#### Abstract

The purpose of this paper is to define a rational orbitally ()-contraction and prove some new results in the context of -metric spaces. Our results extend, generalize, and unify some known results in the literature. As application of our main result, we investigate the solution of Fredholm integral inclusion. We also provide an example to substantiate the advantage and usefulness of obtained results.

#### 1. Introduction

The fixed point theory is a very essential tool for nonlinear analysis of solvability of nonlinear integral equations and others. A suitable selection of a generalized and extended metric space allows to get nontrivial conditions guaranteeing the existence of solutions for a considered equation. Therefore, it is necessary to flourish the fixed point theory in various generalization of metric spaces. One of the famous extensions of metric space is the notion of -metric space which has been given by Bakhtin [1] in 1989. It was properly defined by Czerwik [2] with the aspect of relaxing triangle inequality in metric spaces in 1993 and proved famous Banach Contraction Principle in this generalized metric space. Khamsi and Hussain [3] discussed the topology of -metric space and established fixed point results for KKM mappings in metric type spaces. Van An et al. [4] proved the Stone-type theorem on -metric spaces and obtained a sufficient condition for a -metric space to be metrizable. On the other, Czerwik [5, 6] introduced set-valued mappings in -metric spaces and generalized Nadler’s fixed point theorem. In 2012, Aydi et al. [7, 8] gave fixed point and common fixed point theorems for set-valued quasicontraction mappings and set-valued weak -contraction mappings in the setting of -metric spaces, respectively. Many authors followed the concept of -metric space and established impressive results [9–19].

In 2012, Jleli and Samet [20] introduced a new type of contraction named as -contraction and obtained a fixed point result to generalize the celebrated Banach Contraction Principle in Branciari metric spaces. Ali et al. [21] defined multivalued Suzuki-type -contractions and obtained some generalized fixed point results. Afterwards, Jleli et al. [22] established a new fixed point theorem for -contraction in the setting of Branciari metric spaces and extended the main result of Jleli and Samet [20]. Recently, Alamri et al. [23] adapted Jleli’s approach to the -metric space and obtained some generalized fixed point results. For more details in the direction of -contractions, we refer the reader to [21–30].

In this paper, we define the notion of the rational ()-contraction in -metric spaces and explore the existence of solutions for certain integral problems of Fredholm type as applications of our main results. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of complete -metric spaces. Evidently, the given results generalized some notable results of the literature to -metric spaces.

#### 2. Preliminaries

In this section, we give some fundamental notations, definitions, and lemmas which will be used throughout the paper. Throughout this paper, we denote the set of positive integers and the set of all nonnegative real numbers.

Czerwik [2] gave the notion of the -metric space as follows.

*Definition 1 (see [2]). *Let be a nonempty set and . A function : is called a -metric if these assertions hold: for all ,

(*B*1)

(*B*2)

(*B*3)

The triple is called a *-*metric space.

Now, we give an elementary example of a -metric space, but it is not a metric space as follows.

*Example 2 (see [2]). *Let and be a mapping defined by
Then, is a -metric space which is not a metric space.

A brief but short history for multivalued mappings defined in is given in this way

Let be the family of all bounded and closed subsets of For any we define with

Here, we provide some useful properties of and (see [2, 5, 6]): (1)If and then (2)(3) for any (4)(5)

Moreover, we will always suppose that (6)the function is continuous in its variables

Now, we present the concepts of orbit and orbital continuity of a mapping in the setting of which are generalization and extensions of the same notions for metric spaces given in [31, 32].

*Definition 3. *Let .
(1)An *orbit* of at is any sequence such that for each (2)If, for any , there exists in such that and for each then for each is said to be an *orbit* of at (3) is said to be *-orbitally complete* if any Cauchy subsequence of converges in . Specifically, for , is said to be *-orbitally complete*(4) is said to be *orbitally continuous* at if, for any for each and as gives that as (5)A graph of is constructed as follows:

We need the following property of in our proof:

(G_{p}) is said to be *-orbitally closed* if, for any sequence in , we get whenever and

In 2012, Jleli and Samet [20] gave the notion of -contractions and proved a contemporary fixed point theorem for these contractions in generalized metric spaces. Motivate by Jleli and Samet [20], Alamri et al. [23] present the following definition.

*Definition 4 (see [23]). *Let denote the family of all mappings which satisfy these conditions:

() is nondecreasing

()For any ,

()There exist and such that

()For such that for each and some 0< then for each

They provided the following example.

*Example 5. *Let be a mapping given by . Clearly, satisfies the conditions ()*-*(). Here, we show only the condition (). Suppose that, for each and some we have This implies that
i.e.,
This implies that
It follows that and is nondecreasing, and so, and implies Therefore, (7) implies that
and hence, the condition () holds.

#### 3. Main Results

In this way, we define the notion of rational ()*-*contraction.

*Definition 6. *Let be a *-*metric space. A mapping is called a rational ()-contraction if *0<* and such that
with where

If (9) holds for all for some , then is called a rational orbitally *(**)-*contraction.

Theorem 7. *Let be a -metric space such that is a continuous functional and be a rational orbitally ()-contraction. If the following conditions hold:
*(a)

*is -orbitally complete for some*(b)

*is continuous and is closed, or the property (Gp) holds, then there exists such that*

*Proof. *For any we generate a sequence in as for each

If there exists for which , then is a fixed point of , and so, the proof is completed. Thus, assume that, for each So, we have and for each Then, it follows from (9) with and that
where
Since
it follows from (11) that
Assume that for some . Then, from (14), we get
which is a contradiction with (). Hence, we have
and consequently,
It follows from (17) and () that
Let us represent for Then, for all . Thus, we have
Taking in (19), we get
which implies that
by (). By (), and such that
Assume that For this case, let So, such that
which implies that
Then, we have
where

Now, assume that Let . So, such that
which implies that
where Thus, in all cases, there exist and such that
Hence, by (19) and (28), we get
Taking in (29), we have
and hence, , which yields that is convergent. Thus, is a Cauchy sequence in Since is -orbitally complete, there exists such that
Suppose that is closed. We notice that if such that for each Since is closed and we conclude that , and so, the proof is finished. Hence, we suppose that there exists so that for each with . This implies that for each Then, it follows from (9) with and that
where
Using the continuity of and , so applying the limit of (32) as we get
which is impossible from and By the condition (), we get Since is closed, thus we get . Assume that is -orbitally closed. Since and , we get . Hence, This completes the proof.☐

If for any in Theorem 7, we get following result.

Corollary 8. *Let be a -metric space such that is a continuous functional and be a mapping satisfying the following condition: for some 0< and ,
for all with where
for all . Assume that is -orbitally complete for some . If is closed, for all or the property (Gp) holds, then such that .*

*Example 9. *Let . The -metric is defined by
with coefficient Define given by
If then Let and Then, , and ,
Take , and Then,

Thus, all conditions of Theorem 7 are satisfied and is the required point.

The family contains a wide set of functions; that is, if we take where and , then we can obtain the following corollary from Theorem 7.

Corollary 10. *Let be a -metric space such that is a continuous functional. Assume that be a mapping such that for some and,
*

*with where for all . Suppose that is -orbitally complete for some . If is closed for all or the property (Gp) holds, then there exists such that .*

If we replace self-mapping on the place of multivalued mapping in Theorem 7, we can get following results as consequences.

Corollary 11. *Let be a -metric space such that is a continuous functional and be a mapping such that is -orbitally complete at some . Assume that there exist , and such that
for all with where
**If is continuous, then there exists such that .*

#### 4. Applications

In this section, we solve the Fredholm integral inclusion: given in the start of this paper.

Let on be given by

Then, is a complete -metric space.

Assume that for these conditions hold: (a) is continuous(b) such that and where and (c) such that

Theorem 12. *Under the conditions (a)–(c), the integral inclusion given in (46) has a solution in .*

*Proof. *Let Define -metric as in (47) and a multivalued mapping by
Let . For a mapping , there exists such that
Then,
Hence, Since is continuous and , the ranges of both functions are bounded. This shows that is also bounded.

Now, we show that (9) holds for any on with some , and , i.e.,
for all . Let be such that
It follows that such that
Now, for all , it follows from (b) that
It follows that such that