Some Coincidence and Common Fixed-Point Results on Cone -Metric Spaces over Banach Algebras with Applications to the Infinite System of Integral Equations

Abstract

In this article, common fixed-point theorems for self-mappings under different types of generalized contractions in the context of the cone -metric space over the Banach algebra are discussed. The existence results obtained strengthen the ones mentioned previously in the literature. An example and an application to the infinite system of integral equations are also presented to validate the main results.

1. Introduction and Preliminaries

Gähler [1] proposed the definition of 2-metric spaces as a generalization of an ordinary metric space. He defined that geometrically represents the area of a triangle with vertices . 2-metric is not a continuous function of its variables. This was one of the key drawbacks of the 2-metric space while an ordinary metric is a continuous function.

Keeping these drawbacks in mind, Dhage [2], in his PhD thesis, proposed a concept of the -metric space as a generalized version of the 2-metric space. He also defined an open ball in such spaces and studied other topological properties of the mentioned structure. According to him, represented the perimeter of a triangle. He stated that the -metric induced a Hausdorff topology and in the -metric space, the family of all open balls forms a basis for such topology.

Later, Mustafa and Sims [3] illustrate that the topological structure of Dhage’s -metric is invalid. Then, they revised the -metric and expanded the notion of a metric in which each triplet of an arbitrary set is given a real number called as the -metric space [4].

In addition, the definition of the -metric space is proposed by Sedghi et al. [5] as an updated version of Dhage’s -metric space. Later, they analyzed and found that -metric and -metric have shortcomings. Later, they proposed a new simplified sturcture called the -metric space [6].

On the other hand, by swapping the real numbers with the ordered Banach space and established cone metric space, Huang and Zhang [7] generalized the notion of a metric space and demonstrated some fixed-point results of contractive maps using the normality condition in such spaces. Rezapour and Hamlbarani [8] subsequently ignored the normality assumption and obtained some generalizations of the Huang and Zhang [7] results. However, it should be noted that the equivalence between cone metric spaces and metric spaces has been developed in recent studies by some scholars in the context of the presence of fixed points in the mapping involved. Liu and Xu [9] proposed the concept of a cone metric space over the Banach algebra in order to solve these shortcomings by replacing the Banach space with the Banach algebra. This became an interesting discovery in the study of fixed-point theory since it can be shown that cone metric spaces over the Banach algebra are not equal to metric spaces in terms of the presence of the fixed points of mappings. Among these generalizations, by generalizing the cone 2-metric spaces [10] over the Banach algebra and -metric spaces [11], Fernandez et al. [12] examined cone -metric spaces over the Banach algebra with the constant . In the setting of the new structure, they proved some fixed-point theorems under different types of contractive mappings and showed the existence and uniqueness of a solution to a class of system of integral equations as an application.

Recently, in 2020, Islam et al. [13] initiated the notion of the cone -metric space over the Banach algebra with constant which is a generalization of the definition of Fernandez et al. [12]. They proved some fixed-point theorems under -admissible Hardy-Rogers contractions which generalize many of the results from the existence literature, and as an application, they proved results which guarantee the existence of solution of an infinite system of integral equations.

In 1973, Hardy and Rogers [14] proposed a new definition of mappings called the contraction of Hardy-Rogers that generalizes the theory of the Banach contraction and the theorem of Reich [15] in a metric space setting. For other related work about the concept of Hardy-Rogers contractions, see, for instance, [16, 17] and the references therein.

We recollect certain essential notes, definitions required, and primary results consistent with the literature.

Definition 1 (see [18]). Consider the Banach algebra which is real, and the multiplication operation is defined under the below properties (for all ):
(a₁)
(a₂) and
(a₃)
(a₄)
Unless otherwise stated, we will assume in this article that is a real Banach algebra. If occurs, we call the unit of , so that . We call a unital in this case. If an inverse element exists, the element is said to be invertible, so that . The inverse of in such case is unique and is denoted by . We require the following propositions in the sequel.

Proposition 2 (see [18]). Consider the unital Banach algebra with unit , and let be the arbitrary element. If the spectral radius , i.e., then is invertible. In fact,

Remark 3. We see from [18] that for all with unit .

Remark 4 (see [19]). In Proposition 2, by replacing “” by , then the conclusion remains the same.

Remark 5 (see [19]). If , then as .

Definition 6. Consider the Banach algebra with unit element , zero element , and . Then, is a cone in if:
(b₁)
(b₂)
(b₃) for all
(b₄)
(b₅)
Define the partial order relation in w.r.t by if and only if ; 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. If is least and positive in the above, then it is the normal constant of [7].

Definition 7 (see [7, 9]). Let and be the mapping:
(c₁) For all , and if and only if
(c₂) For all ,
(c₃) For all ,
Then, over the Banach algebra with cone metric is a cone metric space.
In [20], over the Banach algebra with constant , the cone -metric space is introduced as a generalization of the cone metric space over the Banach algebra while in Mitrovic and Hussain [16], over the Banach algebra with parameter , the concept of cone -metric spaces is introduced.

Definition 8 (see [16]). Let and be the mapping:
(e₁) For all , and if and only if
(e₂) For all ,
(e₃) There is , , and for all ,
Then, over the Banach algebra with cone -metric is a cone -metric space. Note that if we take , then it reduces to the cone metric space over the Banach algebra .

Definition 9 (see [1]). Let and :
(f₁) There is a point for such that , if at least two of are not equal
(f₂) if and only if at least two of are equal
(f₃) for all , where stands for all permutations of
(f₄) for all
Then, is a -metric space with -metric .

Definition 10 (see [12]). Let , , and be a real number:
(g₁) There is a point for such that if at least two of are not equal
(g₂) if and only if at least two of are equal
(g₃) for all , where stands for all permutations of
(g₄) for all
Then, over the Banach algebra with parameter is a cone -metric space. By taking , it became a cone 2-metric space. We refer the reader to [21] for other details about the cone 2-metric space over the Banach algebra .
Islam et al. [13] initiated the concept of the cone -metric space over the Banach algebra with parameter .

Definition 11 (see [13]). Let and :
(h₁) There is a point for such that if at least two of are not equal
(h₂) iff at least two of are equal
(h₃) for all , where stands for all permutations of
(h₄) for all with ,
Then, over the Banach algebra with parameter is a cone -metric space. By taking , it reduces to a cone 2-metric space.

Example 12 (see [13]). Let . For each , . Then, is a Banach algebra with unit as a constant function, and multiplication is defined pointwise. Let . Then, is a cone in . Let . For all , define as where is the square of the area of the triangle formed by and defined by . Consider That is, , showing that is not a cone 2-metric, because for with , , and . But for is a cone -metric space over the Banach algebra .

Definition 13 (see [13]). Consider that is a cone -metric space over the Banach algebra with , and let be a sequence in ; then,
(i₁) is said to converge to if for every there is such that for all . That is, (i₂) If for every there exists such that for all , then we say that is a Cauchy sequence
(i₃) is complete if every Cauchy sequence is convergent in

Definition 14 (see [22]). Let a sequence be in ; then, sequence is a -sequence, if for each there is such that for all .

Lemma 15 (see [23]). Consider the Banach algebra and . Also, consider a -sequence in and where is arbitrary; then, is a -sequence.

Lemma 16 (see [23]). Consider the Banach algebra and . Let and be -sequences in . Then, for arbitrary , we have which is also a -sequence.

Lemma 17 (see [23]). Consider the Banach algebra and . Let such that as . Then, is a -sequence.

Lemma 18 (see [19]). Let be the Banach algebra, be their unit element, and . If commutes, then
(k₁)
(k₂)

Lemma 19 (see [19]). Consider the Banach algebra and . If , then

Lemma 20 (see [24]). Consider the Banach algebra , is their unit element, and . Let and . If , then is a -sequence.

Lemma 21 (see [25]). Let be a cone.
(l₁) If , , and , then
(l₂) If , , and , then
(l₃) For any , with and

Lemma 22 (see [26]). Consider the Banach algebra and .
(n₁) If and , then
(n²) If and for , then

Lemma 23 (see [21]). Consider the Banach algebra and . Let , and suppose that is an arbitrary given vector such that for any , then for any .

Lemma 24 (see [27]). Consider the Banach algebra and . Let for each ; then, .

Lemma 25 (see [27]). Consider the Banach algebra and . If as , then for each , there is with , such that .

Definition 26 (see [28]). Let and be self-maps of a set . If for some , then for and , is known as a coincidence point and is known as a point of coincidence of and .

Definition 27 (see [29]). The mappings are said to be weakly compatible, whenever and for any .

Lemma 28 (see [28]). Let the mappings and be weakly compatible self-maps of a set . If and have a unique point of coincidence , then is the unique common fixed point of and .

2. Main Results

In this section, in the framework of the cone -metric space over Banach algebras with parameter , we prove some common fixed-point results.

Proposition 29. Let over the Banach algebra be the complete cone -metric space and be a cone in . If a sequence in converges to , then we have the following: (i) is a -sequence for all (ii)For any , is a -sequence for all

Proof. Since the proof is easy, so we left it.
Now, we here state and prove our first main results which generalize and extend many of the conclusions from the existence literature.☐

Theorem 30. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Set , , , and , . Then, by (8), we have Choose to be arbitrary. Since and for each , there exists such that and . Continuing this process, we can define by and .
Denote and for . Now, we show that is a Cauchy sequence.
From (9), we know that It means that . Therefore, (10) becomes that is, where . Similarly, it is not difficult to show that As , therefore (14) becomes that is, where . Set , and using inequalities (13) and (16), we deduce that In this case, for all , similar to (17) and (18), we have Therefore, for all , , and using the above inequalities, we have Continuing this process, we get That is, where .
Similarly, we have where .
Similar to the above, one can easily get that where and .
From Lemmas 18 and 19, we have that Since , therefore in the light of Remark 5 and Lemma 25, as , and so, for every , there exists such that for all ; that is, the sequence is a -sequence. By Lemma 15, the sequences , , , and are also -sequences. Therefore, for any with , there exists such that, for any , we have for all and for all . Thus, from Lemma 24, it means that . This implies that is a Cauchy sequence in .
If is complete for each , there exists such that So we can find a such that (if is complete for each , there exists ; then, the conclusion remains the same). Now, we show that . By (9), we have That is, Therefore, it follows from Proposition 29 and Lemmas 15 and 16 that where is a -sequence in . In addition, from Proposition 2 and it means that is invertible. In this case, we have for any , which together with Lemma 23 implies that , for any , , and as is invertible. Therefore, by Lemma 24, we have for any . Namely, for any . That is, .
At the same time, as , there exists such that .
Now, we show that . From (9), we have that That is, Hence, by Lemma 20, we know that , and so . Therefore, and .
Next, if we assume is complete for each , there exists such that So, we can find such that (if is complete for each , there exists ; then, the conclusion remains the same).
Now, we show that . By (9), we get that That is, Therefore, it follows from Proposition 29 and Lemmas 15 and 16 that where is a -sequence in . In addition, from Proposition 2 and it means that is invertible. In this case, we have for any , which together with Lemma 23 implies that for any , , and as is invertible. Therefore, by Lemma 24, we have for any . Namely, for any . That is, .
At the same time, as , there exists such that . Now, we show that . From (9), we have That is, Hence, by Lemma 20, we know that , and so . Therefore, and .
Finally, we show that and , , and have a unique point of coincidence in . Assume that there is another point such that ; then, That is, Hence, by Lemma 20, we have that , and so ; that is, is the unique point of coincidence of and .
Similarly, we also have which is the unique point of coincidence of and by induction.
So, according to Lemma 28, is the unique common fixed point of and for each . Therefore, is the unique common fixed point of , and .
Now, it is left to show that is the unique common fixed point of , , , and .
As , so we have , that is, . But is unique; therefore, for .
Also, as , so we have , that is, . But is unique; therefore, for . Similarly, and for . Thus, the four families of mappings , , , and have a unique common fixed point.☐

Remark 31. Theorem 30 of this paper extends and improves Theorem 2.1 of [30] from cone metric spaces to cone -metric spaces; also, it extends and improves Theorem 3.2 of [17] and Theorem 3.1 of [31] from one family and two families, respectively, to four families of mappings.

We obtain a series of new common fixed-point results using Theorem 30 for four families of mappings in the context of cone -metric spaces over Banach algebras, which generalize and improve many known results from the existence literature.

Corollary 32. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Let in Theorem 30.☐

Corollary 33. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Taking in Theorem 30, one can get the desired result.☐

Remark 34. We can have Theorem 3.1 in [21], when and are the same mapping and and are the identity mappings. Therefore, Theorem 3.1 of [21] is a special case of Corollary 33. Also, Corollary 33 of this paper generalizes Theorem 2.1 of [10] from the cone 2-metric space to the cone -metric space and extends Theorem 6.1 in [12].

Corollary 35. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and is a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. One can the result taking , and in Theorem 30.☐

Remark 36. Corollary 35 of this paper extends Theorem 6 in [32]; therefore, Theorem 6 in [32] is a special case of Corollary 35.

Corollary 37. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. One can get the result taking , and in Theorem 30.☐

Corollary 38. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Let in Theorem 30.☐

Corollary 39. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with , , and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Let in Theorem 30.☐

Corollary 40. Let over the Banach algebra be a cone -metric space with and be a cone in . Let , , , and be four families of self-mappings on . For all , if a sequence exists of nonnegative integers, such that for all , where with and commute. If , , and one of , , , and are a complete subspace of for each , then , , , and have a unique point of coincidence in . Moreover, if and are weakly compatible, respectively, then , , , and have a unique common fixed point.

Proof. Let in Theorem 30.☐

From the above corollary, we obtain the following.

Corollary 41. Let over the Banach algebra be a complete cone -metric space with and be a cone in . Let be the family of self-mapping on . For all and for all , where with and commute. Then, have a unique common fixed point.

Proof. Taking , , and which are identity mappings in Corollary 40, then we can obtain the required result.☐

We finish this section with an example that will demonstrate the consequence of Theorem 30.

Example 42. Let . For each , . The multiplication is defined by . Then, is a Banach algebra with unit element . Let . Then, is a cone in .

Let and define as follows: where is the square of the area of the triangle . We have

That is, , which shows that is not a cone 2-metric, because for with , , and . But for the parameter is a cone -metric space over the Banach algebra .

Now, we define mappings by

We have

We define mappings () by

We have

Similarly, we define mappings () by

Then, it is not difficult to show that and . Choose , , and . Clearly,

also and .

Now, considering the contractive condition (8), we have that is, which means that that is,

which shows that , and so , which is true for all . Hence, condition (8) is true for all and , where , , , and . All other conditions of Theorem 30 are satisfied. By Theorem 30, , , , and have a unique common fixed point for all .

3. Application to the Infinite System of Integral Equations

We give here a couple of auxiliary facts that will be needed in our further considerations. Let with norm be a real Banach algebra. Let , and denote by the space consisting of all continuous functions defined on interval with values in the Banach algebra . The space will be equipped with .

Let and define by where and for all . Then, is a complete cone -metric space over the Banach algebra.

We consider the infinite system of integral equations of the form where . Let . We redefine the above infinite system of integral equations as

for all and . Clearly, by using Corollary 41, the existence of solution to (65) is equivalent to the existence of a common fixed point of .

We assume that (i) are continuous for each (ii) are continuous and for each (iii) are continuous for each such that

for all and .

Theorem 43. Under the assumptions (i)–(iii), the infinite system of integral equations (65) has a unique solution in .

Proof. Take with norm , and multiplication is defined by the following way: Let . It is clear that is a normal cone and is a Banach algebra with unit element .
Consider the family of mapping defined by (66). Let . From (64), we deduce that Therefore, Now, all the assumptions of Corollary 41 are fulfilled and the family of mapping has a unique common fixed point in , which means that the infinite system of integral equations (65) has a unique solution in .☐

Data Availability

Data are available upon request or included within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors make equal contributions and read and supported the last original copy.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.