Open Access

On ( α , ψ ) -contractive multi-valued mappings

Fixed Point Theory and Applications20132013:137

https://doi.org/10.1186/1687-1812-2013-137

Received: 14 February 2013

Accepted: 10 May 2013

Published: 28 May 2013

Abstract

In this paper, we generalize the contractive condition for multi-valued mappings given by Asl, Rezapour and Shahzad in 2012. We establish some fixed point theorems for multi-valued mappings from a complete metric space to the space of closed or bounded subsets of the metric space satisfying generalized ( α , ψ ) -contractive condition.

MSC:47H10, 54H25.

Keywords

α-admissible α -ψ-contractive mappinggeneralized ( α , ψ ) -contractive mapping

1 Introduction

Samet et al. [1] introduced the notion of α-ψ-contractive self-mappings of a metric space. Recently, Asl et al. [2] introduced the notion of α -ψ-contractive mappings to extend the notion α-ψ-contractive mappings. In this paper, we generalize the notion of α -ψ-contractive mappings and prove some fixed point theorems for such mappings.

Let Ψ be a family of nondecreasing functions, ψ : [ 0 , ) [ 0 , ) such that n = 1 ψ n ( t ) < for each t > 0 , where ψ n is the n th iterate of ψ. It is known that for each ψ Ψ , we have ψ ( t ) < t for all t > 0 and ψ ( 0 ) = 0 for t = 0 [1]. Let ( X , d ) be a metric space. A mapping G : X X is called α-ψ-contractive if there exist two functions α : X × X [ 0 , ) and ψ Ψ such that α ( x , y ) d ( G x , G y ) ψ ( d ( x , y ) ) for each x , y X . A mapping G : X X is called α-admissible [1] if α ( x , y ) 1 α ( G x , G y ) 1 . We denote by N ( X ) the space of all nonempty subsets of X, by B ( X ) the space of all nonempty bounded subsets of X and by CL ( X ) the space of all nonempty closed subsets of X. For A N ( X ) and x X , d ( x , A ) = inf { d ( x , a ) : a A } . For every A , B B ( X ) , δ ( A , B ) = sup { d ( a , b ) : a A , b B } . When A = { x } , we denote δ ( A , B ) by δ ( x , B ) . For every A , B CL ( X ) , let
H ( A , B ) = { max { sup x A d ( x , B ) , sup y B d ( y , A ) } if the maximum exists ; otherwise .

Such a map H is called generalized Hausdorff metric induced by d. Let ( X , , d ) be an ordered metric space and A , B X . We say that A r B if for each a A and b B , we have a b . We give a few definitions and the result due to Asl et al. [2] for convenience.

Definition 1.1 [2]

Let ( X , d ) be a metric space and let α : X × X [ 0 , ) be a mapping. A mapping G : X CL ( X ) is α -admissible if α ( x , y ) 1 α ( G x , G y ) 1 , where α ( G x , G y ) = inf { α ( a , b ) : a G x , b G y } .

Definition 1.2 [2]

Let ( X , d ) be a metric space. A mapping G : X CL ( X ) is called α -ψ-contractive if there exist two functions α : X × X [ 0 , ) and ψ Ψ such that
α ( G x , G y ) H ( G x , G y ) ψ ( d ( x , y ) )
(1.1)

for all x , y X .

Theorem 1.3 [2]

Let ( X , d ) be a complete metric space, let α : X × X [ 0 , ) be a function, let ψ Ψ be a strictly increasing map and T be a closed-valued, α -admissible and α -ψ-contractive multi-function on X. Suppose that there exist x 0 X and x 1 G x 0 such that α ( x 0 , x 1 ) 1 . Assume that if { x n } is a sequence in X such that α ( x n , x n + 1 ) 1 for all n and x n x , then α ( x n , x ) 1 for all n. Then G has a fixed point.

2 Main results

We begin this section by introducing the following definition.

Definition 2.1 Let ( X , d ) be a metric space and let G : X CL ( X ) be a mapping. We say that G is generalized ( α , ψ ) -contractive if there exists ψ Ψ such that
α ( G x , G y ) d ( y , G y ) ψ ( d ( x , y ) )
(2.1)

for each x X and y G x , where α ( G x , G y ) = inf { α ( a , b ) : a G x , b G y } .

Note that an α -ψ-contractive mapping is generalized ( α , ψ ) -contractive. In case when ψ Ψ is strictly increasing, generalized ( α , ψ ) -contractive is called strictly generalized ( α , ψ ) -contractive. The following lemma is inspired by [[3], Lemma 2.2].

Lemma 2.2 Let ( X , d ) be a metric space and B CL ( X ) . Then, for each x X with d ( x , B ) > 0 and q > 1 , there exists an element b B such that
d ( x , b ) < q d ( x , B ) .
(2.2)
Proof It is given that d ( x , B ) > 0 . Choose
ϵ = ( q 1 ) d ( x , B ) .
Then, by using the definition of d ( x , B ) , it follows that there exists b B such that
d ( x , b ) < d ( x , B ) + ϵ = q d ( x , B ) .

 □

Lemma 2.3 Let ( X , d ) be a metric space and G : X CL ( X ) . Assume that there exists a sequence { x n } in X such that lim n d ( x n , G x n ) = 0 and x n x X . Then x is a fixed point of G if and only if the function f ( ξ ) = d ( ξ , G ξ ) is lower semi-continuous at x.

Proof Suppose f ( ξ ) = d ( ξ , G ξ ) is lower semi-continuous at x, then
d ( x , G x ) lim inf n f ( x n ) = lim inf n d ( x n , G x n ) = 0 .

By the closedness of G it follows that x G x . Conversely, suppose that x is a fixed point of G, then f ( x ) = 0 lim inf n f ( x n ) . □

Theorem 2.4 Let ( X , d ) be a complete metric space and let G : X CL ( X ) be an α -admissible strictly generalized ( α , ψ ) -contractive mapping. Assume that there exist x 0 X and x 1 G x 0 such that α ( x 0 , x 1 ) 1 . Then x is a fixed point of G if and only if f ( ξ ) = d ( ξ , G ξ ) is lower semi-continuous at x.

Proof By the hypothesis, there exist x 0 X and x 1 G x 0 such that α ( x 0 , x 1 ) 1 . If x 0 = x 1 , then we have nothing to prove. Let x 0 x 1 . If x 1 G x 1 , then x 1 is a fixed point. Let x 1 G x 1 . Since G is α -admissible, so α ( G x 0 , G x 1 ) 1 , we have
0 < d ( x 1 , G x 1 ) α ( G x 0 , G x 1 ) d ( x 1 , G x 1 ) .
(2.3)
For given q > 1 by Lemma 2.2, there exists x 2 G x 1 such that
0 < d ( x 1 , x 2 ) < q d ( x 1 , G x 1 ) .
(2.4)
It follows from (2.3), (2.4) and (2.1) that
0 < d ( x 1 , x 2 ) < q ψ ( d ( x 0 , x 1 ) ) .
(2.5)
It is clear that x 1 x 2 and α ( x 1 , x 2 ) 1 . Thus α ( G x 1 , G x 2 ) 1 . Since ψ is strictly increasing, by (2.5), we have
ψ ( d ( x 1 , x 2 ) ) < ψ ( q ψ ( d ( x 0 , x 1 ) ) ) .
Put q 1 = ψ ( q ψ ( d ( x 0 , x 1 ) ) ) ψ ( d ( x 1 , x 2 ) ) , then q 1 > 1 . If x 2 G x 2 , then x 2 is a fixed point. Let x 2 G x 2 , then by Lemma 2.2, there exists x 3 G x 2 such that
0 < d ( x 2 , x 3 ) < q 1 d ( x 2 , G x 2 ) q 1 α ( G x 1 , G x 2 ) d ( x 2 , G x 2 ) q 1 ψ ( d ( x 1 , x 2 ) ) = ψ ( q ψ ( d ( x 0 , x 1 ) ) ) .
It is clear that x 2 x 3 , α ( x 2 , x 3 ) 1 and ψ ( d ( x 2 , x 3 ) ) < ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) . Now put q 2 = ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) ψ ( d ( x 2 , x 3 ) ) . Then q 2 > 1 . If x 3 G x 3 , then x 3 is a fixed point. Let x 3 G x 3 . Then by Lemma 2.2 there exists x 4 G x 3 such that
0 < d ( x 3 , x 4 ) < q 2 d ( x 3 , G x 3 ) q 2 α ( G x 2 , G x 3 ) d ( x 3 , G x 3 ) q 2 ψ ( d ( x 2 , x 3 ) ) = ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) .
By continuing the same process, we get a sequence { x n } in X such that x n + 1 G x n . Also, x n x n + 1 , α ( x n , x n + 1 ) 1 and 0 < d ( x n , x n + 1 ) < ψ n 1 ( q ψ ( d ( x 0 , x 1 ) ) ) or
0 < d ( x n , G x n ) < ψ n 1 ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.6)
For each m > n , we have
d ( x n , x m ) i = n m 1 d ( x i , x i + 1 ) < i = n m 1 ψ i 1 ( q ψ ( d ( x 0 , x 1 ) ) ) .
Since ψ Ψ , it follows that { x n } is a Cauchy sequence in X. Thus there is x X such that x n x . Letting n in (2.6), we have
lim n d ( x n , G x n ) = 0 .
(2.7)

The rest of the proof follows from Lemma 2.3. □

Example 2.5 Let X = R be endowed with the usual metric d. Define G : X CL ( X ) and α : X × X [ 0 , ) by
G x = { [ x , ) if  x 0 , ( , x 2 ] if  x < 0
(2.8)
and
α ( x , y ) = { 1 if  x , y 0 , 0 otherwise .
(2.9)
Let ψ ( t ) = t 2 for all t 0 . For each x X and y G x , we have
α ( G x , G y ) d ( y , G y ) = 0 1 2 d ( x , y ) .

Hence G is a strictly generalized ( α , ψ ) -contractive mapping. Clearly, G is α -admissible. Also, we have x 0 = 1 and x 1 = 1 G x 0 such that α ( x 0 , x 1 ) = 1 . Therefore, all conditions of Theorem 2.4 are satisfied and G has infinitely many fixed points. Note that Theorem 1.3 in Section 1 is not applicable here. For example, take x = 1 and y = 1 .

Corollary 2.6 Let ( X , , d ) be a complete ordered metric space, ψ Ψ be a strictly increasing map and G : X CL ( X ) be a mapping such that for each x X and y G x with x y , we have
d ( y , G y ) ψ ( d ( x , y ) ) .
(2.10)
Also, assume that
  1. (i)

    there exist x 0 X and x 1 G x 0 such that x 0 x 1 ,

     
  2. (ii)

    if x y , then G x r G y .

     

Then x is a fixed point of G if and only if f ( ξ ) = d ( ξ , G ξ ) is lower semi-continuous at x.

Proof Define α : X × X [ 0 , ) by
α ( x , y ) = { 1 if  x y , 0 otherwise .

By using condition (i) and the definition of α, we have α ( x 0 , x 1 ) = 1 . Also, from condition (ii), we have x y implies G x r G y ; by using the definitions of α and r , we have α ( x , y ) = 1 implies α ( G x , G y ) = 1 . Moreover, it is easy to check that G is a strictly generalized ( α , ψ ) -contractive mapping. Therefore, by Theorem 2.4, x is a fixed point of G if and only if f ( ξ ) = d ( ξ , G ξ ) is lower semi-continuous at x. □

Definition 2.7 Let ( X , d ) be a metric space and G : X B ( X ) be a mapping. We say that G is a generalized ( α , ψ , δ ) -contractive mapping if there exists ψ Ψ such that
α ( G x , G y ) δ ( y , G y ) ψ ( d ( x , y ) )
(2.11)

for each x X and y G x , where α ( G x , G y ) = inf { α ( a , b ) : a G x , b G y } .

Lemma 2.8 Let ( X , d ) be a metric space and G : X B ( X ) . Assume that there exists a sequence { x n } in X such that lim n δ ( x n , G x n ) = 0 and x n x X . Then { x } = G x if and only if the function f ( ξ ) = δ ( ξ , G ξ ) is lower semi-continuous at x.

Proof Suppose that f ( ξ ) = δ ( ξ , G ξ ) is lower semi-continuous at x, then
δ ( x , G x ) lim inf n f ( x n ) = lim inf n δ ( x n , G x n ) = 0 .

Hence, { x } = G x because δ ( A , B ) = 0 implies A = B = { a } . Conversely, suppose that { x } = G x . Then f ( x ) = 0 lim inf n f ( x n ) . □

Theorem 2.9 Let ( X , d ) be a complete metric space and let G : X B ( X ) be an α -admissible generalized ( α , ψ , δ ) -contractive mapping. Assume that there exist x 0 X and x 1 G x 0 such that α ( x 0 , x 1 ) 1 . Then there exists x X such that { x } = G x if and only if f ( ξ ) = δ ( ξ , G ξ ) is lower semi-continuous at x.

Proof By the hypothesis of the theorem, there exist x 0 X and x 1 G x 0 such that α ( x 0 , x 1 ) 1 . Assume that x 0 x 1 , for otherwise, x 0 is a fixed point. Let x 1 G x 1 . As G is α -admissible, we have α ( G x 0 , G x 1 ) 1 . Then
δ ( x 1 , G x 1 ) α ( G x 0 , G x 1 ) δ ( x 1 , G x 1 ) ψ ( d ( x 0 , x 1 ) ) .
(2.12)
Since G x 1 , there is x 2 G x 1 . Then
0 < d ( x 1 , x 2 ) δ ( x 1 , G x 1 ) .
(2.13)
From (2.12) and (2.13), we have
0 < d ( x 1 , x 2 ) ψ ( d ( x 0 , x 1 ) ) .
(2.14)
Since ψ is nondecreasing, we have
ψ ( d ( x 1 , x 2 ) ) ψ 2 ( d ( x 0 , x 1 ) ) .
(2.15)
As x 2 G x 1 , we have α ( x 1 , x 2 ) 1 . Since G x 2 , there is x 3 G x 2 . Assume that x 2 x 3 , for otherwise, x 2 is a fixed point of G. Then
0 < d ( x 2 , x 3 ) δ ( x 2 , G x 2 ) α ( G x 1 , G x 2 ) δ ( x 2 , G x 2 ) ψ ( d ( x 1 , x 2 ) ) ψ 2 ( d ( x 0 , x 1 ) ) .
(2.16)
Since ψ is nondecreasing, we have
ψ ( d ( x 2 , x 3 ) ) ψ 3 ( d ( x 0 , x 1 ) ) .
(2.17)
By continuing in this way, we get a sequence { x n } in X such that x n + 1 G x n and x n x n + 1 for n = 0 , 1 , 2 , 3 ,  . Further we have
0 < d ( x n , x n + 1 ) δ ( x n , G x n ) ψ n ( d ( x 0 , x 1 ) ) .
(2.18)
For each m > n , we have
d ( x n , x m ) i = n m 1 d ( x i , x i + 1 ) i = n m 1 ψ i ( d ( x 0 , x 1 ) ) .
Since ψ Ψ , it follows that { x n } is a Cauchy sequence in X. As X is complete, there exists x X such that x n x . Letting n in (2.18), we have
lim n δ ( x n , G x n ) = 0 .
(2.19)

The rest of the proof follows from Lemma 2.8. □

Example 2.10 Let X = { 0 , 2 , 4 , 6 , 8 , 10 , } be endowed with the usual metric d. Define G : X B ( X ) and α : X × X [ 0 , ) by
G x = { { ( x 2 ) , x } if  x 0 , { 0 } if  x = 0
and
α ( x , y ) = { 0 if  x = y 0 , 1 if  x = y = 0 , 1 4 otherwise .
Let ψ ( t ) = t 2 for all t 0 . For each x X and y G x , we have
α ( G x , G y ) δ ( y , G y ) 1 2 ( d ( x , y ) ) .

Hence G is a generalized ( α , ψ , δ ) -contractive mapping. Clearly, G is α -admissible. Also, we have x 0 = 0 X and x 1 = 0 G 0 such that α ( x 0 , x 1 ) = 1 . Therefore, all conditions of Theorem 2.9 are satisfied and G has infinitely many fixed points.

Corollary 2.11 Let ( X , , d ) be a complete ordered metric space, ψ Ψ and G : X B ( X ) be a mapping such that for each x X and y G x with x y , we have
δ ( y , G y ) ψ ( d ( x , y ) ) .
(2.20)
Also, assume that
  1. (i)

    there exists x 0 X such that { x 0 } 1 G x 0 , i.e., there exists x 1 G x 0 such that x 0 x 1 ,

     
  2. (ii)

    if x y , then G x r G y .

     

Then there exists x X such that { x } = G x if and only if f ( ξ ) = δ ( ξ , G ξ ) is lower semi-continuous at x.

Proof Define α : X × X [ 0 , ) by
α ( x , y ) = { 1 if  x y , 0 otherwise .

By using condition (i) and the definition of α, we have α ( x 0 , x 1 ) = 1 . Also, from condition (ii), we have x y implies G x r G y , by using the definitions of α and r , we have α ( x , y ) = 1 implies α ( G x , G y ) = 1 . Moreover, it is easy to check that G is a generalized ( α , ψ , δ ) -contractive mapping. Therefore, by Theorem 2.9, there exists x X such that { x } = G x if and only if f ( ξ ) = δ ( ξ , G ξ ) is lower semi-continuous at x. □

Declarations

Acknowledgements

Authors are grateful to referees for their suggestions and careful reading.

Authors’ Affiliations

(1)
Centre for Advanced Mathematics and Physics, National University of Sciences and Technology H-12
(2)
Department of Mathematics, Quaid-i-Azam University

References

  1. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014MathSciNetView ArticleGoogle Scholar
  2. Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212. doi:10.1186/1687–1812–2012–212Google Scholar
  3. Kamran T: Mizoguchi-Takahashi’s type fixed point theorem. Comput. Math. Appl. 2009, 57: 507–511.MathSciNetView ArticleGoogle Scholar

Copyright

© Ali and Kamran; 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.