Journal of Function Spaces
Volume 2016 (2016), Article ID 7827040, 16 pages
http://dx.doi.org/10.1155/2016/7827040

Research Article

Fixed Point Theorems for Cyclic Contractions in -Algebra-Valued -Metric Spaces

Chakkrid Klin-eam1,2 and Prondanai Kaskasem¹

¹Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
²Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand

Received 10 November 2015; Revised 28 December 2015; Accepted 6 January 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Chakkrid Klin-eam and Prondanai Kaskasem. 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.

We study fundamental properties of -algebra-valued -metric space which was introduced by Ma and Jiang (2015) and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in Ma and Jiang, 2015.

1. Introduction

Firstly, we begin with the basic concept of -algebras. A real or a complex linear space is algebra if vector multiplication is defined for every pair of elements of satisfying two conditions such that is a ring with respect to vector addition and vector multiplication and for every scalar and every pair of elements , A norm on is said to be submultiplicative if for all In this case is called normed algebra. A complete normed algebra is called Banach algebra. An involution on algebra is conjugate linear map on such that and for all . is called -algebra. A Banach -algebra is -algebra with a complete submultiplicative norm such that for all . -algebra is Banach -algebra such that . There are many examples of -algebra, such as the set of complex numbers, the set of all bounded linear operators on a Hilbert space , , and the set of -matrices, . If a normed algebra admits a unit , for all , and , we say that is a unital normed algebra. A complete unital normed algebra is called unital Banach algebra. For properties in -algebras, we refer to [1–3] and the references therein.

Let be a complete metric space. The well-known Banach’s contraction principle, which appeared in the Ph.D. dissertation of S. Banach in 1920, runs as follows: a mapping is said to be a contraction if there exists such that

Then, has a unique fixed point in which was published in 1922 [4]. Banach’s contraction principle has become one of the most important tools used for the existence of solutions of many nonlinear problems in many branches of science and has been extensively studied in many spaces which are more general than metric space by serveral mathematictians; see, for example, quasimetric spaces [5, 6], dislocated metric spaces [7], dislocated quasimetric spaces [8], -metric spaces [9–11], -metric spaces [12–14], metric-type spaces [15, 16], metric-like spaces [17], -metric-like spaces (or dislocated -metric spaces) [18, 19], quasi -metric spaces [20], and dislocated quasi--metric spaces [21]. Note that the Banach contraction principle requires that mapping satisfies the contractive condition that each point of and ranges of are positive real numbers. Consider the operator equation

where is subset of the set of linear bounded operators on Hilbert space , , and is positive linear bounded operators on Hilbert space . Then, we convert the operator equation to the mapping which is defined by

Observe that the range of mapping is not real numbers but it is linear bounded operators on Hilbert space . Therefore, the Banach contraction principle can not be applied with this problem. Afterward, does such mapping have a fixed point which is equivalent to the solution of operator equation? In 2014, Ma et al. [22] introduced new spaces, called -algebra-valued metric spaces, which are more general than metric space, replacing the set of real numbers by -algebras, and establish a fixed point theorem for self-maps with contractive or expansive conditions on such spaces, analogous to the Banach contraction principle. As applications, existence and uniqueness results for a type of integral equation and operator equation are given and were able to solve the above problem if satisfy .

Later, many authors extend and improve the result of Ma et al. For example, in [23], Batul and Kamran generalized the notation of -valued contraction mappings by weakening the contractive condition introduced by Ma et al. (the mapping is called -valued contractive type mappings) and establish a fixed point theorem for such mapping which is more generalized than the result of Ma et al.; in [24], Shehwar and Kamran extend and improve the result of Ma et al. [22] and Jachymski [25] by proving a fixed point theorem for self-mappings on -valued metric spaces satisfying the contractive condition for those pairs of elements from the metric space which form edges of a graph in the metric space. In 2015, Ma and Jiang [26] introduced a concept of -algebra-valued -metric spaces which generalize an ordinary -algebra-valued metric space and give some fixed point theorems for self-map with contractive condition on such spaces. As applications, existence and uniqueness results for a type of operator equation and an integral equation are given.

Generally, in order to use the Banach contraction principle, a self-mapping must be Lipschitz continuous, with the Lipschitz constant . In particular, must be continuous at all elements of its domain. That is one major drawback. Next, many authors could find contractive conditions which imply the existence of fixed point in complete metric space but not imply continuity. We refer to [27, 28] (Kannan-type mappings) and [29] (Chatterjea-type mapping).

Theorem 1 (see [27]). If is a complete metric space and mapping satisfies

where and , then has a unique fixed point.

Theorem 2 (see [29]). If is a complete metric space and mapping satisfies

where and , then has a unique fixed point.

In 2003, Kirk et al. [30] introduced the following notation of a cyclic representation and characterized the Banach contraction principle in context of a cyclic mapping as follows.

Theorem 3. Let be nonempty closed subsets of a complete metric space . Assume that a mapping satisfies the following conditions:

(i) for all and .(ii)There exists such that for all and for

Then, has a unique fixed point.

In 2011, Karapinar and Erhan [31] introduced Kannan-type cyclic contraction and Chatterjea-type cyclic contraction. Moreover, they derive some fixed point theorems for such cyclic contractions in complete metric spaces as follows.

Theorem 4 (fixed point theorem for Kannan-type cyclic contraction). Let and be nonempty subsets of metric spaces and a cyclic mapping satisfies

where . Then, has a unique fixed point in .

Theorem 5 (fixed point theorem for Chatterjea-type cyclic contraction). Let and be nonempty subsets of a metric spaces and a cyclic mapping satisfies

where . Then, has a unique fixed point in .

The purpose of this paper is to study fundamental properties of -algebra-valued -metric space which was introduced by Ma and Jiang [26] and give some fixed point theorems for cyclic mapping with contractive and expansive condition on such space analogous to the results presented in [26].

2. Preliminaries

In this section, we recollect some basic notations, defintions, and results that will be used in main result. Firstly, we begin with the concept of -metric spaces.

Definition 6 (see [12, 13]). Let be a nonempty set. A mapping is called -metric if there exists a real number such that, for every , we have

(i),(ii) if and only if ,(iii),(iv)

In this case, the pair is called a -metric space.

The class of -metric spaces is larger than the calass of metric spaces, since a -metric space is a metric when in the fourth condition in the above definition. There exist many examples in some work showing that the class of -metric is efficiently larger than those metric spaces (see also [12, 14, 32, 33]).

Example 7 (see [12]). The set with , where , together with the function ,

where , is a -metric space with coefficient . Observe that the result holds for the general case with , where is a Banach space.

Example 8 (see [12]). The space of all real functions , , such that , together with the function

is a -metric space with .

Example 9 (see [33]). Let be a metric space and , where is natural numbers. Then, is a -metric with .

The notation convergence, compactness, closedness, and completeness in -metric space are given in the same way as in metric space.

Next, we give concept of spectrum of element in -algebra .

Definition 10 (see [3]). We say that is invertible if there is an element such that . In this case, is unique and written . The set

is a group under multiplication. We define spectrum of an element to be the set

Theorem 11 (see [3]). Let be a unital Banach algebra and let be an element of such that . Then, and

Theorem 12 (see [3]). Let be a unital -algebra with a unit , then

(1),(2)For any , (3)For any ,

All over this paper, means a unital -algebra with a unit . is set of real numbers and is the set of nonnegetive real numbers. is matrix with entries .

Definition 13 (see [3]). The set of hermitain elements of is denoted by ; that is, . An element in is positive element which is denoted by , where means the zero element in if and only if and is a subset of nonnegative real numbers. We define a partial ordering by using definition of positive element as if and only if . The set of positive elements in is denoted by .

The following are definitions and some properties of positive element of a -algebra .

Lemma 14 (see [3]). The sum of two positive elements in a -algebra is a positive element.

Theorem 15 (see [3]). If a is an arbitrary element of a -algebra , then is positive.

We summarise some elementary facts about in the following results.

Theorem 16 (see [3]). Let be a -algebra:

(1)The set is closed cone in [a cone in a real or complex vector space is a subset closed under addition and under scalar multiplication by ].(2)The set is equal to .(3)If , then .(4)If is unital and and are positive invertible elements, then .

Theorem 17 (see [3]). Let be a -algebra. If and , then for any both and are positive elements and .

Lemma 18 (see [3]). Suppose that is a unital -algebra with a unit :

(1)If with , then is invertible and .(2)Suppose that with and ; then, .(3)Define Let ; if with and is invertible operator, then

Definition 19 (see [34]). A matrix is Hermitian if , where is a conjugate transpose matrix of . A Hermitian matrix is positive definite if for all nonzero , and it is positive semidefinite if for all nonzero

In 2014, Ma et al. [22] introduced the concept of -algebra-valued metric space by using the concept of positive elements in . The following is definition of -algebra-valued metric.

Definition 20 (see [22]). Let be a nonempty set. A mapping is called -algebra-valued metric on if it satisfies the following conditions:

(1) for all .(2) if and only if .(3) for all .(4) for all .

Then, is called a -algebra-valued metric on and is called a -algebra-valued metric space.

We know that range of mapping in metric space is the set of real numbers which is -algebra; then, -algebra-valued metric space generalizes the concept of metric spaces, replacing the set of real numbers by . In such paper, Ma et al. state the notation of convergence, Cauchy sequence, and completeness in -algebra-valued metric space. For detail, a sequence in a -algebra-valued metric space is said to converge to with respect to if for any there is such that for all We write it as A sequence is called a Cauchy sequence with respect to if for any there is such that for all The is said to be a complete -algebra-valued metric space if every Cauchy sequence with respect to is convergent. Moreover, they introduce definition of contractive and expansive mapping and give some related fixed point theorems for self-maps with -algebra-valued contractive and expansive mapping, analogous to Banach contraction principle. The following is the definition of contractive mapping and the related fixed point theorem.

Definition 21 (see [22]). Suppose that is a -algebra-valued metric space. A mapping is called -algebra-valued contractive mapping on , if there is an with such that

Theorem 22 (see [22]). If is a complete -algebra-valued metric space and satisfies Defintion 21, then has a unique fixed point in .

In the same way, the concept of expansive mapping is defined in the following way.

Definition 23 (see [22]). Let be a nonempty set. A mapping is a -algebra-valued expansive mapping on , if satisfies

(1),(2), for all ,

where is an invertible element and

The following is the related fixed point theorem for -algebra-valued expansive mapping.

Theorem 24 (see [22]). Let be a complete -algebra-valued metric space. If a satisfies Defintion 23, then has a unique fixed point in .

3. Fundamental Properties of -Algebra-Valued -Metric Spaces

In this section, we begin with the concept of -algebra-valued -metric space which was introduced by Ma and Jiang [26] as follows.

Definition 25 (see [26]). Let be a nonempty set. A mapping is called -algebra-valued -metric on if there exists such that satisfies following conditions:

(1) for all .(2) if and only if .(3) for all .(4) for all .

Then, is called a -algebra-valued -metric space.

Remark 26. If , then a -algebra-valued -metric spaces are -algebra-valued metric spaces. In particular, if is set of real numbers and , then the -algebra-valued -metric spaces is the metric spaces.

Definition 27 (see [26]). Let be a -algebra-valued -metric space. A sequence in is said to converge to if and only if for any there exists such that, for all , . Then, is said to be convergent with respect to and is called limit point of . We denote it by
A sequence is called a Cauchy seqeunce with respect to if and only if for any there exists such that, for all , .
We say is a complete -algebra-valued -metric space if every Cauchy sequence with respect to is convergent sequence with respect to

The following is an example of complete -algebra-valued -metric space.

Example 28 (see [26]). Let and let . Define

where and for all are constants and is a natural number such that .  A norm on is defined by

where . The involution is given by , conjugate transpose of matrix :

It is easy to verify is a -algebra-valued -metric space and is a complete -algebra-valued -metric space be completeness of .

Proof. An element is positive element; denote it by

We define a partial ordering on as follows:

where mean the zero matrix in . Firstly, it clears that is partially order relation. Next, we show that is a -algebra-valued -metric space. Let It is easy to see that satifies conditions (), (), and () of Definition 25. We will only show condition () where with

Since function is convex function for all and , this implies that

and hence for all . We substitute and ; then,

Hence, setting and , we obtain that

implies that each eigenvalue of is nonnegative. Since each eigenvalue of a positive semidefinite matrix is a nonnegative real number, we have that is positive semidefinite; that is, , that is, , where and by . But is impossible for all . Hence, is -algebra-valued -metric spaces but not -algebra-valued metric spaces.
Finally, we show that is a complete -algebra-valued -metric space. Suppose that is a Cauchy sequence with respect to Then, for any , there exists such that for all ; that is,

for all Therefore,

for all Hence, is a Cauchy sequnce in . By completeness of , there exists such that ; that is, . Then, we have that

converges to 0 as . Therefore, is convergent with respect to and converging to , so is a complete -algebra-valued -metric space.

Next, we disscus some fundamental properties of -algebra-valued -metric spaces.

Theorem 29. Let be -algebra-valued -metric space. If is a convergent sequence with respect to , then is Cauchy sequence with respect to .

Proof. Assume that is a convergent sequence with respect to ; then, there exists a such that Let , there is such that, for all ,

For , we get that

By Theorem 16, for , we have

This implies that is Cauchy sequence with respect to

Definition 30. A subset of a -algebra-valued -metric space is bounded with respect to if there exists and a nonnegetive real number such that

Theorem 31. Let be a -algebra-valued -metric space and let be a sequence in and . Then,

(1) if and only if ,(2)a convergent sequence in is bounded with respect to and its limit is unique,(3)a Cauchy sequence in is bounded with respect to .

Proof. (1) Assume that . Let is given. Then, there exists such that

This implies that Conversely, assume that . Then, for any , there exists such that

that is,
(2) Let be a convergent sequence with respect to . Suppose that . Then, taking , we can find such that

Let . Setting . This implies that

Next, suppose that and . Consider, ; by Theorem 16, we have

From (), letting , we obtian that ; that is
(3) Assume that is a Cauchy sequence with respect to . In particular, ; there exists such that

Let , . Then,

Set . Then, we get that

Theorem 32. Let be a convergent sequence in a -algebra-valued -metric space and . Then, every subsequence of   is convergent and has the same limit .

Proof. Let be given. Then, there exists such that

Since is an increasing sequence of natural numbers, it is easily proved (by induction) that . Hence, if , we also have so that

Therefore, subsequence also converges to .

Theorem 33. Let be a -algebra-valued -metric space. Then, every subsequence of a Cauchy sequence is Cauchy sequence.

Proof. Let be a subsequence of Cauchy sequence in a -algebra-valued -metric space. Then, for every , there is such that, for all , we have . Similar to the facts in proof of previous theorem, we have and Hence, we obtain that . Therefore, is Cauchy sequence.

Theorem 34. Let be a -algebra-valued -metric space and let be a Cauchy sequence with respect to . If contains its convergent subsequence, then is convergent sequence.

Proof. Let . Since is a Cauchy sequence with respect to , there exists an such that

Let be a convergent subsequence of and . Then, there exists such that

Let For , we have

By Theorem 16, we also have

Therefore, as .

Theorem 35. Let be a -algebra-valued -metric space. Suppose that and are convergent with respect to and converge to and , respectively. Then, converges to

Proof. Let Since and , there exist such that

Since , by Theorem 16, we have

Therefore, .

Theorem 36. Let be a -algebra-valued -metric space. Suppose that and are convergent with respect to and converge to and , respectively. Then,

In particular, if , then we have Moreover, for any , we have

Proof. By defintion of -algebra-valued -metric space, it easy to see that

Using Theorem 16, we have

Taking the lower limit as in the first inequality and the upper limit as in the second inequality, this completes the first result. In particular, if , we have

Taking the limit as in this inequality, we obtain that . Since

by Theorem 16, we have

Again taking the lower limit as in the first inequality and the upper limit as in the second inequality, we obtain the second desired result.

Definition 37. Let be a -algebra-valued -metric space. A subset of is called a closed set if a sequence in and with respect to imply .

4. Fixed Point Theorems for Cyclic Contractions

Theorem 38. Let and be nonempty closed subset of a complete -algebra-valued -metric space . Assume that is cyclic mapping that satisfies

where with . Then, has a unique fixed point in .

Proof. Let be any point in . Since is cyclic mapping, we have and . Using the contractive condition of mapping , we get

For all , we have

where Consider, for any such that ; then,

From Theorem 16, we have

Since , we have as . Therefore, is Cauchy sequence with respect to . By the completeness of , there exists an element such that .
Since is a sequence in and is a sequence in , we obtain that both sequences converge to the same limit . Since and are closed set, this implies that .
Next, we will complete the proof by showing that is a unique fixed point of . Since

by Theorem 16, we obtain that

We have ; that is, is a fixed point of .
Suppose that is fixed point of and . Since

we have

This is a contradiction. Therefore, which implies that the fixed point is unique.

Example 39. Let be a set of real numbers and with , where are entries of the matrix . Then, is a -algebra-valued -metric space with , where the involution is given by ,

and partial ordering on is given as

Define a mapping by

It is clear that is not continuous at all elements of . Therefore, Theorem 22 cannot imply the existence of fixed point of mapping .
Suppose that and . Firstly, we will show that is cyclic mapping. Let ; that is, . Then, . Again, let ; that is, . Indeed, we consider

this implies that . For any and , since and , we have and . Hence, we obtain that

Next, we consider

Then, we have

where . Then, . Thus, satisfies contraction of Theorem 38 implying that has a unique fixed point in ; that is,

Corollary 40. Suppose that is a -algebra-valued -metric space. Assume that is called a -algebra-valued -contractive mapping on ; that is, satisfies

where with . Then, has a unique fixed point in .

Proof. Putting , by Theorem 38, this implies that has a unique fixed point in

Theorem 41. Suppose that is a complete -algebra-valued -metric space. Assume that a mapping satisfies

(1);(2) for all ,

where is an invertible element and such that is a -algebra-valued -expansive mapping on . Then, has a unique fixed point in .

Proof. We will begin to prove this theorem by showing that is injective. Let be an element in such that ; that is, . Assume that . We have

This implies that Since is invertible, we have which leads to contradiction. Thus, is injective. By the first condition of mapping , we obtain that is bijective which implies that is invertibe and is bijective.
Next, we will show that has a unique fixed point in . In fact, since is -algebra-valued -expansive and invertible mapping, we substitute with , in the second condition of , respectively, which implies that

That is

Since and are positive elements in , and . By condition (2) of Theorem 12 and Theorem 17, we have

Therefore, is -contractive mapping. Using Corollary 40, there exists a unique such that , which means it has a unique fixed point such that

Theorem 42 (cyclic Kannan-type). Let and be nonempty closed subset of a complete -algebra-valued -metric space . Assume that is cyclic mapping that satisfies

where with . Then, has a unique fixed point in .

Proof. Without loss of generality, we can assume that . Since and , by the second condition of Lemma 18, we have .
Let be any element in . Since is cyclic mapping, we have and . Consider

that is,

Since and , by the first condition of Lemma 18, we have that is invertible and . From the third condition of Lemma 18, we have

Similarly, we get that

Since and , the second condition of Lemma 18, we have

that is,

Hence,

Continuing this process, we have

where and . Next, we will show that is Cauchy sequence with respect to . Consider for any and that we have

From Theorem 16, we get that

Consider

Therefore, as . Therefore, is Cauchy sequence with respect to . By the completeness of , there exists an element such that .
Since is a sequence in and is a sequence in , we obtain that both sequences converge to the same limit . Since and are closed set, this implies . Next, we will show that is a unique fixed point of . Consider

by Theorem 16 and submultiplicative, we obtian that

Letting , we get that

and so

This implies that ; that is, and so . That is, is fixed point of . Now if is another fixed point of and , then

which leads to contradiction. Therefore, ; we complete the proof.

Example 43. Let and with where are entries of the matrix . Then, is a -algebra-valued -metric space with , where the involution is given by :

and partial ordering on is given as

Suppose that and . Define a mapping by . Firstly, we will show that is cyclic mapping. Let be an element in ; that is, . Then, implies . Similarly, let , so . Then, . Hence,
For any and , we consider

Then, we have

where . Then, . Thus, satisfies contraction of Theorem 42 implying that has a unique fixed point in ; that is,

Theorem 44 (cyclic Chatterjea-type). Let and be nonempty closed subset of a complete -algebra-valued -metric space . Assume that is cyclic mapping that satisfies

where with . Then, has a unique fixed point in .

Proof. Without loss of generality, we can assume that . Since and , by the second condition of Lemma 18, we have .
Let be any element in , Since is cyclic mapping, we have and . Consider

that is,

Since and , from the second condition of Lemma 18, we get that . Since and , by the first condition of Lemma 18, we have and with . From the third condition of Lemma 18, we have

Similarly, we get that

Since and , the second condition of Lemma 18, we have

that is,

Hence,

Continuing this process, we have

where and . Next, we will show that is Cauchy sequence with respect to . Consider for any and ; we have

From Theorem 16, we get that

Consider

Therefore, as . Therefore, is Cauchy sequence with respect to . By the completeness of , there exists an element such that .
Since is a sequence in and is a sequence in , we obtain that both sequences converge to the same limit . Since and are closed set, this implies .
Next, we will complete the proof by showing that is a unique fixed point of . Since

by Theorem 16, we have

Letting , we get that

and so

This implies that ; that is, and so . That is, is fixed point of . Now if is another fixed point of and , then

From Theorem 16, we get that

which leads to a contradiction. Therefore, which implies that the fixed point is unique.

Example 45. Let and with , where are entries of the matrix . Then, is a -algebra-valued -metric space with , where the involution is given by :

and partial ordering on is given as

Suppose that and . Define a mapping by . Firstly, we will show that is cyclic mapping. Let ; that is, . Then, implies . Similarly, let , so . Then, . Hence,
Now, we will show that satisfies the contraction of Theorem 44. Consider

and so

Then, we have

where . Then, . Thus, satisfies contraction of Theorem 44 implying that has a unique fixed point in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Science Achievement Scholarship of Thailand and Faculty of Science, Naresuan University.

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