Abstract
The notion of C-class functions in Banach algebras is introduced. By using such concept, a new fixed point theorem is established. An example to illustrate main theorem is given. Finally, applications of our main result to cyclic mappings and weak contraction type mappings are given.
1. Introduction and Preliminaries
Huang and Zhang [1] introduced cone metric spaces which are generalizations of metric spaces, and they extended Banach’s contraction principle to such spaces, whereafter many authors (for example, [2–6] and references therein) studied fixed point theorems in cone metric spaces.
Recently, Liu and Xu [7] introduced the notion of cone metric spaces over Banach algebras, which is a modification of the concept of cone metric spaces over real Banach spaces [1], and proved the existence of fixed points for mappings defined on such spaces, and they gave an example that fixed point results in metric spaces and in cone metric spaces are not equivalent.
Very recently, Chandok et al. [8] introduced the concept of TAC-contractive mappings by using the notions of C-class functions and cyclic ()-admissible mappings and established corresponding fixed point theorems in metric spaces.
In the paper, we introduce the notions of C-class functions and cyclic ()-admissible mappings in Banach algebras. By using such concepts, we introduce a new contraction. We obtain a new fixed point theorem and give an example to illustrate main result. Finally, we give applications of our main result to cyclic mappings and weak contraction type mappings in cone metric spaces over Banach algebras.
A Banach space is called a (real) Banach algebra (with unit) if there exists a multiplication that has the followings properties:
for all ,(1);(2) and ;(3);(4)there exists such that ;(5);(6).
An element is called invertible if there exists such that .
Proposition 1 (see [9]). Let be a Banach algebra, and let . If the spectral radius of is less than 1, i.e., then , where is the set of all invertible elements of and
Remark 2. Let be a Banach algebra.
Then the following are satisfied:(1)for all , ;(2)if the condition is replaced by in Proposition 1, then the conclusion holds.
Consistent with Liu and Xu [7], the following definitions will be needed in the sequel.
Let be a Banach algebra. A subset of is called cone if the following conditions are satisfied:(1) is a nonempty and closed subset of and ;(2), whenever and ;(3);(4).
Remark 3. Let be a Banach algebra, and be a cone.
If and , then .
Given a cone , we define a partial ordering with respect to by if and only if . We write to indicate that but .
For , stand for , where is the interior of . A cone is called normal if there exists a number such that, for all , whenever .
A cone is called regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that for some then there exists such that
Equivalently, a cone is regular if and only if every decreasing sequence which is bounded from below is convergent.
It is well known that every regular cone is normal.
From now on, we assume that is a Banach algebra, is a cone with , and is partial ordering with respect to .
Let be a nonempty set, and let be a map such that(1) for all and if and only if ;(2) for all ;(3) for all .
Then is called a cone metric on (with a Banach algebra ) and is called cone metric space (with a Banach algebra ).
Remark 4. If and , where is the set of all complex numbers, then we have that (1);(2)the complex valued metric [10] is a cone metric.
A sequence of points in a cone metric space converges to a point (denoted by or ) if for any with , there exists such that for all , . A sequence of points in a cone metric space is Cauchy if for any with , there exists such that for all , . A cone metric space is called complete if every Cauchy sequence is convergent.
Remark 5. Let be a cone metric space and be a sequence and . Then the following are satisfied: (1) if and only if for any , there exists such that for all , ;(2) is a Cauchy sequence if and only if for any , there exists such that for all , .
Note that if , then . The converse is true if is a normal cone. Also, if is a normal cone, then is a Cauchy sequence in if and only if .
Let be a cone metric space.
Let be the class of all continuous function satisfying the following: () ; () for all ; () either or for all and .
Lemma 6 (see [11]). Let be a cone metric space with regular cone such that for . Let be a sequence, and let .
If the sequence is not a Cauchy sequence, then there exist and two subsequences and of such that is the smallest index for which , and Moreover if , then we have (1);(2).
Let be a cone metric space.
A map is called continuous at if for any containing , there exists containing such that , where is the topology on induced by the cone metric . That is,
,
,
.
If is continuous at each point , then it is called continuous.
Note that is continuous if and only if it is sequentially continuous, i.e., for any sequence with (see [12]).
A point is called cluster point of if for every with ,
Let be the family of all continuous function such that () is strictly increasing, i.e., ; () .
Note that if , then
Alizadeh et al. [13] introduced the concept of cyclic ()-admissible mappings in .
We extend this concept to cones as follows.
Let be functions, where is a set. We say that is a cyclic ()-admissible mapping if (1);(2).
Example 7. Let be a compact Hausdorff topological space, and let be the family of all continuous function from into .
Define the multiplication of and as follows: Then is a Banach algebra with unit .
Let .
Then is a normal cone with normal constant .
Let , and let be functions defined by Define as follows: Then we have and Hence is cyclic ()-admissible.
2. Fixed Points
Let be a family of all continuous functions such that(1);(2) either or .
Then we say that is -class function.
Note that .
Example 8. Let be a Banach algebra, and let be a cone.(1)Let , where with . Since and , from Remark 4. Hence , and hence . So Thus . If , then , and so . Since , . Hence . Thus .(2)Let , where is a continuous function such that and . Then . Let . Then . If , then , which is a contradiction. Hence . Thus .(3)Let , where . Then . If , then , and so . Hence .(4)Let , where is continuous. Since and , . Hence , and hence Thus . If , then . Since , . Hence .(5)Let , where is continuous. Since and , . Hence , and hence . Thus . If , then . We have , because . Hence , Thus .
From now on, let be a cone metric space with cone metric and regular cone such that for with
Theorem 9. Let be complete, and let be a mapping such that for all with , where , , and .
Also, suppose that the following are satisfied:(1) is cyclic -admissible;(2)There exists such that and ;(3)Either is continuous or if is a sequence with and is a cluster point of , then Then has a fixed point.
Moreover, if for all fixed points of , then has a unique fixed point.
Proof. Define a sequence in by for all .
If there exists an integer such that , then ; i.e., is a fixed point of .
Hence we assume that
Since , from (1) we have . Again from (1) we obtain .
Inductively, we have Since is cyclic -admissible and , . Similarly, we obtain Hence Since and , Also, Hence, And so it follows from (14) thatSince is strictly increasing, Hence is a decreasing sequence.
Since is regular, there exists such that By Lemma 3.1 of [14], we have Assume that .
By taking limit in (23) and using continuity of , and , we obtain which implies So and so either or . Hence , which is a contradiction.
Hence Thus we have We now show that is a Cauchy sequence.
On the contrary, assume that is not a Cauchy sequence.
By Lemma 6, there exists and two subsequences and of such that is the smallest index for which, for all , , (5) and (6) hold.
It follows from (19) that , and so from (14) with and we haveBy Letting and by using Lemma 6 (1) and (2), and using continuity of , and , we have which implies Hence which implies either or . Thus , and so , which is a contradiction.
Therefore, is a Cauchy sequence.
It follows from the completeness of that there exists If is continuous, then , and so .
Assume that (15) holds.
Then we have Since is ()-admissible,
Using (19) we have From (14) we haveLetting in the inequality (39) and using continuity of , and , we have which implies Thus Hence For the uniqueness of fixed point, assume that (16) holds and that is another fixed point of .
Then from (16) we have and
Hence
It follows from (14) that which implies Thus which implies So , and hence .
Example 10. Let , and let .
Define the multiplication of and as follows: Then is a Banach algebra with unit , is regular cone, and .
Let , and let be a mapping defined by Then is a complete cone metric space, and
Define a mapping by Let , , and .
We define two functions by
Obviously, is ()-admissible.
Let .
Then we have
Hence with
Condition (2) of Theorem 9 is satisfied with .
We now show condition (15) of Theorem 9.
Let .
Then and . Hence Then , and so is a cluster point of .
Then we obtain Hence condition (15) of Theorem 9 is satisfied.
Thus, all hypotheses of Theorem 9 are satisfied and has a fixed point .
Corollary 11. Let be complete, and let be a mapping such that for all , where , , and .
Also, suppose that conditions (1), (2), and (3) of Theorem 9 hold.
Then has a fixed point.
Moreover, if (16) holds, then has a unique fixed point.
3. Discussion
We give an application of Theorem 9 to prove existence of fixed points for cyclic mappings.
Theorem 12. Let be complete, and let and be two closed subsets of such that , and let be a mapping such that Assume that for all and , where , , and .
Then has a unique fixed point in .
Proof. Define by Then we have that Hence (57) implies (13).
It follows from (56), (58), and (59) that and Thus is cyclic -admissible.
Because , there exists an in . By definition of functions and , and . Hence condition (2) of Theorem 9 is satisfied.
Let be a sequence such that
Then Since is closed, and so . Hence condition (3) of Theorem 9 holds.
Thus all conditions of Theorem 9 are satisfied. It follows from Theorem 9 that has a fixed point in ; say .
If , then from (56) . Thus . Similarly, we have , when .
Let be any two fixed point of .
Then , and so . It follows from Theorem 9 that has a unique fixed point in .
Corollary 13. Let be complete, and let and be two closed subsets of such that , and let be a mapping such that Assume that for all and , where and .
Then has a unique fixed point in .
Corollary 14. Let be complete, and let and be two closed subsets of such that , and let be a mapping such that Assume that for all and , where and .
Then has a unique fixed point in .
Proof. From (67) we have for all and , where , , and .
By Corollary 13 with , has a unique fixed point in .
Remark 15. If and , then Corollary 14 reduces to Theorem 1.1 of [15].
Remark 16. Corollary 14 is a generalization of Theorem 2.1 of [7] (resp., Theorem 3.1 of [16]) to cyclic mappings under assumption of regularity.
4. Applications
By applying -class functions of Example 8 to Theorem 9, we derive some existing fixed point results in the literature and have generalizations of well-known fixed point theorems in metric spaces to cone metric spaces.
In particular, by taking , we have the following result.
Theorem 17. Let be complete, and let be a cyclic -admissible mapping such that for all with , where and such that Suppose that and , where .
Assume that either (1) is continuous or(2)if is a sequence such that and , then Then has a fixed point in . Further if for all fixed points of , then has a unique fixed point.
Proof. Let , where .
Then is a C-class function.
From (69) and (70) we have that for all with Hence for all with .
It follows that (13) holds with . All conditions of Theorem 9 hold, and so we have a desired result.
Corollary 18. Let be complete, and let be a cyclic -admissible mapping such that for all with , where .
Suppose that and , where .
Assume that either (1) is continuous or(2)if is a sequence such that and , then Then has a fixed point in . Further if for all fixed points of , then has a unique fixed point.
By taking and in Theorem 17, we have the following result.
Corollary 19. Let be complete, and let be a mapping such that for all , where and with .
Then has a unique fixed point in .
Corollary 20. Let be complete, and let be a mapping such that for all , where .
Then has a unique fixed point in .
Remark 21. By taking a real Banach space in Corollary 19 (resp., Corollary 20), we have Theorem 2.1 of [10] (resp., Theorem 2.1 of [4]).
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that he has no conflicts of interest.