Open Access

# Cyclic generalized contractions and fixed point results with applications to an integral equation

Fixed Point Theory and Applications20122012:217

DOI: 10.1186/1687-1812-2012-217

Accepted: 12 November 2012

Published: 28 November 2012

## Abstract

We set up a new variant of cyclic generalized contractive mappings for a map in a metric space and present existence and uniqueness results of fixed points for such mappings. Our results generalize or improve many existing fixed point theorems in the literature. To illustrate our results, we give some examples. At the same time as applications of the presented theorems, we prove an existence theorem for solutions of a class of nonlinear integral equations.

MSC:47H10, 54H25.

### Keywords

fixed point cyclic generalized -contraction integral equation

## 1 Introduction and preliminaries

All the way through this paper, by , we designate the set of all real nonnegative numbers, while is the set of all natural numbers.

The celebrated Banach’s [1] contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. This principle has been extended and improved in many ways over the years (see, e.g., [25]). Fixed point theorems have applications not only in various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and other fields. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Owing to the magnitude, generalizations of the Banach fixed point theorem have been explored heavily by many authors. This celebrated theorem can be stated as follows.

Theorem 1.1 ([1])

Let be a complete metric space and T be a mapping of X into itself satisfying
(1)

where k is a constant in . Then T has a unique fixed point .

Inequality (1) implies the continuity of T. A natural question is whether we can find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity.

On the other hand, cyclic representations and cyclic contractions were introduced by Kirk et al. [6]. A mapping is called cyclic if and , where A, B are nonempty subsets of a metric space . Moreover, T is called a cyclic contraction if there exists such that for all and . Notice that although a contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of this theorem.

Definition 1.1 (See [6, 7])

Let be a metric space. Let p be a positive integer, be nonempty subsets of X, , and . Then Y is said to be a cyclic representation of Y with respect to T if
1. (i)

, are nonempty closed sets, and

2. (ii)

.

Following the paper in [6], a number of fixed point theorems on a cyclic representation of Y with respect to a self-mapping T have appeared (see, e.g., [3, 715]).

In this paper, we introduce a new class of cyclic generalized -contractive mappings, and then investigate the existence and uniqueness of fixed points for such mappings. Our main result generalizes and improves many existing theorems in the literature. We supply appropriate examples to make obvious the validity of the propositions of our results. To end with, as applications of the presented theorems, we achieve fixed point results for a generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.

## 2 Main results

In this section, we introduce two new notions of a cyclic contraction and establish new results for such mappings.

In the sequel, we fixed the set of functions by such that
1. (i)

is nondecreasing, continuous, and for every ;

2. (ii)

ψ is nondecreasing, right continuous, and for every .

Define and .

We state the notion of a cyclic generalized -contraction as follows.

Definition 2.1 Let be a metric space. Let p be a positive integer, be nonempty subsets of X and . An operator is said to be a cyclic generalized -contraction for some , , and if
1. (a)

is a cyclic representation of Y with respect to T;

2. (b)
for any , (with ),

where
and

Our first main result is the following.

Theorem 2.1 Let be a complete metric space, , be nonempty closed subsets of X, and . Suppose is a cyclic generalized -contraction mapping for some and . Then T has a unique fixed point. Moreover, the fixed point of T belongs to .

Proof Let (such a point exists since ). Define the sequence in X by
We shall prove that
(2)
If, for some k, we have , then (2) follows immediately. So, we can suppose that for all n. From the condition (a), we observe that for all n, there exists such that . Then, from the condition (b), we have
(3)
On the other hand, we have
and
Suppose that for some . Then , so
and thus
(4)
Similarly, we have
(5)
Thus, from (4) and (5), we get
for all . Now, from

and the property of ψ, we obtain , and consequently (2) holds.

Now, we shall prove that is a Cauchy sequence in . Suppose, on the contrary, that is not a Cauchy sequence. Then there exists for which we can find two sequences of positive integers and such that for all positive integers k,
(6)
Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (6). Then we have
(7)
Using (6), (7), and the triangular inequality, we get
Thus, we have
Passing to the limit as in the above inequality and using (2), we obtain
(8)
On the other hand, for all k, there exists such that . Then (for k large enough, ) and lie in different adjacently labeled sets and for certain . Using (b), we obtain
(9)
for all k. Now, we have
(10)
and
(11)
for all k. Using the triangular inequality, we get
which implies from (8) that
(12)
Using (2), we have
(13)
and
(14)
Again, using the triangular inequality, we get
Passing to the limit as in the above inequality, using (14) and (12), we get
(15)
Similarly, we have
Passing to the limit as , using (2) and (12), we obtain
(16)
Similarly, we have
(17)
Now, it follows from (12)-(16) and the continuity of φ that
(18)
and
(19)
Passing to the limit as in (9), using (17), (18), (19), and the condition (ii), we obtain

which is a contradiction. Thus, we proved that is a Cauchy sequence in .

Since is complete, there exists such that
(20)
We shall prove that
(21)

From the condition (a), and since , we have . Since is closed, from (20), we get that . Again, from the condition (a), we have . Since is closed, from (20), we get that . Continuing this process, we obtain (21).

Now, we shall prove that is a fixed point of T. Indeed, from (21), since for all n there exists such that , applying (b) with and , we obtain
(22)
for all n. On the other hand, we have
and
Passing to the limit as in the above inequality and using (20), we obtain that
(23)
Passing to the limit as in (22), using (23) and (20), we get
Suppose that . In this case, we have
which implies that

a contradiction. Then we have , that is, is a fixed point of T.

Finally, we prove that is the unique fixed point of T. Assume that is another fixed point of T, that is, . From the condition (a), this implies that . Then we can apply (b) for and . We obtain
Since and are fixed points of T, we can show easily that and . If , we get

a contradiction. Then we have , that is, . Thus, we proved the uniqueness of the fixed point. □

In the following, we deduce some fixed point theorems from our main result given by Theorem 2.1.

If we take and in Theorem 2.1, then we get immediately the following fixed point theorem.

Corollary 2.1 Let be a complete metric space and satisfy the following condition: there exist , , and such that

for all . Then T has a unique fixed point.

Remark 2.1 Corollary 2.1 extends and generalizes many existing fixed point theorems in the literature [1, 1621].

Corollary 2.2 Let be a complete metric space, , be nonempty closed subsets of X, , and . Suppose that there exist and such that

(a′) is a cyclic representation of Y with respect to T;

(b′) for any , (with ),

Then T has a unique fixed point. Moreover, the fixed point of T belongs to .

Remark 2.2 Corollary 2.2 is similar to Theorem 2.1 in [7].

Remark 2.3 Taking in Corollary 2.2 with , we obtain a generalized version of Theorem 1.3 in [6].

Corollary 2.3 Let be a complete metric space, , be nonempty closed subsets of X, , and . Suppose that there exist and such that

(a′) is a cyclic representation of Y with respect to T;

(b′) for any , (with ),

Then T has a unique fixed point. Moreover, the fixed point of T belongs to .

Remark 2.4 Taking in Corollary 2.3 with , we obtain a generalized version of Theorem 3 in [13].

Corollary 2.4 Let be a complete metric space, , be nonempty closed subsets of X, , and . Suppose that there exist and such that

(a′) is a cyclic representation of Y with respect to T;

(b′) for any , (with ),

Then T has a unique fixed point. Moreover, the fixed point of T belongs to .

Remark 2.5 Taking in Corollary 2.4 with , we obtain a generalized version of Theorem 5 in [13].

Corollary 2.5 Let be a complete metric space, , be nonempty closed subsets of X, , and . Suppose that there exist and such that
1. (a)

is a cyclic representation of Y with respect to T;

2. (b)
for any , (with ),

Then T has a unique fixed point. Moreover, the fixed point of T belongs to .

We provide some examples to illustrate our obtained Theorem 2.1.

Example 2.1 Let with the usual metric. Suppose and and . Define such that for all . It is clear that is a cyclic representation of Y with respect to T. Let be defined by and of the form , . For all and , we have

So, T is a cyclic generalized -contraction for any . Therefore, all conditions of Theorem 2.1 are satisfied (), and so T has a unique fixed point (which is ).

Example 2.2 Let with the usual metric. Suppose and and . Define the mapping by

Clearly, we have and . Moreover, and are nonempty closed subsets of X. Therefore, is a cyclic representation of Y with respect to T.

Now, let with and , we have
On the other hand, we have
and
Then we have
Define the function by and of the form , and . Then we have
(24)

Moreover, we can show that (24) holds if or . Similarly, we also get (24) holds for .

Now, all the conditions of Theorem 2.1 are satisfied (with ), we deduce that T has a unique fixed point .

## 3 An application to an integral equation

In this section, we apply the result given by Theorem 2.1 to study the existence and uniqueness of solutions to a class of nonlinear integral equations.

We consider the nonlinear integral equation
(25)

where , and are continuous functions.

Let be the set of real continuous functions on . We endow X with the standard metric

It is well known that is a complete metric space.

Let , such that
(26)
We suppose that for all , we have
(27)
and
(28)
We suppose that for all , is a decreasing function, that is,
(29)
We suppose that
(30)
Finally, we suppose that, for all , for all with ( and ) or ( and ),
(31)

where is a nondecreasing function that belongs to and .

Now, define the set

We have the following result.

Theorem 3.1 Under the assumptions (26)-(31), problem (25) has one and only one solution .

Proof Define the closed subsets of X, and , by
and
Define the mapping by
We shall prove that
(32)
Let , that is,
Using condition (29), since for all , we obtain that
The above inequality with condition (27) implies that

for all . Then we have .

Similarly, let , that is,
Using condition (29), since for all , we obtain that
The above inequality with condition (28) implies that

for all . Then we have . Finally, we deduce that (32) holds.

Now, let , that is, for all ,
This implies, from condition (26), that for all ,
Now, using conditions (30) and (31), we can write that for all , we have
This implies that

where of the form . Using the same technique, we can show that the above inequality holds also if we take .

Now, all the conditions of Theorem 2.1 are satisfied (with ), we deduce that T has a unique fixed point , that is, is the unique solution to (25). □

## Declarations

### Acknowledgements

The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). Moreover, the third author was supported by the Commission on Higher Education (CHE), the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).

## Authors’ Affiliations

(1)
Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

## References

1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math. 1922, 3: 133–181.
2. Chauhan S, Sintunavarat W, Kumam P: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (JCLR) property. Appl. Math. 2012, 3(9):976–982. 10.4236/am.2012.39145View Article
3. Pacurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2011, 12(2):419–428.MathSciNet
4. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
5. Sintunavarat W, Kumam P: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012., 2012: Article ID 84
6. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4(1):79–89.MathSciNet
7. Pacurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002
8. Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40
9. Aydi H, Vetro C, Sintunavarat W, Kumam P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 124
10. Karapınar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24(6):822–825. 10.1016/j.aml.2010.12.016
11. Karapınar E, Sadaranagni K: Fixed point theory for cyclic -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
12. Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. 10.1007/s10957-012-9991-y
13. Petric MA: Some results concerning cyclical contractive mappings. Gen. Math. 2010, 18(4):213–226.MathSciNet
14. Rus IA: Cyclic representations and fixed points. Ann. T. Popoviciu Semin. Funct. Equ. Approx. Convexity 2005, 3: 171–178.
15. Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25(11):1849–1855. 10.1016/j.aml.2012.02.045
16. Bianchini RMT: Su un problema di S. Reich riguardante la teoria dei punti fissi. Boll. Unione Mat. Ital., Ser. IV 1972, 5: 103–108.MathSciNet
17. Boyd DW, Wong JS: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–469. 10.1090/S0002-9939-1969-0239559-9
18. Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.MathSciNet
19. Hardy GE, Rogers TD: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 1973, 16(2):201–206.
20. Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 10: 71–76.
21. Reich S: Some remarks concerning contraction mappings. Can. Math. Bull. 1971, 14: 121–124. 10.4153/CMB-1971-024-9View Article