C-Algebra-Valued Extended Rectangular H_b-Metric Spaces and Presić-Type Fixed Point Theorem with an Application

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 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 [4] as follows.

Theorem 4 (see [4]). 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