#### Abstract

After the establishment of the Banach contraction principle, the notion of metric space has been expanded to more concise and applicable versions. One of them is the conception of -metric, presented by Jleli and Samet. Following the work of Jleli and Samet, in this article, we establish common fixed points results of Reich-type contraction in the setting of -metric spaces. Also, it is proved that a unique common fixed point can be obtained if the contractive condition is restricted only to a subset closed ball of the whole -metric space. Furthermore, some important corollaries are extracted from the main results that describe fixed point results for a single mapping. The corollaries also discuss the iteration of fixed point for Kannan-type contraction in the closed ball as well as in the whole -metric space. To show the usability of our results, we present two examples in the paper. At last, we render application of our results.

#### 1. Introduction and Preliminaries

In recent years, along with -metric presented by Jleli et al. [1], many authors presented interesting generalizations of metric spaces [29]. Jleli and Samet introduced generalized metric spaces, known as -metric spaces, and proved their generality to metric spaces with the help of concrete examples. The idea of -metric spaces was compared with -metric and -relaxed metric spaces, and hence, the Banach contraction principle was established in the frame of -metric spaces.

Banach contraction principle states that any contraction on a complete metric space has a unique ﬁxed point. This principle guarantees the existence and uniqueness of the solution of considerable problems arising in mathematics. Because of its importance for mathematical theory, the Banach contraction principle has been extended and generalized in many directions [10, 11]. The ﬁxed point theory of multivalued contraction mappings using the Hausdorﬀ metric was initiated by Nadler [12], who extended the Banach contraction principle to multivalued mappings. Since then, many authors have studied various ﬁxed point results for multivalued mappings. Nazam et al. [13] proved fixed point theorems for Kannan-type contractions on closed balls in complete partial metric spaces. The abovementioned results and its generalizations are recently investigated for fixed point in the setting of F-metric space (see [1416]).

In this article, we prove fixed point and common fixed points results of Reich-type contractions for single-valued mappings in -metric spaces.

This article is organized into three sections. Section 2 contains a short history of the previous literature that becomes a motivation for this article. There are some basic definitions which help readers to understand our results easily. In Section 3, we established theorems of fixed points and common fixed points of single-valued Reich contractions in -metric spaces. An example is provided to explain our results. Section 4 deals with fixed point theorems of contractions with respect to closed balls in -metric spaces along with an example.

#### 2. Basic Relevant Notions

Definition 1. (see [1]). A self-mapping on a nonempty set is said to be Kannan contraction if there exists a number , , such that, for each , we haveLet with following characteristics:(F1) is strictly increasing(F2)For any sequence, we haveThe collection of all such functions satisfying (F1) and (F2) is denoted by . The concept of -metric is generalized as follows:

Definition 2. (see [1]). Supposeis a nonempty set and . Let the function be such that(d1)For all (d2) (d3) {tn}i=1n⊂X For every for each and for every with , we haveThen, is known as an-metric on A, and the pair is called -metric space.

Example 1. (see [1]). Let (set of natural numbers) and defined byfor all . It can easily be seen that is an -metric with .

Example 2. (see [1]). Let and is defined asfor all . Then, is -metric on

Definition 3. (see [1]). Supposeis a sequence in Then,(i) is -convergent to a point if (ii) is -Cauchy sequence if (iii)The space is -complete if every-Cauchy sequence is -convergent to a point

Definition 4. (see [1]). Let be an -metric space. A subset of is said to be -open if, for every , there is some such that , whereWe say that a subset of is -closed if is -open.

Definition 5. (see [1]). Let be an -metric space and be a nonempty subset of . Then, the following statements are equivalent:(i)-closed.(ii)For any sequence , we have

Theorem 1. (see [1]). Suppose and is an -complete -metric space. be a given mapping. Suppose that there exists such that

Then, has a unique fixed point . Moreover, for any , the sequence defined by  =  is -convergent to

Theorem 2. (see [17]). Suppose is a complete metric space with metric , and let be a function such thatfor all , where are nonnegative integers and satisfy . Then, has a unique fixed point.

Lemma 1. (see [18]). The Banach space along with the metric defined byis an -metric space.

#### 3. Fixed Points of Reich-Type Contractions in Metric Spaces

In this section, we construct fixed point and common fixed points results for single-valued Reich-type and Kannan-type contractions in the setting of -metric space.

Theorem 3. Suppose and is an -complete -metric space. Let be self-mappings such thatfor such for all . Then, and have at most one common fixed point in .

Proof. Suppose is an arbitrary point and define a sequence byUsing (11) and (12), we can writeThis implieswhere
Similarly,Continuing this way, we getwhich yieldsHence,Using (18), we can writeSince , for any , there exists some such thatFurthermore, suppose satisfies (d3) andis fixed. By there is some such thatBy (21), we writeUsing (20), we writeBy (d3) and above equation, we obtainThis shows thatHence, we showed that is -Cauchy sequence in . Since is -complete, there exists such that is -convergent to i.e.,To prove that is the fixed point of , assume . Then,which implies which is a contradiction. Hence, i.e., . Similarly, suppose :i.e.,which is contradiction to the assumption. Therefore, we get . Hence, .
Uniqueness. Assume thatis also a common fixed point of and ,We get (1-a) which is a contradiction. Hence,

Example 3. SupposeLet and are defined byIt can be easily verified that is an -metric and satisfies Fix and Suppose , thenwhere . The inequality (11) holds true. Moreover, it is clear that is the only common fixed point of and .
Taking in Theorem 1, we get the following result of Kannan contractions.
Replacing in Theorem 3, we get the following corollary.

Corollary 1. Suppose and is an -complete -metric space. Let is a self-mapping such thatfor such for all . Then, has at most one fixed point in .
Taking in Corollary 1, we get the following result.

Corollary 2. Suppose and is an -complete -metric space. Let is a self-mapping such thatfor and . Then, has at most one fixed point in .
Besides the above important results, Theorem 3 also led us to the following fixed point result of Kannan-type contraction.

Corollary 3. Suppose and is an -complete -metric space. Let be self-mappings. Suppose that, for such thatfor all , then and have at most one common fixed point in .

Proof. Suppose is an arbitrary point and define a sequence by and ;
Using the contraction and the iteration given above, we can writeThis impliesorwhere . Similarly,Continuing the same way as in Theorem 3, we get the common fixed point of and .
Replacing with , we get the following result of single mapping.

Corollary 4. Suppose and is an -complete -metric space. Let be a self-mapping. Suppose that, forsuch thatfor all , then has at most one fixed point in .

#### 4. Fixed Points of Reich-Type Contractions on -Closed Balls

This portion of the paper deals with the fixed points theorems of Reich-type contractions that hold true only on the closed balls rather than on the whole space .

Definition 6. Let be an -complete -metric space and be self-mappings. Suppose that for Then, the mappings and are called Reich-type contractions on such that

Theorem 4. Suppose and is an -complete -metric space. Let S and be Reich-type F-contractions on . Suppose and the following conditions are satisfied:(a) is -closed(b), for and (c)There exist such as , where Then, and have at most one common fixed point in .

Proof:. Suppose is an arbitrary point and define a sequence by and ; .
We need to show that is in for all . We show it by mathematical induction. By (b), we writeTherefore, . We know by previous theorems thatNow,This implies thati.e., . Suppose for some . Now, if , then by (42), we can writeThis impliesorLet , we getSimilarly, if , thenTherefore, from inequality (50) and (51), we writeandFrom (52) and (53), we writeNow, using (54), we haveUsing (b), we writeUsing (c), we deduce thatHence, by (F1), we notice thatThis implies that . Therefore, for all . Now, we have by (42)Following the same steps of proof of Theorem 3 and using (a), we obtain that the sequence is -convergent to some in can be proved as common fixed point of and in the same way as in Theorem 3.
Taking in Theorem 4, we get the following result of single mappings.

Corollary 5. Suppose, is an -complete -metric space and is a self-mapping. Suppose that for Suppose and the following conditions are satisfied:(a) is -closed(b), for all (c), for and (d)There exists such as , where Then, has at most one fixed point in .

Example 4. Let and. Define byand define It can be easily verified that is an -metric and function satisfies . Fix , then . Clearly, is -closed so condition (a) of Corollary 5 is satisfied. Now, if , then andThis shows that condition (b) is fulfilled. Furthermore, suppose , theni.e.,Hence, condition (d) is satisfied for and Similarly, for all values of , we can find some and such that condition (d) is fulfilled. Now, checking for condition (b), we have two cases:(i)If , thenas  Therefore, for all condition (d) is also satisfied.(ii)If , e.g., and , thenHence, condition (b) holds only for and not on . Moreover, is the fixed point of .

Corollary 6. Suppose and is an -complete -metric space. Let Sare self-mappings and assume that, for and the following conditions are satisfied:(a) is -closed(b), for all (c), for and (d)there exist such as , where Then, and have at most one common fixed point in .

Corollary 7. Suppose and is an -complete -metric space. Let Sare self-mappings and assume that, for and the following conditions are satisfied:(a) is -closed(b), for all (c), for and (d)there exist such as , where Then, and have at most one common fixed point in .
An example can be proved in a similar way as that to the previous examples.

#### 5. Application

This section is concerned with the application of the main result proved in Section 2, in finding a unique common solution of the functional equations that are used in dynamic programming.

The two main components of dynamic programming are decision space (DS) and a state space (SS). The SS includes different states such as transitional states, initial, and action states, while the DS is composed of the steps that are taken for locating the possible solution point of the problem. Optimization and computer programming are based on this system. In particular, a problem of dynamic programming is converted to functional equations aswhere and are Banach spaces such as and and

Suppose and are the DS and SS, respectively. We aim to locate a single common solution point for equations (67) and (68). We denote the set of all bounded real-valued mappings on by . Let be arbitrary member of and say . Then, the duplet is a Banach space with defined by

Let the following conditions holds true:(C1) are bounded.(C2)For and , define byObserve that, and are well-defined whenever the functions , and are bounded.(C3)For , and , we writewhere for and Now, we develop the following theorem.

Theorem 5. Suppose conditions (C1)–(C3) hold true, then there exists a single bounded common solution of equations (67) and (68).

Proof:. From Lemma 1.10, we have is an -complete MS. is defined by (70), and from (C1), we deduce that and are self-mappings on Let be an arbitrary positive number and . Take and such asandThen, using (74) and (77), we obtainAlso, from (75) and (76), we getMerging the above two inequalities, we writefor all . Thus,i.e.,