Abstract

In the current manuscript, the notion of a cone -metric space over Banach’s algebra with parameter is introduced. Furthermore, using -admissible Hardy-Rogers’ contractive conditions, we have proven fixed-point theorems for self-mappings, which generalize and strengthen many of the conclusions in existing literature. In order to verify our key result, a nontrivial example is given, and as an application, we proved a theorem that shows the existence of a solution of an infinite system of integral equations.

1. Introduction and Preliminaries

There are many generalizations in the literature about the concept of metric spaces like -metric spaces [1], 2-metric spaces [2], -metric spaces [3], and weak partial -metric spaces [4]. Gähler incorporated the notion of a 2-metric space in [2]. Recall that a 2-metric is not a continuous function of its variables, whereas a standard metric is. This led Dhage to implement the -metric notion in [5]. In [6, 7] Mustafa and Sims implemented the -metric notion for overcoming -metric flaws. Following that, several fixed-point theorems were proven on -metric spaces (see [8]). The authors in [9] found that fixed-point theorems in -metric spaces can potentially be deduced from metric or quasimetric spaces in a variety of cases. Different researchers have additionally indicated that the fixed-point results about cone metric spaces can be acquired in a few cases by diminishing them to their standard metric partners; see for instance [10–12]. It is worth noting that a 2-metric space was not considered to be topologically equivalent to an ordinary metric in the generalizations described above.

Bakhtin [1] analyzed the phenomenon of a -metric space. After this theory, Czerwik [13] demonstrated the contraction mapping method in -metric spaces which generalized the renowned Banach contraction principle in -metric spaces.

Replacing the set of real numbers by an ordered Banach space, Huang and Zhang [14] generalized the concept of metric spaces and defined the cone metric space, where they studied certain fixed-point results for contractive mapping in the context of cone metric space. Later, Mustafa et al. [15] set the space structure -metric as a generalization of -metric and 2-metric spaces. They illustrated some fixed-point theorems in a partially ordered -metric space under different contractive conditions and provided some smart examples and an application to integral equations for their main outcomes.

Recently, the equivalence of cone metric space and metric space has become an extremely fascinating topic after the work of several researchers discovered that the fixed-point results in a cone metric space are special cases of metric spaces in some cases. They found that is equivalent to any cone metric if the real-valued function is replaced by a nonlinear scalariztion function or by a Minkowski functional . To address these shortcomings, Liu and Xu [16] presented the definition of cone metric space over Banach’s algebra.

Fernandez et al.[17] presented the concept of cone -metric spaces over Banach’s algebra with coefficient as an extension of -metric spaces and cone metric spaces over Banach’s algebras. They also presented many fixed-point results under different contractive conditions in the said structure. As an application, they discussed the existence of solutions to the integral equation.

On the other hand, Hardy and Rogers [18] introduced a new concept of mapping called the Hardy-Rogers contraction which generalize the Banach contraction principle and Reich’s [19] theorem in the setting of metric spaces. Samet et al. [20] initiated the -admissibility of mappings and gave a result of -contractive mapping which generalized the Banach contraction principle. After that, many researchers worked on the Hardy-Rogers contraction and -admissibility of mapping in different settings; for examples, see [21–27] and the references therein.

Motivated by the work done in [17, 18, 20] we study some results for the generalized -admissible Hardy-Roger contractions in cone -metric spaces over Banach’s algebras. We note that some well-known results in the literature can be deduced by using the presented work.

In the sequel, we need the following definitions and results from the existing literature.

Definition 1. (see [28]). Let be a real Banach algebra, and the multiplication operation is defined according to the following properties (for all ):
(𝔞₁) ;
(𝔞₂) and ;
(𝔞₃) ;
(𝔞₄) .
We will presume in the course of this article that is a real Banach algebra, unless otherwise specified. We call the unit of , if there is , such that . In this case, we call a unital. It is said that an element is invertible if an inverse element occurs, such that . In such case, the inverse of is unique and is denoted by . In the sequel, we need the following propositions.

Proposition 2. (see [28]). Let be the unit element of the Banach algebra and be arbitrary. If the spectral radius , that is then, is invertible. In fact

Remark 3. From [28] we see that, for all in the Banach algebra with unit , we have .

Remark 4. (see [29]). In Proposition 2, if we replace “” by , then the conclusion remains true.

Remark 5. (see [29]). If , then as .

Definition 6. Let be the zero element of the unital Banach algebra and . Then, is a cone in if
(𝔟₁) ;
(𝔟₂) ;
(𝔟₃) for all ;
(𝔟₄) ;
(𝔟₅) .
Define a partial order relation in w.r.t. by if and only if and also if but while stands for , where is the interior of . is solid if .
If there is such that for all , we have then, is normal. Ifis the least and positive among those cited above, then it is a normal constant of [14].
Onward, we assume that is a cone in with , and is a partial order with respect to the cone .

Definition 7. (see [16–14]). Let and the mapping be
(𝔠₁) for all , and if and only if ;
(𝔠₂) for all ;
(𝔠₃) for all .
Then is a cone metric and is a cone metric space over the Banach algebra .

Definition 8. (see [13]). Let and be a real number. Then, the mapping is a -metric if, for all , the following holds:
(𝔡₁) if and only if ;
(𝔡₂) ;
(𝔡₃) .
Here, the pair is a -metric space.
The cone -metric space over a Banach algebra with constant is introduced in [30]. Mitrovic and Hussain in [26] introduced the cone -metric space over a Banach algebra with constant .

Definition 9. (see [26]). Let . A function is a cone -metric if
(𝔢₁) for all , and if and only if ;
(𝔢₂) for all ;
(𝔢₃) There exists , such that for all .
Here, the pair is a cone -metric space over . If , then becomes a cone metric space over .

Definition 10. (see [2]). Let , satisfy the following conditions:
(𝔣₁) For , there is a point with at least two of which are not equal, then ;
(𝔣₂) if at least two of are equal;
(𝔣₃) For all where stands for all permutations of ;
(𝔣₄) For all
Then, the function is a -metric and is -metric space.

Definition 11. (see [17]). Let and be a real number. Let satisfy the following:
(𝔤₁) For , there is a point with at least two of which are not equal, then ;
(𝔤₂) if at least two of are equal;
(𝔤₃) For all where stands for all permutations of ;
(𝔤₄) For all .
Then, the function is a cone -metric and is a cone -metric space over the Banach algebra with parameter . It reduces to a cone 2-metric space if we take mentioned above. For other details about the cone 2-metric space over the Banach algebra , we refer the reader to [31].

Definition 12. (see [32]). Let be a sequence in , then
(𝔧₁) is a -sequence, if for each there exists a natural number such that for all ;
(𝔧₂) is a -sequence, if as .

Lemma 13. (see [33]). Let be Banach’s algebra and . Also, let be -sequences in , then for arbitrary , is a -sequence.

Lemma 14. (see [33]). Let be Banach’s algebra and . Let and be -sequences in . Let and be arbitrarily given vectors, then is a -sequence.

Lemma 15. (see [33]). Let be Banach’s algebra and . Let such that as . Then, is a-sequence.

Lemma 16. (see [28]). Let be the unit element of , and , then exists and the spectral radius satisfies If , then is invertible in , moreover, we have

Lemma 17. (see [28]). Let be the unit element of , and . If commute, then
(𝔨₁) ;
(𝔨₂) .

Lemma 18. (see [34]). Let be the unit element of, and . Let , and . If , then is a -sequence.

Lemma 19. (see [34]). Let be the unit element of , and . Let be a complex number, and , then

Lemma 20. (see [35]). Let be a cone.
(𝔩₁) If , , and , then ;
(𝔩₂) If are such that and , then ;
(𝔩₃) If and , then for any fixed , we have .

Lemma 21. (see [32]). Let and .
(𝔪₁) Let . Then is a -sequence if and only if ;
(𝔪₂) Every -sequence in is a -sequence;
(𝔪₃) is normal if and only if each -sequence in is a -sequence.

Lemma 22. (see [36]). Let be a Banach algebra and . Then the following are always true:
(𝔫₁) If and , then ;
(𝔫₂) If and for each , then .

Definition 23. (see [37]). Let a cone -metric space be over the Banach algebra with parameter , , be a solid cone, , and be two mappings. If for any sequence , with for each and as , it follows that for all and for all ; then, we say that is -regular.

2. Results and Discussion

We introduced here the notion of cone -metric space over Banach’s algebra with parameter .

Definition 24. Let and satisfy the following:
(𝔥₁) For , there is a point with at least two of which are not equal, then ;
(𝔥₂) if at least two of are equal;
(𝔥₃) For all where stands for all permutations of ;
(𝔥₄) For all , there exists , such that .
Then, the function is a cone -metric and is a cone -metric space over Banach’s algebra with parameter . It is reduced to a cone 2-metric space if we take mentioned above.

Remark 25. Note that every cone 2-metric space is a cone -metric space with parameter over Banach’s algebra. But the converse is not true.

Example 26. Let . For each , . The multiplication is defined by . Then is a Banach algebra with unit . Let . Then is a cone in .
Let . Define as follows: where . We have That is, , which shows that is not a cone 2-metric, because for with , , and . But the parameter with is a cone -metric space over the Banach algebra .

Example 27. Let . For each , . The multiplication is defined point wise. Then, is a Banach algebra with unit a constant function. Let . Then, is a cone in .
Let . Define as follows: for all , where and is such that . We have That is, , which shows that is not a cone 2-metric, because for with , , and . But the parameter is a cone -metric space over the Banach algebra .

Definition 28. Let a cone -metric space be over the Banach algebra with parameter , and let be a sequence in , then
(𝔦₁) converges to if for every there exists such that for all . We denote it by or (𝔦₂) If for there is such for all , then is a Cauchy sequence.
(𝔦₃) If every Cauchy sequence is convergent in , then is complete.
Next in the framework of cone -metric space over Banach’s algebra, we introduce the notion of -admissibility of mappings [20] and give the consequence of Hardy and Rogers [18] through -admissibility in cone -metric spaces over Banach’s algebras.

Definition 29. Let and be a cone in a Banach algebra . We say is -admissible if and , such that

Definition 30. Let and is a cone -metric space over the Banach algebra . We say is continuous at point , if for every sequence we have as , whenever as . is continuous if it is continuous at every point of .