a -complete -metric space and is weak -contractive multivalued mapping. Assume that the following conditions hold: (i) is a -admissible(ii)There exist and such that (iii) is a -continuousThen, has a fixed point.
Corollary 18. Let be a -complete -metric space and is weak -contractive multivalued mapping. Assume that the following conditions are true: (i) is a -admissible(ii)There exist and such that (iii) is a continuousThen, has a fixed point.
Corollary 19. Let be a -complete -metric space and is weak -contractive multi-valued mapping. Assume that the following conditions are true: (i) is a -admissible(ii)There exist and such that (iii) is a continuousThen, has a fixed point.
Theorem 20. Let be a -complete -metric space and is a weak -contractive multivalued mapping. Assume that the following conditions hold: (i) is a -admissible(ii)There exists and such that (iii)If is sequence in such that for all and as , then for all Then, has a fixed point.
Proof. As in the proof of Theorem 16, we obtain a -Cauchy sequence such that for all and Assume that .
Since is weak -contractive multivalued mapping, we have
where
If we obtain that
Since , we obtain that
which is a contradiction since Thus, is a fixed point of .☐
3. Examples
Example 4. Let , and take the -metric for all . Define , , and by the following:
and . Note that for all ,
Now, for all and , we have
Then,
Then,
Thus, is weak -contractive multivalued mapping, with and .
So, all conditions of Theorem 16 are fulfilled, which implies that has a fixed point, which is (in particular if ).
Example 5. Let , and take the -metric for all . Define , , , and by the following:
and . Note that for all , one has
Now, for all and , we have
Then,
Then,
Thus, is weak -contractive multivalued mapping, with and .
So, all conditions of Theorem 16 are verified, which implies that has a fixed point, which is (in particular if ).
Example 6. Let , and take the -metric for all . Define , , , and by the following:
and . Note that for all , one has
Now, for all and , we have
Then,
Then,
Thus, is weak -contractive multivalued mapping, with and .
So, all conditions of Theorem 16 are verified, which implies that has a fixed point, which is (in particular if ).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.