Abstract
In this article, we introduce the notion of a Chatterjeatype 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 mapping1 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 selfmapping 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
whereand, thenThas a unique fixed point.
The mappings satisfying (1.1) are called Kannantype 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
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 GuerreDelabriere [4] introduced the concept of weakly contractive mappings and proved the existence of fixed points for singlevalued 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 [124] 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 nonempty set and be an operator. By definition, is a cyclic representation of X with respect to T if
It is the aim of this paper to introduce the notion of a cyclic weakly Chatterjeatype 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 Chatterjeatype 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 semicontinuous functions , with , for and .
Definition 2.1 Let be a metric space, m be a natural number, be nonempty subsets of X and . An operator is called a Chatterjeatype cyclic weakly contraction if
(1) is a cyclic representation of Y with respect to T;
Theorem 2.1Letbe a complete metric space, , be nonempty closed subsets ofXand. Suppose thatTis a Chatterjeatype 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 Chatterjeatype cyclic weakly contraction, we have
Since μ is a nondecreasing function, for all , we have
This implies that . Thus is a monotone decreasing sequence of nonnegative 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 semicontinuity 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
Again, by the triangular inequality,
Consider
and
On taking in inequalities (2.5) and (2.6), we have
and
As and lie in different adjacently labeled sets and for certain , using the fact that T is a Chatterjeatype cyclic weakly contraction, we obtain
On taking in (2.9), using (2.7) and (2.8), the continuity of μ and lower semicontinuity 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
Since , we can also find such that
Assume that and . Then there exists such that . Hence for . So, we have
Using (2.10), (2.11) and (2.12), we obtain
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 semicontinuity 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 nonempty closed subsets ofXand. Suppose thatis an operator such that
(1) is a cyclic representation ofYwith respect toT;
for any, , , whereand. ThenThas a fixed point.
If , where , we have the following result.
Corrollary 2.2Letbe a complete metric space, , be nonempty closed subsets ofXand. Suppose thatis an operator such that
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 Lebesgueintegrable mapping on each compact of ;
Corrollary 3.1Letbe a complete metric space, , be nonempty closed subsets ofXand. Suppose thatis an operator such that
(1) is a cyclic representation ofYwith respect toT;
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 Chatterjeatype cyclic weakly contraction, i.e., . To see this fact, we examine three cases.
and
Hence, the given inequality is satisfied.
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

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

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

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

Alber, Y, GuerreDelabriere, 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)

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

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

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

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

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

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

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

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

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)

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

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

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

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)

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)

Haghi, RH, Postolache, M, Rezapour, S: On Tstability of the Picard iteration for generalized φcontraction mappings. Abstr. Appl. Anal.. 2012, Article ID 658971 (2012)

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

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

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

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

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)