Skip to content

Advertisement

  • Research
  • Open Access

Fixed point theorem for weakly Chatterjea-type cyclic contractions

Fixed Point Theory and Applications20132013:28

https://doi.org/10.1186/1687-1812-2013-28

  • Received: 28 November 2012
  • Accepted: 27 January 2013
  • Published:

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 L < 1 . 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 1.1 If ( X , d ) is a complete metric space and the mapping T : X X satisfies
d ( T x , T y ) k [ d ( x , T x ) + d ( y , T y ) ] ,
(1.1)

where 0 < k < 1 2 and x , y X , then T has 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 1.2 If ( X , d ) is a complete metric space and T : X X satisfies
d ( T x , T y ) k [ d ( x , T y ) + d ( y , T x ) ] ,
(1.2)

where 0 < k < 1 2 and x , y X , then T has 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 [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 non-empty set and T : X X be an operator. By definition, X = i = 1 m X i is a cyclic representation of X with respect to T if
  1. (a)

    X i ; i = 1 , , m are non-empty sets;

     
  2. (b)

    T ( X 1 ) X 2 , , T ( X m 1 ) X m , T ( X m ) X 1 .

     

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 μ : [ 0 , ) [ 0 , ) , with μ ( t ) = 0 , if and only if t = 0 , and let Ψ denote the set of all lower semi-continuous functions ψ : [ 0 , ) 2 [ 0 , ) , with ψ ( t 1 , t 2 ) > 0 , for t 1 , t 2 ( 0 , ) and ψ ( 0 , 0 ) = 0 .

Definition 2.1 Let ( X , d ) be a metric space, m be a natural number, A 1 , A 2 , , A m be non-empty subsets of X and Y = i = 1 m A i . An operator T : Y Y is called a Chatterjea-type cyclic weakly contraction if
  1. (1)

    i = 1 m A i is a cyclic representation of Y with respect to T;

     
  2. (2)

    μ ( d ( T x , T y ) ) μ ( 1 2 [ d ( x , T y ) + d ( y , T x ) ] ) ψ ( d ( x , T y ) , d ( y , T x ) )

     

for any x A i , y A i + 1 , i = 1 , 2 , , m , where A m + 1 = A 1 , μ Φ and ψ Ψ .

Theorem 2.1 Let ( X , d ) be a complete metric space, m N , A 1 , A 2 , , A m be non-empty closed subsets of X and Y = i = 1 m A i . Suppose that T is a Chatterjea-type cyclic weakly contraction. Then T has a fixed point z i = 1 n A i .

Proof Let x 0 X . We can construct a sequence x n + 1 = T x n , n = 0 , 1 , 2 ,  .

If there exists n 0 N such that x n 0 + 1 = x n 0 , hence the result. Indeed, we can see that T x n 0 = x n 0 + 1 = x n 0 .

Now, we assume that x n + 1 x n for any n = 0 , 1 , 2 ,  . As X = i = 1 m A i , for any n > 0 , there exists i n { 1 , 2 , , m } such that x n 1 A i n and x n A i n + 1 . Since T is a Chatterjea-type cyclic weakly contraction, we have
μ ( d ( x n + 1 , x n ) ) = μ ( d ( T x n , T x n 1 ) ) μ ( 1 2 [ d ( x n , T x n 1 ) + d ( x n 1 , T x n ) ] ) ψ ( d ( x n , T x n 1 ) , d ( x n 1 , T x n ) ) = μ ( 1 2 d ( x n 1 , x n + 1 ) ) ψ ( 0 , d ( x n 1 , x n + 1 ) ) μ ( 1 2 d ( x n 1 , x n + 1 ) ) .
(2.1)
Since μ is a non-decreasing function, for all n = 1 , 2 ,  , we have
d ( x n + 1 , x n ) 1 2 d ( x n 1 , x n + 1 ) 1 2 [ d ( x n 1 , x n ) + d ( x n , x n + 1 ) ] .
(2.2)

This implies that d ( x n + 1 , x n ) d ( x n , x n 1 ) . Thus { d ( x n + 1 , x n ) } is a monotone decreasing sequence of non-negative real numbers and hence is convergent. Therefore, there exists r 0 such that d ( x n + 1 , x n ) r . Letting n in (2.2), we obtain that lim d ( x n 1 , x n + 1 ) = 2 r .

Letting n in (2.1) and using the continuity of μ and lower semi-continuity of ψ, we obtain that μ ( r ) μ ( r ) ψ ( 0 , 2 r ) . This implies that ψ ( 2 r , 0 ) = 0 , hence r = 0 . Thus we have proved that
d ( x n + 1 , x n ) 0 .

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

Lemma 2.1 For every positive ϵ, there exists a natural number n such that if r , q n with r q 1 ( mod m ) , then d ( x r , x q ) < ϵ .

Proof Assume the contrary. Thus there exists ϵ > 0 such that for any n N , we can find r n > q n n with r n q n 1 ( mod m ) satisfying d ( x r n , x q n ) ϵ .

Now, we take n > 2 m . Then, corresponding to q n n , we can choose r n such that it is the smallest integer with r n > q n satisfying r n q n 1 ( mod m ) and d ( x r n , x q n ) ϵ . Therefore, d ( x r n m , x q n ) < ϵ . By using the triangular inequality, we have
ϵ d ( x q n , x r n ) d ( x q n , x r n m ) + i = 1 m d ( x r n i , x r n i + 1 ) < ϵ + i = 1 m d ( x r n i , x r n i + 1 ) .
Letting n and using d ( x n + 1 , x n ) 0 , we obtain
lim d ( x q n , x r n ) = ϵ .
(2.3)
Again, by the triangular inequality,
ϵ d ( x q n , x r n ) d ( x q n , x q n + 1 ) + d ( x q n + 1 , x r n + 1 ) + d ( x r n + 1 , x r n ) d ( x q n , x q n + 1 ) + d ( x q n + 1 , x q n ) + d ( x q n , x r n ) + d ( x r n , x r n + 1 ) + d ( x r n + 1 , x r n ) .
Letting n and using d ( x n + 1 , x n ) 0 , we get
lim d ( x q n + 1 , x r n + 1 ) = ϵ .
(2.4)
Consider
d ( x q n , T x r n ) = d ( x q n , x r n + 1 ) d ( x q n , x r n ) + d ( x r n , x r n + 1 ) ,
(2.5)
and
d ( x r n , T x q n ) = d ( x r n , x q n + 1 ) d ( x r n , x q n ) + d ( x q n , x q n + 1 ) .
(2.6)
On taking n in inequalities (2.5) and (2.6), we have
lim n d ( x q n , T x r n ) = ϵ ,
(2.7)
and
lim n d ( x r n , T x q n ) = ϵ .
(2.8)
As x q n and x r n lie in different adjacently labeled sets A i and A i + 1 for certain 1 i m , using the fact that T is a Chatterjea-type cyclic weakly contraction, we obtain
μ ( ϵ ) μ ( d ( x q n + 1 , x r n + 1 ) ) = μ ( d ( T x q n , T x r n ) ) μ ( 1 2 [ d ( x q n , T x r n ) + d ( x r n , T x q n ) ] ) ψ ( d ( x q n , T x r n ) , d ( x r n , T x q n ) ) = μ ( 1 2 [ d ( x q n , x r n + 1 ) + d ( x r n , x q n + 1 ) ] ) ψ ( d ( x q n , x r n + 1 ) , d ( x r n , x q n + 1 ) ) .
(2.9)
On taking n in (2.9), using (2.7) and (2.8), the continuity of μ and lower semi-continuity of ψ, we get that
μ ( ε ) μ ( 1 2 [ ε + ε ] ) ψ ( ε , ε ) .

Consequently, ψ ( ε , ε ) 0 , which is contradiction with ε > 0 . Hence the result is proved. □

Now, using Lemma 2.1, we will show that { x n } is a Cauchy sequence in Y. Fix ϵ > 0 . By Lemma 2.1, we can find n 0 N such that r , q n 0 with r q 1 ( mod m )
d ( x r , x q ) ϵ 2 .
(2.10)
Since lim d ( x n , x n + 1 ) = 0 , we can also find n 1 N such that
d ( x n , x n + 1 ) ϵ 2 m
(2.11)

for any n n 1 .

Assume that r , s max { n 0 , n 1 } and s > r . Then there exists k { 1 , 2 , , m } such that s r k ( mod m ) . Hence s r + t = 1 ( mod m ) for t = m k + 1 . So, we have
d ( x r , x s ) d ( x r , x s + j ) + d ( x s + j , x s + j 1 ) + + d ( x s + 1 , x s ) .
(2.12)
Using (2.10), (2.11) and (2.12), we obtain
d ( x r , x s ) ϵ 2 + j × ϵ 2 m ϵ 2 + m × ϵ 2 m = ϵ .
(2.13)

Hence { x n } is a Cauchy sequence in Y. Since Y is closed in X, then Y is also complete and there exists x Y such that lim x n = x .

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

As Y = i = 1 m A i is a cyclic representation of Y with respect to T, the sequence { x n } has infinite terms in each A i for i = { 1 , 2 , , m } . Suppose that x A i , T x A i + 1 and we take a subsequence { x n k } of { x n } with x n k A i . By using the contractive condition, we can obtain
μ ( d ( x n k + 1 , T x ) ) = μ ( d ( T x n k , T x ) ) μ ( 1 2 [ d ( x n k , T x ) + d ( x , T x n k ) ] ) ψ ( d ( x n k , T x ) , d ( x , T x n k ) ) = μ ( 1 2 [ d ( x n k , T x ) + d ( x , x n k + 1 ) ] ) ψ ( d ( x n k , T x ) , d ( x , x n k + 1 ) ) .
Letting n and using the continuity of μ and lower semi-continuity of ψ, we have
μ ( d ( x , T x ) ) μ ( 1 2 d ( x , T x ) ) ψ ( d ( x , T x ) , 0 ) ,

which is a contradiction unless d ( x , T x ) = 0 . Hence x is a fixed point of T.

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

Suppose that x 1 and x 2 ( x 1 x 2 ) are two fixed points of T. Using the contractive condition and the continuity of μ and lower semi continuity of ψ, we have
μ ( d ( x 1 , x 2 ) ) = μ ( d ( T x 1 , T x 2 ) ) μ ( 1 2 [ d ( x 1 , T x 2 ) + d ( x 2 , T x 1 ) ] ) ψ ( d ( x 1 , T x 2 ) , d ( x 2 , T x 1 ) ) = μ ( 1 2 [ d ( x 1 , x 2 ) + d ( x 2 , x 1 ) ] ) ψ ( d ( x 1 , x 2 ) , d ( x 2 , x 1 ) ) = μ ( d ( x 1 , x 2 ) ) ψ ( d ( x 1 , x 2 ) , d ( x 2 , x 1 ) ) μ ( d ( x 1 , x 2 ) ) ,

which is a contradiction unless x 1 = x 2 . Hence the main result is proved. □

If μ ( a ) = a , then we have the following result.

Corrollary 2.1 Let ( X , d ) be a complete metric space, m N , A 1 , A 2 , , A m be non-empty closed subsets of X and Y = i = 1 m A i . Suppose that T : Y Y is an operator such that
  1. (1)

    i = 1 m A i is a cyclic representation of Y with respect to T;

     
  2. (2)

    d ( T x , T y ) 1 2 [ d ( x , T y ) + d ( y , T x ) ] ψ ( d ( x , T y ) , d ( y , T x ) )

     

for any x A i , y A i + 1 , i = 1 , 2 , , m , where A m + 1 = A 1 and ψ Ψ . Then T has a fixed point z i = 1 n A i .

If ψ ( a , b ) = ( 1 2 k ) ( a + b ) , where k [ 0 , 1 2 ) , we have the following result.

Corrollary 2.2 Let ( X , d ) be a complete metric space, m N , A 1 , A 2 , , A m be non-empty closed subsets of X and Y = i = 1 m A i . Suppose that T : Y Y is an operator such that
  1. (1)

    i = 1 m A i is a cyclic representation of Y with respect to T;

     
  2. (2)

    there exists k [ 0 , 1 2 ) such that d ( T x , T y ) k [ d ( x , T y ) + d ( y , T x ) ]

     

for any x A i , y A i + 1 , i = 1 , 2 , , m , where A m + 1 = A 1 . Then T has a fixed point z i = 1 n A i .

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 μ : [ 0 , ) [ 0 , ) satisfying the following hypotheses:
  1. (h1)

    μ is a Lebesgue-integrable mapping on each compact of [ 0 , ) ;

     
  2. (h2)

    for any ϵ > 0 , we have 0 ϵ μ ( t ) > 0 .

     
Corrollary 3.1 Let ( X , d ) be a complete metric space, m N , A 1 , A 2 , , A m be non-empty closed subsets of X and Y = i = 1 m A i . Suppose that T : Y Y is an operator such that
  1. (1)

    i = 1 m A i is a cyclic representation of Y with respect to T;

     
  2. (2)
    there exists k [ 0 , 1 2 ) such that
    0 d ( T x , T y ) α ( s ) d s k 0 d ( x , T y ) + d ( y , T x ) α ( s ) d s
     

for any x A i , y A i + 1 , i = 1 , 2 , , m , where A m + 1 = A 1 and α Λ . Then T has a fixed point z i = 1 n A i .

If we take A i = X , i = 1 , 2 , , m , we obtain the following result.

Corrollary 3.2 Let ( X , d ) be a complete metric space and T : X X be a mapping such that
0 d ( T x , T y ) α ( s ) d s k 0 d ( x , T y ) + d ( y , T x ) α ( s ) d s ,

for any x y X , k [ 0 , 1 2 ) and α Λ . Then T has a fixed point z i = 1 n A i .

Example 3.1 Let X be a subset in endowed with the usual metric. Suppose A 1 = [ 0 , 1 ] , A 2 = [ 0 , 1 2 ] and Y = i = 1 2 A i . Define T : Y Y such that T x = x 5 for all x Y . It is clear that i = 1 2 A i is a cyclic representation of Y with respect to T. Furthermore, if μ : [ 0 , ) [ 0 , ) is given as μ ( t ) = t and ψ : [ 0 , ) 2 [ 0 , ) is given by ψ ( x , y ) = 1 7 ( x + y ) , then ψ Ψ .

Now, we prove that T satisfies the inequality of Chatterjea-type cyclic weakly contraction, i.e., μ ( d ( T x , T y ) ) μ ( 1 2 [ d ( x , T y ) + d ( y , T x ) ] ) ψ ( d ( x , T y ) , d ( y , T x ) ) . To see this fact, we examine three cases.

Case 1. Suppose that x y . Then
μ ( d ( T x , T y ) ) = μ ( | x 5 y 5 | ) = x y 5
(3.1)
and
(3.2)
If y < x 5 , then
x y 5 5 14 [ x y 5 + x 5 y ] = 3 7 ( x y ) .

Hence, the given inequality is satisfied.

If y x 5 , then
x y 5 5 14 [ x y 5 + y x 5 ] = 2 7 ( x + y ) .

Hence the given inequality is satisfied.

Case 2. Suppose that y 5 x y . Then from (3.1) and (3.2), we have
x y 5 5 14 [ x y 5 + y x 5 ] = 2 7 ( x + y ) .

Hence the given inequality is satisfied.

Case 3. Finally, suppose that y 5 x . Then from (3.1) and (3.2), we have
μ ( d ( T x , T y ) ) = μ ( | x 5 y 5 | ) = y x 5
and
x y 5 5 14 [ x y 5 + y x 5 ] = 2 7 ( x + y ) .

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 z = 0 i = 1 2 A i ).

Declarations

Acknowledgements

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

Authors’ 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

References

  1. Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71–76.MathSciNetGoogle Scholar
  2. Kannan R: Some results on fixed points-II. Am. Math. Mon. 1969, 76: 405–408. 10.2307/2316437View ArticleGoogle Scholar
  3. Chatterjea SK: Fixed point theorem. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.MathSciNetGoogle Scholar
  4. Alber Y, Guerre-Delabriere S: Principles of Weakly Contractive Maps in Hilbert Spaces. Oper. Theory Adv. Appl. 8. In New Results in Operator Theory and Its Applications. Edited by: Gohberg I, Lyubich Y. Birkhäuser, Basel; 1997:7–22.View ArticleGoogle Scholar
  5. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleGoogle Scholar
  6. Boyd DW, Wong TSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView ArticleGoogle Scholar
  7. Reich S: Some fixed point problems. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1975, 57: 194–198.Google Scholar
  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., 2012: Article ID 44Google Scholar
  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., 2012: Article ID 359054Google Scholar
  10. Chandok S: Some common fixed point theorems for generalized f -weakly contractive mappings. J. Appl. Math. Inform. 2011, 29: 257–265.MathSciNetGoogle Scholar
  11. Chandok S: Some common fixed point theorems for generalized nonlinear contractive mappings. Comput. Math. Appl. 2011, 62: 3692–3699. 10.1016/j.camwa.2011.09.009MathSciNetView ArticleGoogle Scholar
  12. Chandok S: Common fixed points, invariant approximation and generalized weak contractions. Int. J. Math. Math. Sci. 2012., 2012: Article ID 102980Google Scholar
  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. 2012, 17: 301–306.Google Scholar
  14. Chandok S: Common fixed points for generalized nonlinear contractive mappings in metric spaces. Mat. Vesn. 2013, 65: 29–34.MathSciNetGoogle Scholar
  15. Chandok S: A fixed point result for weakly Kannan type cyclic contractions. Int. J. Pure Appl. Math. 2013, 82(2):253–260.Google Scholar
  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)Google Scholar
  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)Google Scholar
  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. 2013, 7(9):433–440.MathSciNetGoogle Scholar
  19. Haghi RH, Postolache M, Rezapour S: On T-stability of the Picard iteration for generalized φ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971Google Scholar
  20. Karapinar E, Sadarangani K: Fixed point theory for cyclic ( ϕ - ψ ) -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69Google Scholar
  21. Karapinar E, Erhan IM: Best proximity on different type contractions. Appl. Math. Inf. Sci. 2011, 5: 342–353.MathSciNetGoogle Scholar
  22. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory Appl. 2003, 4(1):79–89.MathSciNetGoogle Scholar
  23. Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012, 218(12):6727–6732. 10.1016/j.amc.2011.12.038MathSciNetView ArticleGoogle Scholar
  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., 2012: Article ID 856302Google Scholar

Copyright

© Chandok and Postolache; licensee Springer 2013

This article is published under license to BioMed Central Ltd. 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.

Advertisement