#### Abstract

The purpose of this article is to introduce locally --multivalued contraction and rational Ćirić type --multivalued contraction in the context of -metric spaces and prove some endpoint results. We provide a nontrivial example to show the authenticity of our main result. Our results generalize some well-known results of literature. We also present some endpoint results in both graphic -metric spaces and ordered -metric spaces. As an application of our main result, we investigate the solution of an integral equation.

#### 1. Introduction

In 2010, Amini-Harandi [1] showed that a multivalued mapping has a unique endpoint if and only if this multivalued mapping has the approximate endpoint property. Hussain et al. [2] established some approximate endpoints of the multivalued almost I-contractions in complete metric spaces. Later on, Moradi and Khojasteh [3] proved a result for generalized weak contractive multifunctions.

On the other hand, Samet et al. [4] introduced the notion of -admissibility and --contraction in 2012. Asl et al. [5] extended this notion of -admissibility to -admissibility and proved some results for multivalued mappings. In 2015, Mohammadi and Rezapour [6] improved the -admissibility concept and obtained endpoint of --multivalued contraction. Later on, Choudhury et al. [7] used the notion of -admissibility and proved end point results of multivalued mappings without continuity. Very recently, Isik et al. [8] proved endpoint results for --contraction in the newly introduced space of Jleli and Samet [9] which is named as -metric space (-MS). In this artilce, we give locally --multivalued contraction and rational Ćirić type --multivalued contraction in the framework of -metric space and generalized the main result of Isik et al. [8].

#### 2. Preliminaries

Let and (nonempty subsets of ) be a multivalued mapping. A point is professed to be an endpoint (fixed point) of if . Now, let be a metric space, then is said to satisfy the approximate fixed point property if

Let represents the set of all nonempty, closed, and bounded subsets of . The Hausdorff metric is defined on as follows:

In 2012, Samet et al. [4] used the following set of nondecreasing functions satisfying and introduced --contraction. Clearly, for all ([30]).

Samet et al. [4] also initiated the concept of -admissibility of a single valued mapping in this way.

*Definition 1 (see [4]). *Let and let , then is said to be -admissible if , implies .

They gave the following property of that is is -regular, if for each sequence in with and , then ,

In 2013, Asl et al. [5] extended this concept to multivalued mapping and gave the notion of -admissibility as follows.

*Definition 2 (see [5]). *Let and let , then is said to be -admissible if for all , implies , where , for all .

In 2015, Mohammadi and Rezapour [6] extended the above notion in this way.

*Definition 3 (see [6]). *Let and , then is *-*admissible provided that for all and with , then for all .

They proved endpoint results for --multivalued contraction by using the following property.

A multivalued mapping is said to satisfy the property (), if for all there exists such that . Isik et al. [8] used the property () of Mohammadi and Rezapour [6] to prove their results, that is, for each sequence with for all and , then for all .

For more details in this direction, we refer the readers (see [10–14]).

Recently, Jleli and Samet [9] introduced an interesting generalization of metric space which is called -metric space (-MS) as follows.

Let be the class of such that ()< (), for ,

*Definition 4 (see [9]). *Let , and let . Suppose that there exists and such that

, for all

, for all

for every , for every , and for every with , we have
Then, is called an -MS.

Theorem 5 (see[9]). *Let be an -MS and let . Suppose that these assertions hold:
*(i)

*is -complete*(ii)

*there exists such that*

Then, there exists such that which is unique.

Hussain and Kanwal [15] utilized an -metric space and generalized the above result by considering the notion of --contraction to prove a fixed point theorem. Many researchers (see [16–18]) worked in this newly generalized space.

Very recently, Isik et al. [8] introduced the notion of Hausdorff metric (.,.) on influence by -metric as follows: for all , where and obtained endpoint results for --multivalued contraction in this way.

Theorem 6. *Let be an -MS and be an -admissible mapping which satisfies the property (). Suppose there exists and such that
*

*Also, suppose that these assertions hold:*(i)

*is -complete*(ii)

*(iii)*

*is -regular*

*Then, has an endpoint.*

#### 3. Main Results

*Definition 7. *Let be an -MS. A mapping is called a locally *-**-*multivalued contraction if there exists and such that
for

Now, we state our main result regarding the existence of the endpoint of an --multivalued contraction on the closed ball which is very advantageous in the perception that it needs the contractiveness of the multivalued mapping only on the closed ball instead of the whole space.

Theorem 8. *Let be an -MS and be an -admissible, locally --multivalued contraction such that satisfies the property () and for there exists such that
for all and Also, suppose that the following assertions hold:
*(i)

*is -complete*(ii)

*for an and*(iii)

*is -regular*

*Then, has an endpoint.*

*Proof. *Choose and such that . It follows directly from (10); we have
which implies that

It follows from (10) that

Since satisfies the property (), so such that . Now, from (13), we have This implies that

Since is -admissible, , so follows from (9) that

Continuing this process, we obtain a sequence in such that , and for all . If for some , then we get that . It implies that is an endpoint. Hence, we suppose that for all .

Now, since so

for all . Assume that be such that () is satisfied and fix . By (F), such that

Suppose that be such that . Hence, by (17), (18) and (), we have for Using () and (19), we obtain that where which implies that which implies by () that , for all This proves that is -Cauchy. Because of -completeness of there exists such that . We shall prove that is an endpoint of . We assume on the contrary that . Then . Since is locally -regular, so for all . Then, by (9) and (), we have as Thus,

On the other side, as , that is a contradiction. Hence, .

*Definition 9. *Let be an -MS. A mapping is called a rational Ćirić type *-**-*multivalued contraction if there exists two functions and such that
for where

Theorem 10. *Suppose that be an -MS and be an -admissible and rational Ćirić type --multivalued contraction such that satisfies the property (). Also, suppose that these conditions hold:
*(i)

*is -complete*(ii)

*for an and ;*(iii)

*is continuous*

*Then, has an endpoint.*

*Proof. *Choose and such that . Since satisfies the property (), there exists such that

Since is -admissible, . Continuing this process, we obtain a sequence such that , and for all . If for some , then we get that

It implies that is an endpoint. Hence, we suppose that for all .

Note that for all . If , then which is a contradiction. So, we have which implies

Continuing in this way, we obtain that for all which yields that for Suppose that be arbitrary. Next, let be such that () is satisfied. By (), there exists such that

Suppose that be such that . Hence, by (24), (35) and (), we have for Using () and (36), we obtain that where which implies that

which implies by () that , for all This proves that is -Cauchy. As is -complete, so such that . We shall prove that is an endpoint of . We assume on contrary that . Then, . Now, we have

Note that we used the property () in the above inequality. Taking the limit in both sides of the above inequality and using continuity assumption of , we get which implies that . Hence, as , that is a contradiction. Hence, .

*Example 1. *Consider the set . Suppose that the mapping be given by

So, is an -metric on with and . Now, define by and . Taking , we have where where

Therefore, where for all . Taking for all , satisfies all of the conditions of Theorem 10 and so has an endpoint. Here, .

#### 4. Endpoint Theorem in Graphic -Metric Spaces

In the present section, we will discuss the existence of endpoints on an -MS equiped with a graph , i.e, (-GMS).

Jachymski [19] has obtained an extension of Banach’s contraction principle in metric space equiped with a graph . Afterwars, Dinevari and Frigon [20] proved his results for multivalued mappings. Let be an -MS. A set is said to be a *diagonal* of , and represented by . Let be a graph such that the set that is, the set of its vertices and the set of its edges consists of all loops, i.e., .

*Definition 11. *[21] Let equiped with a graph and . The mapping is said to preserves edges weakly if, for all and with , we get .

We give the following definition from [21] which is required in our proof.

*Definition 12. *Let be an *-GMS.*

The -GMS is called -complete if every Cauchy sequence in with for all converges in .

*Definition 13. *A mapping is called *a**-*continuous mapping if, for any and any sequence with and for all , we have

*Definition 14. *A multivalued mapping is called a rational Ćirić type *-*contraction multivalued mapping if there exist a function such that
where .

Theorem 15. *Suppose that be an -GMS and be a rational Ćirić type -multivalued contraction. Suppose that the following conditions hold:*

*is an -complete -GMS*

*preserves edges weakly*

*there exist and such that*

*is an -continuous multivalued mapping*

*Then, has an endpoint point in .*

*Proof. *This result can be obtain from Theorem 10 if we define a mapping by if and , otherwise.

#### 5. Endpoint Theorem in Ordered -Metric Spaces

In 2004, Ran and Reurings [22] gave the idea of ordered metric space (OMS) by combing classical metric space and partial order on Fixed point results in OMS have many applications in integral and differential equations and other fields of mathematical analysis (see [23, 24]). In this section, we will consider (-OMS), i.e., where is an -MS and is a partial order on and we will derive some new results from Theorems 8 and 10. Remember that is nondecreasing if .

Here, we state the following notion motivated from [25].

*Definition 16. *Let with partial order *on* and . Then, is said to be weakly increasing if, f or all and with , we get that for all .

*Definition 17. *Let be an *-OMS.*

The -OMS is called *-*complete if every Cauchy sequence in with for all converges in .

*Definition 18. *A mapping is said to be *a**-*continuous mapping if, for any and any sequence with and for all , we get

Motivated from [8], we define the notion of an ordered rational iri type *-*multivalued contraction in an -OMS.

*Definition 19. *A multivalued is called an ordered rational *iri* type *-*multivalued contraction if there exists such that
where .

Theorem 20. *Let be an -OMS and be an ordered rational Ćirić type -multivalued contraction. Assume that these hold:*

*is an -complete -OMS*

*is weakly increasing*

*there exist and such that*

*is an -continuous multivalued mapping*

*Then, has an endpoint point in .*

*Proof. *This result can be obtained from Theorem 10 if we define a mapping by if and , otherwise.

#### 6. Suzuki Type Endpoint Results in -MS

In 2008, Suzuki [26] obtained a fixed point result as generalization of the Banach fixed point theorem. In this section, we derive endpoint results for rational Suzuki type -multivalued contraction in -MS as consequence of our result.

Corollary 21. *Let be a complete -MS, and such that implies
where , for all and satisfies the property (). If is continuous, then has an endpoint.*

*Proof. *Define by

It is easy to check that is -admissible.Also, for every and , we have . Hence, . It is very simple to check that where , for all . Therefore, by Theorem 10, has an endpoint.

Corollary 22. *Suppose that be a complete -MS, and such that implies that
where for all and enjoys property (). If is continuous, then has an endpoint.*

#### 7. Application to Nonlinear Integral Equations

Let CB () represents the set of all nonempty closed and bounded subsets of and be the space of all real-valued continuous functions on Clearly, equiped with the -metric given by where is a -complete -metric space (see [15]).

Now, we consider the integral equation

where and is continuous.

Theorem 23. *Suppose that these conditions hold:
*(i)*for all is such that is continuous in *(ii)*there exists which is continuous that satisfy the property such that for any and each there exists such that
*