Abstract

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.

1. Introduction

In 1922, Banach [1] 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 [2], partial JS-metric spaces [3], metric-like spaces [4], -metric spaces [5], rectangular metric spaces [6], controlled and doubled metric spaces [7, 8], generalized -metric spaces [9], and extended rectangular b-metric spaces [10]. 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. [11] 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 [14].

In 2008, Suzuki [15] 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 [16] 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.

2. Preliminaries

In what follows, we collect some relevant definitions and auxiliary results needed in the sequel.

Definition 1 [11]. Let be a nonempty set. The function satisfying
iff ,
,
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 [11]. A sequencein -converges to, with respect to, if

Definition 8 [11]. A sequenceinis-Cauchy iffexists and is finite.

Definition 9 [11]. 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 [11], 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 ) Now 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

is increasing

For all .