Skip to main content

A fixed point theorem for cyclic generalized contractions in metric spaces

A Research to this article was published on 26 February 2013

Abstract

In this paper, we extend a recent result of V. Pata (J. Fixed Point Theory Appl. 10:299-305, 2011) in the frame of a cyclic representation of a complete metric space.

1 Introduction

One of the fundamental result in fixed point theory is the Banach contraction principle. It has various non-trivial applications in many branches of pure and applied sciences (see, for instance, [2, 7, 14] and references cited therein).

Let (X,d) be a metric space and f:XX be an operator. We say that f is a contraction if there exists λ[0,1) such that, for all x,yX,

d ( f ( x ) , f ( y ) ) λd(x,y).
(1.1)

In terms of Picard operator theory (see [13]), Banach contraction principle asserts that if f is a contraction and (X,d) is complete, then f is a Picard operator. This result has been extended to other important classes of maps. Recently, Pata [8] proved that if (X,d) is a complete metric space and f:XX is an operator such that there exists fixed constants γ0, α1 and β[0,α] such that, for every ε[0,1] and every x,yX,

d ( f ( x ) , f ( y ) ) (1ε)d(x,y)+γ ε α ψ(ε) [ 1 + x + y ] β
(1.2)

(where ψ:[0,1][0,) is an increasing function vanishing with continuity at zero and x:=d(x, x 0 ), with arbitrary x 0 X), then f has a unique fixed point in X.

Remark 1.1 (see [8])

The condition (1.2) is weaker than the contraction condition (1.1). In fact, if

d ( f ( x ) , f ( y ) ) λd(x,y),for every x,yX and some λ[0,1),

then it can be verified that, for every x,yX, we have

d ( f ( x ) , f ( y ) ) (1ε)d(x,y)+γ ε 1 + θ [ 1 + x + y ] ,for every θ>0,

where

γ=γ(θ,λ)= θ θ ( 1 + θ ) 1 + θ 1 ( 1 λ ) θ .

Remark 1.2 (see [8])

The function f:[1,)[1,) defined as

f(x)=2+x2 x +4 x 4

has a unique fixed point x =1, but fails to be a contraction on any neighborhood both of 1 and of ∞.

Kirk, Srinivasan and Veeramani [6] obtained an extension of Banach’s fixed point theorem for mappings satisfying cyclical contractive conditions. Some generalizations of the results given in [6], using the setting of so-called fixed point structures, are presented in I. A. Rus [12]. In [10], Păcurar and Rus established a fixed point theorem for cyclic φ-contractions and they further discussed fixed point theory in metric spaces. In [3], Karapinar proved a fixed point theorem for cyclic weak φ-contraction mappings. Some other recent results concerning this topic are given in [1, 4, 5, 9, 11].

In the present paper, we obtain a fixed point theorem for a generalized contraction in the sense of the assumption (1.2), defined on a cyclic representation of a complete metric space.

2 Main results

We need first to recall a known concept.

Definition 2.1 ([3])

Let X be a nonempty set, m be a positive integer and f:XX an operator. Then, we say that i = 1 m A i is a cyclic representation of X with respect to f if:

  1. (i)

    X= i = 1 m A i , where A i are nonempty sets for each i{1,,m};

  2. (ii)

    f( A 1 ) A 2 ,,f( A m 1 ) A m ,f( A m ) A 1 .

Let (X,d) be a complete metric space. Selecting an arbitrary x 1 X, we denote

x:=d(x, x 1 ),for all xX.

Our main result is as follows.

Theorem 2.2 Let (X,d) be a complete metric space, m be a positive integer, A 1 ,, A m be closed nonempty subsets of X, Y:= i = 1 m A i , ψ:[0,1][0,) be an increasing function vanishing with continuity at zero, and f:YY be an operator. Assume that:

  1. 1.

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

  2. 2.

    For every ε[0,1], x A i , and y A i + 1 (i{1,,m}, where A m + 1 = A 1 ), we have

    d ( f ( x ) , f ( y ) ) (1ε)d(x,y)+γ ε α ψ(ε) [ 1 + x + y ] β ,
    (2.1)

where γ0, α1 and β[0,α] are fixed constants.

Then, we have the following conclusions:

  1. (i)

    f is a Picard operator, i.e., f has a unique fixed point x i = 1 m A i and the Picard iteration sequence { f n ( x ) } n N converges to x , for any initial point xY;

  2. (ii)

    the following estimates hold:

    d ( x n , x ) x , n 2 ; d ( x n , x 1 ) 2 x , n 2 .

Proof (i) For convenience of notation, if j>m, define A j = A i where i=jmodm and 1im. Let x 1 A 1 . Starting from x 1 , let { x n } n 1 be the Picard iteration defined by the sequence

x n =f( x n 1 )= f n 1 ( x 1 ),n2,

and set c n = x n . Assume x n x n + 1 for all n. By (2.1), we have

d( x n , x n + 1 )d( x n 1 , x n )d( x 1 , x 2 )= c 2 .
(2.2)

First, we prove that the sequence ( c n ) n N is bounded. By (2.2) we get that

c n d ( x n , x n + 1 ) + d ( x n + 1 , x 2 ) + d ( x 2 , x 1 ) d ( x n + 1 , x 2 ) + 2 c 2 = d ( f ( x n ) , f ( x 1 ) ) + 2 c 2 .

Since x 1 A 1 and x n A n , from (2.1), we obtain that

c n ( 1 ε ) d ( x n , x 1 ) + γ ε α ψ ( ε ) [ 1 + x n + x 1 ] β + 2 c 2 = ( 1 ε ) c n + γ ε α ψ ( ε ) [ 1 + c n ] β + 2 c 2 ( 1 ε ) c n + a ε α ψ ( ε ) c n α + b ,

where c 1 = x 1 =d( x 1 , x 1 )=0, βα, and for some a,b>0. Thus,

ε c n a ε α ψ(ε) c n α +b.

If there is a subsequence ( c n k ) k N , the choice ε= ε k = ( 1 + b ) c n k leads to the contradiction

1a ( 1 + b ) α ψ( ε k )0.

Therefore, the sequence ( c n ) is bounded.

From (2.2) we obtain that the sequence {d( x n , x n + 1 )} is nonincreasing and then it is convergent to the real number

lim n d( x n , x n + 1 )=r=inf { d ( x n 1 , x n ) : n = 2 , 3 , } .

Now we show that r=0. Assume that r>0. Let x n A n and x n + 1 A n + 1 . By (2.1), we have

r d ( x n , x n + 1 ) = d ( f ( x n 1 ) , f ( x n ) ) ( 1 ε ) d ( x n 1 , x n ) + γ ε α ψ ( ε ) [ 1 + x n 1 + x n ] β ( 1 ε ) d ( x n 1 , x n ) + K ε ψ ( ε ) ,

for some K>0. Letting n, we obtain

rKψ(ε),for every ε[0,1],

which implies r=0. This leads to a contradiction, therefore

lim n d( x n , x n + 1 )=0.

For p1, suppose there exists j, 0jm1, such that (n+p)n+j=1modm, i.e., p+j=1modm. Now, let p be fixed, j=0 and let

q n = n α d( x n , x n + p ).

So, we have

q n + 1 = ( n + 1 ) α d( x n + 1 , x n + 1 + p )= ( n + 1 ) α d ( f ( x n ) , f ( x n + p ) ) .

Since p=1modm, x n and x n + p lie in different sets A i and A i + 1 , for some 1im. Then by (2.1) we have

q n + 1 = ( n + 1 ) α (1ε)d( x n , x n + p )+C ( n + 1 ) α ε α ψ(ε),
(2.3)

where C=supγ ( 1 + 2 c n ) β <. Choosing for each n

ε=1 ( n n + 1 ) α α n + 1 ,

the relation (2.3) becomes

q n + 1 n α d( x n , x n + p )+C α α ψ ( α n + 1 ) = q n +C α α ψ ( α n + 1 ) .

Since q 0 =0, it follows that

q n = k = 1 n ( q k q k 1 ) k = 1 n C α α ψ ( α k ) =C α α k = 1 n ψ ( α k ) .

Consequently,

d( x n , x n + p )C ( α n ) α k = 1 n ψ ( α k ) .

This shows that { x n } is a Cauchy sequence in the complete metric space (Y,d) and, thus, it is convergent to a point yY= i = 1 m A i . The case j0 similar.

On the other hand, the sequence { x n } has an infinite number of terms in each A i , for every i{1,,m}. Since (Y,d) is complete, in each A i , i{1,,m} we can construct a subsequence of { x n } which converges to y. Since each A i is closed for i{1,,m}, we get that y i = 1 m A i . Then i = 1 m A i and we can consider the restriction

which satisfies the conditions of Theorem 1 in [8], since i = 1 m A i is also closed and complete. From this result, it follows that g has a unique fixed point, say x i = 1 m A i .

We claim now that for any initial value xY, we get the same limit point x i = 1 m A i . Indeed, for xY= i = 1 m A i , by repeating the above process, the corresponding iterative sequence yields that g has a unique fixed point, say z i = 1 m A i . Since x , z i = 1 m A i , we have x , z A i for all i{1,,m} and, hence, d( x ,z) and d(f( x ),f(z)) are well defined. We can write (2.1) in the form

d ( x , z ) =d ( f ( x ) , f ( z ) ) (1ε)d ( x , z ) +Kεψ(ε),

for some K>0. Suppose that ε=0. Then we have

d ( f ( x ) , f ( z ) ) d ( x , z ) .

If equality occurs, the relation

d ( x , z ) Kψ(ε)

is valid for every ε[0,1], which implies d( x ,z)=0. Thus, x is the unique fixed point of f for any initial value xY.

To prove that the Picard iteration converges to x , let us consider x 1 Y= i = 1 m A i . Then there exists i 0 {1,,m} such that x n A i 0 . As x i = 1 m A i it follows that x A i 0 + 1 as well. By the continuity of f, we obtain

d ( f n 1 ( x 1 ) , x ) =d ( f ( x n 1 ) , x ) =d ( x n , x ) = lim p d( x n , x n + p )C ( α n ) α k = 1 nψ ( α k ) .

Letting n, it follows that ( x n ) x , i.e., the Picard iteration converges to the unique fixed point of f for any initial point x 1 Y.

  1. (ii)

    Since x is a fixed point and x i = 1 m A i , we obtain that

    d ( x n , x ) =d ( f ( x n 1 ) , f ( x ) ) d ( x n 1 , x ) d ( x 1 , x ) = x .
    (2.4)

By (2.4), it follows that

d( x n , x 1 )d ( x n , x ) +d ( x , x 1 ) x +d ( x , x 1 ) 2 x .

 □

In view of Remark 1.1, we immediately obtain the following corollary.

Corollary 2.3 (Kirk, Srinivasan, Veeramani [2], Theorem 1.3])

Let (X,d) be a complete metric space, m be a positive integer, A 1 ,, A m be closed nonempty subsets of X, Y:= i = 1 m A i and f:YY be an operator. Assume that:

  1. (i)

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

  2. (ii)

    there exists λ[0,1) such that, for any x A i , y A i + 1 , where A m + 1 = A 1 , we have

    d ( f ( x ) , f ( y ) ) λd(x,y).

Then f has a unique fixed point x i = 1 m A i .

Finally, we will prove a periodic point theorem. For this purpose, notice first that if f satisfies (1.2) with constants α, β, γ and function ψ, and if f(x)x for each xX, then its m-iterate f m also satisfies the condition (1.2) with constants α, β, and function ψ. Indeed, let us suppose that f satisfies (1.2) with constants α, β, γ. Then, for every ε[0,1], we have

d ( f 2 ( x ) , f 2 ( y ) ) ( 1 ε ) d ( f ( x ) , f ( y ) ) + γ ε α ψ ( ε ) [ 1 + f ( x ) + f ( y ) ] β ( 1 ε ) [ ( 1 ε ) d ( x , y ) + γ ε α ψ ( ε ) ( 1 + x + y ) β ] + γ ε α ψ ( ε ) [ 1 + f ( x ) + f ( y ) ] β ( 1 ε ) [ ( 1 ε ) d ( x , y ) + γ ε α ψ ( ε ) ( 1 + x + y ) β ] + γ ε α ψ ( ε ) [ 1 + x + y ] β = ( 1 ε ) 2 d ( x , y ) + ( 1 ε ) γ ε α ψ ( ε ) ( 1 + x + y ) β + γ ε α ψ ( ε ) [ 1 + x + y ] β = ( 1 ε ) 2 d ( x , y ) + ( 2 ε ) γ ε α ψ ( ε ) ( 1 + x + y ) β ( 1 ε ) d ( x , y ) + 2 γ ε α ψ ( ε ) ( 1 + x + y ) β .

Thus, we immediately get that, for mN with m2, we have

d ( f m ( x ) , f m ( y ) ) (1ε)d(x,y)+mγ ε α ψ(ε) ( 1 + x + y ) β .

Notice also that if i = 1 m A i is a cyclic representation of X with respect to f, then each A i (i{1,2,,m}) is an invariant set with respect to f m . Using these two remarks, we get the following periodic point theorem.

Theorem 2.4 Let (X,d) be a complete metric space, m be a positive integer, A 1 ,, A m be nonempty subsets of X, Y:= i = 1 m A i , ψ:[0,1][0,) be an increasing function vanishing with continuity at zero and f:YY be an operator such that f(x)x for each xY. Assume that:

  1. 1.

    i = 1 m A i is a cyclic representation of Y with respect to f.

  2. 2.

    There exists i 0 {1,,m} such that A i 0 is closed.

  3. 3.

    For every ε[0,1] and each x,y A i 0 , we have

where γ0, α1 and β[0,α] are fixed constants.

Then, f m has a fixed point.

Proof Notice that, by the above considerations, f m is a self mapping on A i 0 and it satisfies the condition (1.2) with constants α, β, and function ψ. Thus, by Theorem 1 in [8] we get the conclusion. □

References

  1. Agarwal RP, Alghamdi M, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40

    Google Scholar 

  2. de Bakker JW, de Vink EP: Denotational models for programming languages: applications of Banach’s fixed point theorem. Topol. Appl. 1998, 85(1–3):35–52. 10.1016/S0166-8641(97)00140-5

    MathSciNet  Article  Google Scholar 

  3. Karapinar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016

    MathSciNet  Article  Google Scholar 

  4. Karapinar E, Erhan IM, Ulus AY: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 2012, 6: 239–244.

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  7. Kunze HE, La Torre D: Solving inverse problems for differential equations by the collage method and application to an economic growth model. Int. J. Optim. Theory Methods Appl. 2009, 1(1):26–35.

    MathSciNet  Google Scholar 

  8. Pata V: A fixed point theorem in metric spaces. J. Fixed Point Theory Appl. 2011, 10: 299–305. 10.1007/s11784-011-0060-1

    MathSciNet  Article  Google Scholar 

  9. Păcurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2011, 12: 419–428.

    MathSciNet  Google Scholar 

  10. Păcurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002

    MathSciNet  Article  Google Scholar 

  11. Petruşel G: Cyclic representations and periodic points. Stud. Univ. Babeş-Bolyai, Math. 2005, 51: 107–112.

    Google Scholar 

  12. Rus IA: Cyclic representations and fixed points. Ann. Tiberiu Popoviciu Semin. 2005, 3: 171–178.

    Google Scholar 

  13. Rus IA: Picard operators and applications. Sci. Math. Jpn. 2003, 58: 191–219.

    MathSciNet  Google Scholar 

  14. Serov VS, Schüermann HW, Svetogorova E: Application of the Banach fixed-point theorem to the scattering problem at a nonlinear three-layer structure with absorption. Fixed Point Theory Appl. 2010., 2010: Article ID 439682

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Naseer Shahzad.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2013-39.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Alghamdi, M.A., Petruşel, A. & Shahzad, N. A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory Appl 2012, 122 (2012). https://doi.org/10.1186/1687-1812-2012-122

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2012-122

Keywords

  • Point Theorem
  • Fixed Point Theorem
  • Nonempty Subset
  • Fixed Point Theory
  • Unique Fixed Point