#### Abstract

In this article, we introduce the concept of C-algebra-valued extended rectangular H_{b}-metric spaces and use it to prove the Banach contraction principle (BCP). Also, we extend this concept to Presić type contractions and use a method to shorten the proof. We also take an application to show the usefulness of our main result.

#### 1. Introduction

One of the most important topics in mathematics is metric spaces, which many authors have extended to different spaces (see [1–16] for most information). In [3], the concept of rectangular metric spaces was introduced in which the triangular inequality of metric spaces was replaced by another inequality called rectangular inequality. Later, George et al. in [5], introduced the concept of rectangular -metric space, which generalizes the concepts of metric spaces, rectangular metric spaces, and -metric spaces.

In this paper, inspired by Branchiari’s work, we generalize the definitions of -metric [12] and extended rectangular -metric space [2] to a metric, that we will call -metric space. Next, we introduce the concept of -algebra-valued extended rectangular -metric spaces and use it to prove the Banach contraction principle. Also, in the next section, we generalize this concept to Presić type contractions [15] and use a method to shorten the proof.

Let be a unital -algebra with the unity element , then, we denote , be the zero element, and . An involution on is a conjugate linear map on such that

for all . The pair is called a -algebra. A Banach -algebra is a -algebra together with a complete submultiplicative norm such that for all . A -algebra is a Banach -algebra such that for all . Let be a Hilbert space and the set of all bounded linear operators on , then, is a -algebra with the operator norm. An element in is self-adjoint or hermitian if . Let be the set of all self-adjoint elements in , and , where is spectrum of . An element of a -algebra is positive if is hermitian and . We write to mean that is positive. Using the positive element, one can define a partial ordering on as follows: if and only if . It is clear that if and , then, , and if are invertible, then, .

*Definition 1 (see [12]). *Let be a nonempty set and be a function satisfying the following properties: (G1) if ; (G2) for all with ; (G3) for all with ; (G4) (symmetry in all three variables); (G5) for all (rectangle inequality).Then, the function is called a -metric on and the pair is called a -metric space.

Proposition 1 (see [12]). *Every -metric space defines a metric space by , for all .*

*Definition 2 (see [9]). *Let be a nonempty set. Suppose the mapping satisfies (for all ):(i) and if and only if ;(ii);(iii).Then, is called a -algebra-valued metric on , and is called a -algebra-valued metric space.

*Definition 3 (see [16]). *Let be a nonempty set and be the permutation group on . If is a mapping satisfying the following properties:(1);(2) for all ;(3) for all with ;(4) for all .Then, is called a -algebra-valued -metric on , and is called a -algebra-valued -metric space.

Proposition 2 (see [16]). *Let be a -algebra-valued -metric space, , . The following statements are equivalent:*(1)

*;*(2)

*as ;*(3)

*as .*

Proposition 3 (see [16]). *Let be a -algebra-valued -metric space. The following statements are equivalent:*(1)*The sequence is -Cauchy;*(2)*For every , there is such that , for all .*

*Definition 4 (see [3]). *Let be a nonempty set. A mapping is called a rectangular metric on , if satisfies the following (for all and all distinct ):(1); if and only if ;(2);(3).Then, the pair is called a rectangular metric space.

*Definition 5 (see [5]). *Let be a nonempty set. A mapping is called a rectangular -metric on , if satisfies the following (for all and all distinct ):(1); if and only if ;(2);(3).Then, the pair is called a rectangular -metric space with the coefficient .

*Definition 6 (see [2]). *Let be a nonempty set and . A mapping is called an extended rectangular -metric on , if satisfies the following for all and all distinct :(1); if and only if ;(2);(3)Then, the pair is called an extended rectangular -metric space.

*Definition 7 (see [1]). *Let be a nonempty set and be a given real number. Suppose that satisfies ( 1) if ; ( 2) for all with ; ( 3) for all with ; ( 4) (symmetry in all three variables); ( 5) for all (triangle inequality).Then, is called a generalized -metric and the pair is called a generalized -metric space or -metric space.

The aim of this paper is to prove the existence and uniqueness of fixed points for the sequence of self-mappings on , where is a -algebra-valued extended rectangular -metric space.

#### 2. Main Results

Before starting the main theorem, we first introduce the following definitions.

*Definition 8. *Let be a nonempty set, and such that . Suppose that the mapping satisfies the following properties: (G1) if ; (G2) for all with ; (G3) for all with ; (G4) (symmetry in all three variables); (G5) for all , and for all distinct points (rectangle inequality).Then, the function is called a -algebra-valued extended rectangular -metric on , and is called a -algebra-valued extended rectangular -metric space with the coefficient .

*Example 1. *Let . is defined byfor all and , we haveIn the last part of the above inequality, we use the fact that for all , we have . Hence, is a -algebra-valued extended rectangular -metric space with

*Definition 9. *Let be a -algebra-valued extended rectangular -metric space. Then,(i)A sequence is -convergent to in if That is, for each , there exists such that for all ;(ii)A sequence is called -Cauchy if, for each , there exists such that for all .

*Definition 10. *The triple is called complete -algebra-valued extended rectangular -metric space, if every -Cauchy sequence in is -convergent in .

*Definition 11. *Let and be -algebra-valued extended rectangular -metric spaces, and be a function. Then, is called -continuous at a point if for every , there is such that and implies . A function is -continuous at if and only if it is -continuous at all points of .

*Definition 12. *Suppose that be a -algebra-valued extended rectangular -metric space. The sequence is called contractive, if there is an element with such thatNow, we establish the main result of this manuscript as follows.

Theorem 1. *Let be a complete -algebra-valued extended rectangular -metric space, be a sequence of contractive mappings on , and , then, has a unique fixed point in .*

*Proof. *Choose , and setBy condition (8), we getwhere . Now, we show that is an -Cauchy sequences in , for this, we prove two cases as follows:

(1) Let with . Using (G5) for any , we haveSo,Note that the above inequality is dominated byFrom hypothesis, we have . So, by the ratio test, we haveThat is, the series is convergent for every . LetAs a result, by the above inequality, we have(2) Let with .So,Therefore, again by the ratio test, we haveHence, we will haveTherefore, this indicates that the sequence is -Cauchy in . Since is complete, then, such that , that is,From continuity of , we haveSo, , therefore, is a fixed point of . Now, suppose that is another fixed point of . Sinceone hasSo, it is a contradiction. Hence, and .

#### 3. Presić Fixed Point Theorems

In this section, we obtain some results for Presić contractive mappings on the -algebra-valued extended rectangular -metric space.

Theorem 2 (see [17]). *Let be a complete metric space and let so thatwhere . Then, there exists such that . Also, for every , the sequence converges to .*

The BCP has expanded and generalized in various ways (see [10, 14]). Presić in [15] reached the following result.

Theorem 3 (see [15]). *Let be a complete metric space and let , where is a positive integer. Suppose that**for all , where and . Then, has a unique fixed point , that is, . In addition, for all arbitrary points , the sequence defined by , converges to .*

It is obvious that for , Theorem 3 coincides with the BCP. The above theorem has been generalized by Ćirić and Presić [4] as follows.

Theorem 4 (see [4]). *Let be a complete metric space and , where is a positive integer. Suppose that**for all , where . Then, has a fixed point . Also, for all points , the sequence defined by , converges to . The fixed point of is unique if*