Journal of Function Spaces
Volume 2017 (2017), Article ID 5054603, 6 pages
https://doi.org/10.1155/2017/5054603

Research Article

Fixed Point Results for Weak φ-Contractions in Cone Metric Spaces over Banach Algebras and Applications

Biwen Li¹ and Huaping Huang²

¹School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
²School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Correspondence should be addressed to Huaping Huang; mathhhp@163.com

Received 8 October 2017; Accepted 26 November 2017; Published 13 December 2017

Academic Editor: Manuel De la Sen

Copyright © 2017 Biwen Li and Huaping Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using a nontrivial proof method, the purpose of this paper is to obtain some fixed point results for weak -contractions in cone metric spaces over Banach algebras. Several examples and applications to the existence and uniqueness of a solution to two classes of equations are also given.

1. Introduction

Fixed point theory is without doubt one of the most important tools of modern mathematics as attested by Browder [1], who is considered as one of the pioneers in the development of the nonlinear functional analysis. The flourishing field of fixed point theory started in the early days of topology through the work of Poincaré [2], Lefschetz-Hopf, and Leray-Schauder, for example. Fixed point theory is widely used in different areas such as ordinary and partial differential equations, economics, logic programming, convex optimization, and control theory. In metric fixed point theory, successive approximations are rooted in the work of Cauchy, Fredholm, Liouville, Lipschitz, Peano, and Picard. It is well accepted among experts of this subarea that Banach is responsible for laying the ground for an abstract framework well beyond the scope of elementary differential and integral equations. In 1922, Banach [3] proved the most influential and celebrating fixed point theorem, which was Banach fixed point theorem (i.e., Banach contraction principle). Since then, fixed point theory has had a rapid development. In [4], Huang and Zhang introduced cone metric space and generalized Banach fixed point theorem in such spaces. Subsequently, many people were interested in fixed point results in cone metric spaces (see [5–10] and the references therein). In [11–13], Rus and Berinde introduced the notion of -contraction and also generalized Banach fixed point theorem in usual metric spaces. Recently, Liu and Xu [14] introduced the concept of a cone metric space over Banach algebra, which is an interesting generalization of classic metric spaces. From then on, many authors focused on the investigation of fixed point in such spaces (see [15–20]). Stimulated and motivated by the previous work, throughout this paper, we introduce weak -contractions in the setting of cone metric spaces over Banach algebras and present some fixed point theorems for weak -contractions. Our results improve and weaken the conditions of the vector-valued version of Banach fixed point theorem. To the best of our knowledge, our methods are new. In addition, by using our results, we give the existence and uniqueness of a solution to elementary equations and to integral equations.

The following notions and facts will be needed in this paper.

Definition 1 (see [12]). A function is called a comparison if it satisfies the following two conditions:

(1) is monotone nondecreasing; that is, .(2) converges to as .

Remark 2. By Definition 1, it is sufficient to get that for each , , and .

Definition 3 (see [12]). Let be a metric space. A mapping is called a -contraction if there exists a comparison such that

The following theorem generalizes Banach fixed point theorem.

Theorem 4 (see [12]). Let be a metric space and be a -contraction. Then has a unique fixed point in . Moreover, for any , the iterative sequence converges to the fixed point.

In the following, we consider our results in the framework of cone metric spaces over Banach algebras. For the reader who is unfamiliar with cone metric space over Banach algebra, we recall some of its notions and results as follows.

Definition 5 (see [14]). Let be a Banach algebra with a unit and a zero element . A nonempty closed subset of is called a cone if the following conditions hold:

(1).(2).(3).(4).

A cone is called a solid cone if , where stands for the interior of .
On this basis, we define a partial ordering with respect to by if and only if . We shall write to indicate that int. We shall also write as the norm on . A cone is called normal if there is a number such that, for all , implies .
In the sequel, unless otherwise specified, we always suppose that is a Banach algebra with a unit , is a solid cone in , and and are partial orderings with respect to . We always write and as the set of all natural numbers and the set of all real numbers, respectively.

Definition 6 (see [14]). Let be a nonempty set and be a Banach algebra. A mapping is called a cone metric if it satisfies

(i), ;(ii);(iii).

In this case, the pair is called a cone metric space over Banach algebra.

Definition 7 (see [8]). A sequence in a Banach algebra is said to be a -sequence if, for each , there exists such that for all .
The introduction of -sequence is an interesting increase since, by using -sequence, many intricate concepts may be simplified. For example, Definitions  2–4 of [4] are abbreviated with the following.

Definition 8 (see [17]). Let be a cone metric space over Banach algebra and be a sequence in . We say that

(i) converges to if is a -sequence;(ii) is a Cauchy sequence if is a -sequence for ;(iii) is complete if every Cauchy sequence in is convergent.

Lemma 9 (see [9]). Let be a Banach algebra and . Then

(1) if or ;(2) if for each .

Lemma 10 (see [21]). Let be a Banach algebra with its unit . Then the spectral radius of equals .

Lemma 11 (see [16]). Let be a cone in a Banach algebra , and be two -sequences in , and be vectors; then is a -sequence in .

Lemma 12 (see [15]). Let be a cone and with . Then is a -sequence.

2. Main Results

In this section, we introduce weak -contractions in the framework of a cone metric space over Banach algebra and obtain some corresponding fixed point theorems. Moreover, we present some examples to illustrate the superiority of the results.

Definition 13. Let be a Banach algebra and be a cone in . A mapping is called a weak comparison if the following conditions hold:

(i) is nondecreasing with respect to ; namely, , .(ii) is a -sequence in .(iii)if is a -sequence in , then is also a -sequence in .

Remark 14. By Definition 13, it suffices to show that . Indeed, by (i) of Definition 13, we have . Since is a -sequence, then, by Lemma 9, it may be verified that .

Remark 15. If and , then Definition 13 is reduced to Definition 1. In other words, Definition 13 is a generalization of Definition 1.

The following examples are trivial, whose proofs are straightforward and are therefore omitted.

Example 16. Let be a Banach algebra, be a cone in , and . Take , where . Then by Lemmas 11 and 12, is a weak comparison.

Example 17. Let be a compact set of and , where denotes the set of all continuous functions on . Let and define a mapping by . Then is a weak comparison.

Definition 18. Let be a cone metric space over Banach algebra. Let be a cone and be a weak comparison. Then a mapping is called a weak -contraction if

Remark 19. By Remark 15, Clearly, Definition 18 generalizes Definition 3.

The following theorem generalizes Theorem 4 and its proof method is nontrivial.

Theorem 20. Let be a complete cone metric space over Banach algebra and be a weak -contraction. Then has a unique fixed point in . Moreover, for any , the iterative sequence converges to the fixed point.

Proof. Choose and put . For any , by (ii) of Definition 13, there exists such that . Making the most of (2), it is not hard to verify that

Since is a -sequence, then, by Lemma 9, is also a -sequence. Hence, there exists such that

Let

Choosing , by (3) and (5), it is established that

This means that is -invariant. Accordingly, for any , we have .
Making the best of (2), it ensures us that

Now that are -sequences, then, by Lemma 11, is a -sequence. Next, by Lemma 9, is also a -sequence. So, for the above , there exists such that

Denote , for ; let

where stands for the integer part. Because

we have

Then, by Lemma 9, we claim that is a Cauchy sequence. Since is complete, then there exists such that .
We prove that is the fixed point of . Actually, by (2), we arrive at

Now that is a -sequence, it follows immediately from (iii) of Definition 13 that is a -sequence. Thus, by Lemma 11, we speculate that is also a -sequence. Next by Lemma 9, it may be verified that ; that is, .
Finally, we prove that the fixed point is unique. Assume that is another fixed point of ; then, by (2) and the monotonicity of , we deduce that

Since is a -sequence, then, by Lemma 9, it is obvious that . As a result, .

Corollary 21. Let be a complete cone metric space over Banach algebra and be a cone in . Suppose that and is a mapping satisfying

If , then has a unique fixed point in . Moreover, for any , the iterative sequence converges to the fixed point.

Proof. Let ; then, by Theorem 20, we get the desired result.

Remark 22. Corollary 21 is called the vector-valued version of Banach fixed point theorem. It generalizes Theorem  2.1 of [14] because it deletes the assumption of normality of cones of Theorem  2.1 of [14].

The following example shows the superiority of Theorem 20.

Example 23. Under the conditions of Example 17, let and define a mapping by . Then is a complete cone metric space over Banach algebra. Define a mapping by

Clearly, is not a contraction; that is, there is no with such that the contractive condition of Theorem  2.1 of [14] holds.
However, for all , we have

By Example 17, we know that is a weak -contraction and then, by Theorem 20, has a unique fixed point.

3. Applications

In this section, we give some applications to prove the existence and uniqueness of a solution to some equations. First of all, we consider the following elementary equations:

where is a constant.

Theorem 24. The elementary equation (17) has a unique solution in .

Proof. Let and . Take . Define a norm of by . Define a multiplication of by

For any , define a mapping by

then is a complete cone metric space over Banach algebra. Define a mapping by

For all , by mean value theorem of differentials, there exists belonging to numbers between and , such that

Choose with . Clearly, is a weak comparison. Thus, is a weak -contraction. By Theorem 20, has a unique fixed point in . That is to say, the elementary equation (17) has a unique solution in .
Secondly, we prove an existence theorem for a solution of the following nonlinear integral equation by using our results in the previous section.

where ,   (the set of all continuous functions from into ), , and are given mappings.

Theorem 25. Let . If there exists with such that, for all and ,

then the integral equation (22) has a unique solution in .

Proof. Let with the same norm, the same multiplication, and the same cone as stated in the proof of Theorem 24. Then is a normal cone and is a Banach algebra with a unit . Let . We endow with the cone metric

for all . It is clear that is a complete cone metric space over Banach algebra . Define the mapping by

for all . Then the existence of a solution to (22) is equivalent to the existence of fixed point of . Indeed, by utilizing (23)–(25), we have

where and .
It is easy to get that is a weak comparison and is a weak -contraction. Accordingly, all the conditions of Theorem 20 are satisfied and then has a unique fixed point in . In other words, the integral equation (22) has a unique solution in .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors have equal contribution in writing this paper. All authors read and approved the final paper.

Acknowledgments

The research was partially supported by the Funding Program of Higher School Outstanding Youth Scientific and Technological Innovation Team in Hubei Province of China (no. T2014212).

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