Research

# Fixed point theorem for weakly Chatterjea-type cyclic contractions

Sumit Chandok1 and Mihai Postolache2*

Author Affiliations

1 Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar, 143001, India

2 Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independenţei, Bucharest, 060042, Romania

For all author emails, please log on.

Fixed Point Theory and Applications 2013, 2013:28  doi:10.1186/1687-1812-2013-28

 Received: 28 November 2012 Accepted: 27 January 2013 Published: 11 February 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we introduce the notion of a Chatterjea-type cyclic weakly contraction and derive the existence of a fixed point for such mappings in the setup of complete metric spaces. Our result extends and improves some fixed point theorems in the literature. Example is given to support the usability of the result.

MSC: 41A50, 47H10, 54H25.

##### Keywords:
fixed point; cyclic contraction mapping

### 1 Introduction and preliminaries

It is well known that the fixed point theorem of Banach, for contraction mappings, is one of the pivotal results in analysis. It has been used in many different fields of mathematics but suffers from one major drawback. More accurately, in order to use the contractive condition, a self-mapping T must be Lipschitz continuous, with the Lipschitz constant . In particular, T must be continuous at all points of its domain.

A natural question arises:

Could we find contractive conditions which will imply the existence of a fixed point in a complete metric space but will not imply continuity?

Kannan [1,2] proved the following result giving an affirmative answer to the above question.

Theorem 1.1Ifis a complete metric space and the mappingsatisfies

(1.1)

whereand, thenThas a unique fixed point.

The mappings satisfying (1.1) are called Kannan-type mappings.

A similar type of contractive condition has been studied by Chatterjea [3]. He proved the following result.

Theorem 1.2Ifis a complete metric space andsatisfies

(1.2)

whereand, thenThas a unique fixed point.

In Theorems 1.1 and 1.2, there is no the requirement for the continuity of T.

Alber and Guerre-Delabriere [4] introduced the concept of weakly contractive mappings and proved the existence of fixed points for single-valued weakly contractive mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [5] proved the fixed point theorem which is one of the generalizations of Banach’s contraction mapping principle because the weakly contractions contain contractions as a special case, and he also showed that some results of [4] are true for any Banach space. In fact, weakly contractive mappings are closely related to the mappings of Boyd and Wong [6] and of Reich types [7].

Fixed point problems involving different types of contractive type inequalities have been studied by many authors (see [1-24] and the references cited therein).

In [22], Kirk et al. introduced the following notion of a cyclic representation and characterized the Banach contraction principle in the context of a cyclic mapping.

Definition 1.1[22]

Let X be a non-empty set and be an operator. By definition, is a cyclic representation of X with respect to T if

(a) ; are non-empty sets;

(b) .

It is the aim of this paper to introduce the notion of a cyclic weakly Chatterjea-type contraction and then derive a fixed point theorem for such cyclic contractions in the framework of complete metric spaces.

### 2 Main results

To state and prove our main results, we will introduce our notion of a Chatterjea-type cyclic weakly contraction in a metric space. In this respect, let Φ denote the set of all monotone increasing continuous functions , with , if and only if , and let Ψ denote the set of all lower semi-continuous functions , with , for and .

Definition 2.1 Let be a metric space, m be a natural number, be non-empty subsets of X and . An operator is called a Chatterjea-type cyclic weakly contraction if

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

(2)

for any , , , where , and .

Theorem 2.1Letbe a complete metric space, , be non-empty closed subsets ofXand. Suppose thatTis a Chatterjea-type cyclic weakly contraction. ThenThas a fixed point.

Proof Let . We can construct a sequence ,  .

If there exists such that , hence the result. Indeed, we can see that .

Now, we assume that for any  . As , for any , there exists such that and . Since T is a Chatterjea-type cyclic weakly contraction, we have

(2.1)

Since μ is a non-decreasing function, for all  , we have

(2.2)

This implies that . Thus is a monotone decreasing sequence of non-negative real numbers and hence is convergent. Therefore, there exists such that . Letting in (2.2), we obtain that .

Letting in (2.1) and using the continuity of μ and lower semi-continuity of ψ, we obtain that . This implies that , hence . Thus we have proved that

Now, we show that is a Cauchy sequence. For this purpose, we prove the following result first.

Lemma 2.1For every positiveϵ, there exists a natural numbernsuch that ifwith, then.

Proof Assume the contrary. Thus there exists such that for any , we can find with satisfying .

Now, we take . Then, corresponding to , we can choose such that it is the smallest integer with satisfying and . Therefore, . By using the triangular inequality, we have

Letting and using , we obtain

(2.3)

Again, by the triangular inequality,

Letting and using , we get

(2.4)

Consider

(2.5)

and

(2.6)

On taking in inequalities (2.5) and (2.6), we have

(2.7)

and

(2.8)

As and lie in different adjacently labeled sets and for certain , using the fact that T is a Chatterjea-type cyclic weakly contraction, we obtain

(2.9)

On taking in (2.9), using (2.7) and (2.8), the continuity of μ and lower semi-continuity of ψ, we get that

Consequently, , which is contradiction with . Hence the result is proved. □

Now, using Lemma 2.1, we will show that is a Cauchy sequence in Y. Fix . By Lemma 2.1, we can find such that with

(2.10)

Since , we can also find such that

(2.11)

for any .

Assume that and . Then there exists such that . Hence for . So, we have

(2.12)

Using (2.10), (2.11) and (2.12), we obtain

(2.13)

Hence is a Cauchy sequence in Y. Since Y is closed in X, then Y is also complete and there exists such that .

Now, we will prove that x is a fixed point of T.

As is a cyclic representation of Y with respect to T, the sequence has infinite terms in each for . Suppose that , and we take a subsequence of with . By using the contractive condition, we can obtain

Letting and using the continuity of μ and lower semi-continuity of ψ, we have

which is a contradiction unless . Hence x is a fixed point of T.

Now, we will prove the uniqueness of the fixed point.

Suppose that and () are two fixed points of T. Using the contractive condition and the continuity of μ and lower semi continuity of ψ, we have

which is a contradiction unless . Hence the main result is proved. □

If , then we have the following result.

Corrollary 2.1Letbe a complete metric space, , be non-empty closed subsets ofXand. Suppose thatis an operator such that

(1) is a cyclic representation ofYwith respect toT;

(2)

for any, , , whereand. ThenThas a fixed point.

If , where , we have the following result.

Corrollary 2.2Letbe a complete metric space, , be non-empty closed subsets ofXand. Suppose thatis an operator such that

(1) is a cyclic representation ofYwith respect toT;

(2) there existssuch that

for any, , , where. ThenThas a fixed point.

### 3 Applications

Other consequences of our results, for mappings involving contractions of integral type, are given in the following. In this respect, denote by Λ the set of functions satisfying the following hypotheses:

(h1) μ is a Lebesgue-integrable mapping on each compact of ;

(h2) for any , we have .

Corrollary 3.1Letbe a complete metric space, , be non-empty closed subsets ofXand. Suppose thatis an operator such that

(1) is a cyclic representation ofYwith respect toT;

(2) there existssuch that

for any, , , whereand. ThenThas a fixed point.

If we take , , we obtain the following result.

Corrollary 3.2Letbe a complete metric space andbe a mapping such that

for any, and. ThenThas a fixed point.

Example 3.1 Let X be a subset in ℝ endowed with the usual metric. Suppose , and . Define such that for all . It is clear that is a cyclic representation of Y with respect to T. Furthermore, if is given as and is given by , then .

Now, we prove that T satisfies the inequality of Chatterjea-type cyclic weakly contraction, i.e., . To see this fact, we examine three cases.

Case 1. Suppose that . Then

(3.1)

and

(3.2)

If , then

Hence, the given inequality is satisfied.

If , then

Hence the given inequality is satisfied.

Case 2. Suppose that . Then from (3.1) and (3.2), we have

Hence the given inequality is satisfied.

Case 3. Finally, suppose that . Then from (3.1) and (3.2), we have

and

Hence the given inequality is satisfied.

Therefore, all the conditions of Theorem 2.1 are satisfied, and so T has a fixed point (which is ).

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

### Acknowledgements

The authors are thankful to learned referee(s) for suggestions.

### References

1. Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc.. 60, 71–76 (1968)

2. Kannan, R: Some results on fixed points-II. Am. Math. Mon.. 76, 405–408 (1969). Publisher Full Text

3. Chatterjea, SK: Fixed point theorem. C. R. Acad. Bulgare Sci.. 25, 727–730 (1972)

4. Alber, Y, Guerre-Delabriere, S: Principles of Weakly Contractive Maps in Hilbert Spaces. In: Gohberg I, Lyubich Y (eds.) New Results in Operator Theory and Its Applications, pp. 7–22. Birkhäuser, Basel (1997)

5. Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal.. 47, 2683–2693 (2001). Publisher Full Text

6. Boyd, DW, Wong, TSW: On nonlinear contractions. Proc. Am. Math. Soc.. 20, 458–464 (1969). Publisher Full Text

7. Reich, S: Some fixed point problems. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat.. 57, 194–198 (1975)

8. Aydi, H, Karapınar, E, Postolache, M: Tripled coincidence point theorems for weak φ-contractions in partially ordered metric spaces. Fixed Point Theory Appl.. 2012, Article ID 44 (2012)

9. Aydi, H, Shatanawi, W, Postolache, M, Mustafa, Z, Tahat, N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal.. 2012, Article ID 359054 (2012)

10. Chandok, S: Some common fixed point theorems for generalized f-weakly contractive mappings. J. Appl. Math. Inform.. 29, 257–265 (2011)

11. Chandok, S: Some common fixed point theorems for generalized nonlinear contractive mappings. Comput. Math. Appl.. 62, 3692–3699 (2011). Publisher Full Text

12. Chandok, S: Common fixed points, invariant approximation and generalized weak contractions. Int. J. Math. Math. Sci.. 2012, Article ID 102980 (2012)

13. Chandok, S, Kim, JK: Fixed point theorem in ordered metric spaces for generalized contractions mappings satisfying rational type expressions. J. Nonlinear Funct. Anal. Appl.. 17, 301–306 (2012)

14. Chandok, S: Common fixed points for generalized nonlinear contractive mappings in metric spaces. Mat. Vesn.. 65, 29–34 (2013)

15. Chandok, S: A fixed point result for weakly Kannan type cyclic contractions. Int. J. Pure Appl. Math.. 82(2), 253–260 (2013)

16. Chandok, S: Some common fixed point results for generalized weak contractive mappings in partially ordered metric spaces. J. Nonlinear Anal. Optim. (2013, in press)

17. Chandok, S, Karapinar, E: Some common fixed point results for generalized rational type weak contraction mappings in partially ordered metric spaces. Thai J. Math. (2013, in press)

18. Chandok, S, Khan, MS, Rao, KPR: Some coupled common fixed point theorems for a pair of mappings satisfying a contractive condition of rational type without monotonicity. Int. J. Math. Anal.. 7(9), 433–440 (2013)

19. Haghi, RH, Postolache, M, Rezapour, S: On T-stability of the Picard iteration for generalized φ-contraction mappings. Abstr. Appl. Anal.. 2012, Article ID 658971 (2012)

20. Karapinar, E, Sadarangani, K: Fixed point theory for cyclic -contractions. Fixed Point Theory Appl.. 2011, Article ID 69 (2011)

21. Karapinar, E, Erhan, IM: Best proximity on different type contractions. Appl. Math. Inf. Sci.. 5, 342–353 (2011)

22. Kirk, WA, Srinivasan, PS, Veeramani, P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory Appl.. 4(1), 79–89 (2003)

23. Olatinwo, MO, Postolache, M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput.. 218(12), 6727–6732 (2012). Publisher Full Text

24. Zhou, X, Wu, W, Ma, H: A contraction fixed point theorem in partially ordered metric spaces and application to fractional differential equations. Abstr. Appl. Anal.. 2012, Article ID 856302 (2012)