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

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.

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 ψΨ, 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:XX is called α-ψ-contractive if there exist two functions α:X×X[0,) and ψΨ such that α(x,y)d(Gx,Gy)ψ(d(x,y)) for each x,yX. A mapping G:XX is called α-admissible [1] if α(x,y)1α(Gx,Gy)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 AN(X) and xX, d(x,A)=inf{d(x,a):aA}. For every A,BB(X), δ(A,B)=sup{d(a,b):aA,bB}. When A={x}, we denote δ(A,B) by δ(x,B). For every A,BCL(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,BX. We say that A r B if for each aA and bB, we have ab. 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:XCL(X) is α -admissible if α(x,y)1 α (Gx,Gy)1, where α (Gx,Gy)=inf{α(a,b):aGx,bGy}.

Definition 1.2 [2]

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

α (Gx,Gy)H(Gx,Gy)ψ ( d ( x , y ) )
(1.1)

for all x,yX.

Theorem 1.3 [2]

Let (X,d) be a complete metric space, let α:X×X[0,) be a function, let ψΨ 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:XCL(X) be a mapping. We say that G is generalized ( α ,ψ)-contractive if there exists ψΨ such that

α (Gx,Gy)d(y,Gy)ψ ( d ( x , y ) )
(2.1)

for each xX and yGx, where α (Gx,Gy)=inf{α(a,b):aGx,bGy}.

Note that an α -ψ-contractive mapping is generalized ( α ,ψ)-contractive. In case when ψΨ 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 BCL(X). Then, for each xX with d(x,B)>0 and q>1, there exists an element bB such that

d(x,b)<qd(x,B).
(2.2)

Proof It is given that d(x,B)>0. Choose

ϵ=(q1)d(x,B).

Then, by using the definition of d(x,B), it follows that there exists bB such that

d(x,b)<d(x,B)+ϵ=qd(x,B).

 □

Lemma 2.3 Let (X,d) be a metric space and G:XCL(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 xX. Then x is a fixed point of G if and only if the function f(ξ)=d(ξ,Gξ) is lower semi-continuous at x.

Proof Suppose f(ξ)=d(ξ,Gξ) is lower semi-continuous at x, then

d(x,Gx) lim inf n f( x n )= lim inf n d( x n ,G x n )=0.

By the closedness of G it follows that xGx. 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:XCL(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,Gξ) 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 )<qd( x 1 ,G x 1 ).
(2.4)

It follows from (2.3), (2.4) and (2.1) that

0<d( x 1 , x 2 )<qψ ( 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 (qψ(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 (qψ(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 ψΨ, it follows that { x n } is a Cauchy sequence in X. Thus there is xX 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:XCL(X) and α:X×X[0,) by

Gx={ [ 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 t0. For each xX and yGx, we have

α (Gx,Gy)d(y,Gy)=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 =1G 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, ψΨ be a strictly increasing map and G:XCL(X) be a mapping such that for each xX and yGx with xy, we have

d(y,Gy)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 xy, then Gx r Gy.

Then x is a fixed point of G if and only if f(ξ)=d(ξ,Gξ) 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 xy implies Gx r Gy; by using the definitions of α and r , we have α(x,y)=1 implies α (Gx,Gy)=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,Gξ) is lower semi-continuous at x. □

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

α (Gx,Gy)δ(y,Gy)ψ ( d ( x , y ) )
(2.11)

for each xX and yGx, where α (Gx,Gy)=inf{α(a,b):aGx,bGy}.

Lemma 2.8 Let (X,d) be a metric space and G:XB(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 xX. Then {x}=Gx if and only if the function f(ξ)=δ(ξ,Gξ) is lower semi-continuous at x.

Proof Suppose that f(ξ)=δ(ξ,Gξ) is lower semi-continuous at x, then

δ(x,Gx) lim inf n f( x n )= lim inf n δ( x n ,G x n )=0.

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

Theorem 2.9 Let (X,d) be a complete metric space and let G:XB(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 xX such that {x}=Gx if and only if f(ξ)=δ(ξ,Gξ) 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 ψΨ, it follows that { x n } is a Cauchy sequence in X. As X is complete, there exists xX 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:XB(X) and α:X×X[0,) by

Gx={ { ( 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 t0. For each xX and yGx, we have

α (Gx,Gy)δ(y,Gy) 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 =0X and x 1 =0G0 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, ψΨ and G:XB(X) be a mapping such that for each xX and yGx with xy, we have

δ(y,Gy)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 xy, then Gx r Gy.

Then there exists xX such that {x}=Gx if and only if f(ξ)=δ(ξ,Gξ) 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 xy implies Gx r Gy, by using the definitions of α and r , we have α(x,y)=1 implies α (Gx,Gy)=1. Moreover, it is easy to check that G is a generalized ( α ,ψ,δ)-contractive mapping. Therefore, by Theorem 2.9, there exists xX such that {x}=Gx if and only if f(ξ)=δ(ξ,Gξ) is lower semi-continuous at x. □

References

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

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Ali, M.U., Kamran, T. On ( α ,ψ)-contractive multi-valued mappings. Fixed Point Theory Appl 2013, 137 (2013). https://doi.org/10.1186/1687-1812-2013-137

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Keywords

  • α-admissible
  • α -ψ-contractive mapping
  • generalized ( α ,ψ)-contractive mapping