Open Access

A fixed point theorem for cyclic generalized contractions in metric spaces

Fixed Point Theory and Applications20122012:122

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

Received: 13 March 2012

Accepted: 4 July 2012

Published: 23 July 2012

The Research to this article has been published in Fixed Point Theory and Applications 2013 2013:39

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 : X X be an operator. We say that f is a contraction if there exists λ [ 0 , 1 ) such that, for all x , y X ,
d ( f ( x ) , f ( y ) ) λ d ( 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 : X X is an operator such that there exists fixed constants γ 0 , α 1 and β [ 0 , α ] such that, for every ε [ 0 , 1 ] and every x , y X ,
d ( f ( x ) , f ( y ) ) ( 1 ε ) 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 ) ) λ d ( x , y ) , for every  x , y X  and some  λ [ 0 , 1 ) ,
then it can be verified that, for every x , y X , we have
d ( f ( x ) , f ( y ) ) ( 1 ε ) 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 + x 2 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 : X X 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  x X .

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

where γ 0 , α 1 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 x Y ;

     
  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 = j mod m and 1 i m . 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 ) , n 2 ,
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 , β α , 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
1 a ( 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
r K ψ ( 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 p 1 , suppose there exists j, 0 j m 1 , such that ( n + p ) n + j = 1 mod m , i.e., p + j = 1 mod m . 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 = 1 mod m , x n and x n + p lie in different sets A i and A i + 1 , for some 1 i m . Then by (2.1) we have
q n + 1 = ( n + 1 ) α ( 1 ε ) d ( x n , x n + p ) + C ( n + 1 ) α ε α ψ ( ε ) ,
(2.3)
where C = sup γ ( 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 y Y = i = 1 m A i . The case j 0 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 x Y , we get the same limit point x i = 1 m A i . Indeed, for x Y = 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 ) ) ( 1 ε ) d ( x , z ) + K ε ψ ( 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 ) K ψ ( 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 x Y .

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 n ψ ( 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 : Y Y 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 ) ) λ d ( 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 x X , 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 m N with m 2 , we have
d ( f m ( x ) , f m ( y ) ) ( 1 ε ) d ( x , y ) + m γ 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 : Y Y be an operator such that f ( x ) x for each x Y . 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 γ 0 , α 1 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. □

Notes

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University
(2)
Department of Mathematics, Babeş-Bolyai University
(3)
Department of Mathematics, King Abdulaziz University

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