Fixed Point Theory and Applications for Function SpacesView this Special Issue
On Suzuki and Wardowski-Type Contraction Multivalued Mappings in Partial Symmetric Spaces
The purpose of this paper is to provide some fixed-point results for Suzuki and Wardowski-type contraction multivalued mappings in partial symmetric spaces. We give some examples to support and substantiate the developed notions and obtained results. Also, we use one of our main results to establish the existence and uniqueness of the solution for a system of integral inclusions.
In 1922, Banach  formulated a very famous, fruitful, useful, and core fixed-point result known as the Banach contraction principle on a complete metric space. This celebrated result has been generalized and extended in several directions by various authors on some generalized metric spaces such as partial metric spaces , partial JS-metric spaces , metric-like spaces , -metric spaces , rectangular metric spaces , controlled and doubled metric spaces [7, 8], generalized -metric spaces , and extended rectangular b-metric spaces . Sometimes, one may come across a situation wherein the full force of metric conditions is not required in the enunciation of fixed-point results. Motivated by this observation, several researchers established fixed and common fixed-point results in symmetric spaces which did not require triangular inequality, i.e., a symmetric on a nonempty set is a mapping which satisfies iff , and for all .
Very recently, Asim et al.  introduced the class of partial symmetric spaces and proved some related fixed-point results for single-valued and multivalued mappings. The theory of multivalued mappings plays an important role in many areas of mathematics due to its diverse applications, namely in differential equations, integral equations, optimization problems, game theory, control theory, economics, etc. Aydi et al. [12, 13] proved fixed and common fixed-point results for multivalued maps in partial Hausdorff metric spaces to generalize fixed-point results due to Nadler .
In 2008, Suzuki  introduced a class of new types of contraction mappings and established fixed-point theorems for such mappings, which are genuine improvements of the Banach contraction principle. Thereafter, many authors attempted to extend such results for multivalued maps (employing the Pompieu-Hausdorff metric). In 2012, Wardowski  initiated the notion of a new kind of nonlinear contractions, namely -contractions and proved some related fixed-point results. In recent years, the idea of -contractions has been generalized and improved in several ways and directions. For more details, see [17–21].
In our paper, based on Suzuki and Wardowski-type contractions, we consider multivalued mappings, and we establish some related fixed-point results in the setting of partial symmetric spaces. Some examples are also furnished to demonstrate the utility of our results. As an application of Theorem 16, we establish the existence and uniqueness of the solution for a system of integral inclusions.
In what follows, we collect some relevant definitions and auxiliary results needed in the sequel.
Definition 1 . Let be a nonempty set. The function satisfying
is said to be a partial symmetric, and the pair is called a partial symmetric space.
A partial symmetric space reduces to a symmetric space if , for all . Observe that every symmetric space is a partial symmetric space. The converse is not true as it is shown in the following examples.
Example 2. Letand definebywhereand
Example 3. Letand definebywhere.
Example 4. Letand definebywhere
Example 5. Letand definebywhere.
Let be a partial symmetric space. The -open ball with a center and a radius is defined by
Similarly, the -closed ball with a center and a radius is defined by
The family of -open balls denoted by forms on a basis of some topology .
Lemma 6. Letbe a topological space andbe given mapping. If f is continuous; then, for every convergent sequenceto a pointin, the sequenceconverges to. The converse holds ifis metrizable.
We require some more definitions, namely, a convergent sequence, a Cauchy sequence, and a complete partial symmetric space, in our forthcoming discussions.
Definition 7 . A sequencein -converges to, with respect to, if
Definition 8 . A sequenceinis-Cauchy iffexists and is finite.
Definition 9 . A partial symmetric spaceis-complete if each-Cauchy sequenceinis-convergent with respect toto a pointsuch that
Let be a partial symmetric space and be the set of all nonempty, -closed, and bounded subsets of Moreover, for and , we define
Let be the partial Pompieu-Hausdorff symmetric, that is
3. Main Results
Given that , then for every , we define where and .
Now, we present the following lemma proved in , which is needed in the sequel.
Lemma 10. Letbe a partial symmetric space and. Then, for alland, there isso that
Next, let be the nonincreasing function defined by
Theorem 11. Let be a complete partial symmetric space and .
Assume the following:
There exists such that f satisfies the condition for all , where is defined by (14) and There is so that
The function is lower semicontinuous.
Then, has a unique fixed point.
Proof. By , there exists such that (15) holds. Let be such that For such , define a sequence by . If we assume that for some , then is a fixed point of , and we are done. Now, suppose that for each , and so . From Lemma 10 (with ) and condition (15) (for any arbitrary and for all ), we have
Since , we get
From (15) and (17), we have
The above inequality is true for all ; therefore, by conditions and (12), we have
By repeating this process, we have (for all )
Since and , then
so that is a -Cauchy sequence in . In view of the -completeness of , there exists such that -converges to . Thus
Assume that is lower semicontinuous. Then
Therefore, , that is, . Hence, is a fixed point of .
Next, let us show that has a unique fixed point. Suppose there are such that and . Then, by conditions and , we have which implies (by Lemma 10 with and ) a contradiction, so that , which implies that . Hence, has a unique fixed point. This completes the proof.☐
Example 12. Take. The partial symmetricis defined byThen,is a-complete symmetric space. Note thatandare bounded sets in. In fact, if, thenHence,is closed. NextHence,is also closed. Now, definebyClearly, by a routine calculation, one can easily show thatis lower semicontinuous. To prove the contractive conditionof Theorem 11, we need the following.
Case 13. Let. ThenFor or , we have. For, we have
Case 14. Let. Then, for all .
For or , we haveFor, we have
Case 15. Let. ThenFor, we haveFor, we getFor, we haveHence, the contractive conditionof Theorem11is satisfied. Observe thathas a unique fixed point (namely,).
Denote by the class of functions such that
For all .
The related fixed-point Wardowski-type result for multivalued mappings in the setting of partial symmetric spaces is as follows.
Theorem 16. Let be a complete partial symmetric space and . Assume that (i)There are and such that(ii)There is so that (iii)The function is lower semicontinuous.Then, has a unique fixed point.
Proof. For such , take . Define the sequence by . If we assume that for some , then is a fixed point of . Suppose for all , so . Since is a generalized multivalued -contraction, we have (for all )
The above inequality is true for all . Therefore, by conditions and (12), we have
By continuing this process, we find that
Now, for all , we have
On making the limit as in (45), we have
Owing to conditions and , we have
so that is a -Cauchy sequence in X. In view of the -completeness of , there is in order that -converges to . Thus, we have
Assume that is lower semicontinuous. Then, we have
Therefore, implies that . Hence, is a fixed point of .
For the uniqueness part, suppose there exist such that and . Thus, by the condition , we have which is a contradiction. Therefore, , that is, . It completes the proof.☐
Example 17. Letbe equipped with the partial symmetricdefined byThen, is a-complete symmetric space. Note thatandare bounded sets in. In factHence,is closed. NextHence,is also closed with respect to the partial symmetric.
Define and byand for , respectively. ThenHence,is lower semicontinuous. Now, we will show that the contractive conditionof Theorem16is satisfied.
Case 18. Let (say and for all ). HereThat is
Case 19. Letand. ThenAlsoFor, one writesTherefore, all the conditions of Theorem16are satisfied, andhas a unique fixed point (namely,).
4. An Application to an Integral Inclusion
Let be the set of all continuous real valued functions defined on . Now, we consider the following integral inclusion of Volterra type: where is a multivalued operator and is the set of nonempty compact and convex subsets of .
Then, is a -complete partial symmetric space.
Now, we are ready to present our result as follows:
Theorem 20. Suppose that for allthere exist a continuous functionandwithsuch thatThen, the integral inclusion (61) has a unique solution.
Proof. Let us define the multivalued operator by
Let and consider . For a multivalued operator , there exists a continuous operator (by Michael’s selection theorem) such that , for all . It implies that . Thus, the operator is nonempty. Note that is closed. For more details, see .
Firstly, we check the condition of Theorem 16. Let be such that . Then, there exists , for all such that.
Besides, for all , we have Consequently, there exists such that Let us consider the multivalued operator defined by Since is a lower semicontinuous, it follows that there exists such that , for all . Then, we have Hence, we have On interchanging the job of and , we have By using we have Thus, all the hypotheses of Theorem 16 are satisfied. Hence, the operator has exactly one fixed point, that is, the Volterra integral inclusion (61) has a unique solution.☐
Remark 21. Consider the following differential inclusion (for):where is a multivalued operator. Observe that, for , (71) is equivalent to (61). Also,satisfies the condition (63) of Theorem20. Then, applying Theorem20, the differential inclusion (71) has a unique solution.
Example 22. For all, letwhere(resp.,) is upper semicontinuous (resp., lower semicontinuous) on. For , assume the following:For all, there exist a continuous functionandsuch thatandHence,is a compact and convex valued operator. Thus,satisfies the condition (63) of Theorem20(for), so (74) has a unique solution.
We are concerned with Suzuki and Wardowski-type contraction multivalued mappings in the setting of partial symmetric spaces. We also studied a system of integral inclusions. It would be interesting to work on more generalized contraction mappings involving simulation or control functions.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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