#### Abstract

In this article, we introduce the concept of C-algebra-valued extended rectangular Hb-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  for most information). In , 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 , 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  and extended rectangular -metric space  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  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 ). 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 ). Every -metric space defines a metric space by , for all .

Definition 2 (see ). 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 ). 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 ). Let be a -algebra-valued -metric space, , . The following statements are equivalent:(1);(2) as ;(3) as .

Proposition 3 (see ). 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 ). 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 ). 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 ). 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 ). 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 ifThat is, for each , there exists such thatfor all ;(ii)A sequence is called -Cauchy if, for each , there exists such thatfor 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 ). 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  reached the following result.

Theorem 3 (see ). Let be a complete metric space and let , where is a positive integer. Suppose thatfor 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  as follows.

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