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A new approach to (α,ψ)-contractive nonself multivalued mappings

Abstract

In this paper, we introduce the notions of α-admissible and α-ψ-contractive type condition for nonself multivalued mappings. We establish fixed point theorems using these new notions along with a new condition. Moreover, we have constructed examples to show that our new condition is different from the corresponding existing conditions in the literature.

MSC: 47H10, 54H25.

1 Introduction and preliminaries

In the last decades, metric fixed point theory has been appreciated by a number of authors who have extended the celebrated Banach fixed point theorem for various contractive mapping in the context of different abstract spaces; see, for example, [132]. Among them, we mention the interesting fixed point theorems of Samet et al. [20]. In this paper [20], the authors introduced the notions of α-ψ-contractive mappings and investigated the existence and uniqueness of a fixed point for such mappings. Further, they showed that several well-known fixed point theorems can be derived from the fixed point theorem of α-ψ-contractive mappings. Following this paper, Karapınar and Samet [21] generalized the notion α-ψ-contractive mappings and obtained a fixed point for this generalized version. On the other hand, Asl et al. [22] characterized the notions of α-ψ-contractive mapping and α-admissible mappings with the notions of α -ψ-contractive and α -admissible mappings to investigate the existence of a fixed point for a multivalued function. Afterward, Ali and Kamran [23] generalized the notion of α -ψ-contractive mappings and obtained further fixed point results for multivalued mappings. Some results in this direction in the context of various abstract spaces were also given by the authors [2428, 3336]. The purpose of this paper is to prove fixed point theorems for nonself multivalued (α,ψ)-contractive type mappings using a new condition.

Let Ψ be the family of functions ψ:[0,)[0,), known in the literature as Bianchini-Grandolfi gauge functions (see, e.g., [3032]), satisfying the following conditions:

( ψ 1 ) ψ is nondecreasing;

( ψ 2 ) n = 1 + ψ n (t)< for all t>0, where ψ n is the n th iterate of ψ.

Notice that such functions are also known as (c)-comparison functions in some sources (see, e.g., [29]).

It is easily proved that if ψΨ, then ψ(t)<t for any t>0 and ψ(0)=0 for t=0 (see, e.g., [20, 29]). Let (X,d) be a metric space. A mapping G:XX is called α-ψ-contractive type 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 [20] if

α(x,y)1α(Gx,Gy)1.

We denote by N(X) the space of all nonempty 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,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 a generalized Hausdorff metric induced by d. We use the following lemma in our results.

Lemma 1.1 [23]

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).
(1.1)

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.

2 Main results

We begin this section with the following definition which is a modification of the notion of α-admissible.

Definition 2.1 Let (X,d) be a metric space and let D be a nonempty subset of X. A mapping G:DCL(X) is called α-admissible if there exists a mapping α:D×D[0,) such that

α(x,y)1α(u,v)1

for each uGxD and vGyD.

Definition 2.2 Let (X,d) be a metric space and let D be a nonempty subset of X. We say that G:DCL(X) is an (α,ψ)-contractive type mapping on D if there exist α:D×D[0,) and ψΨ satisfying the following conditions:

  1. (i)

    GxD for all xD,

  2. (ii)

    for each x,yD, we have

    α(x,y)H(GxD,GyD)ψ ( M ( x , y ) ) ,
    (2.1)

where M(x,y)=max{d(x,y), d ( x , G x ) + d ( y , G y ) 2 , d ( x , G y ) + d ( y , G x ) 2 }.

Note that if ψΨ in the above definition is a strictly increasing function, then G:DCL(X) is said to be a strictly (α,ψ)-contractive type mapping on D.

Theorem 2.3 Let (X,d) be a metric space, let D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly (α,ψ)-contractive type mapping on D. Assume that the following conditions hold:

  1. (i)

    G is an α-admissible map;

  2. (ii)

    there exist x 0 D and x 1 G x 0 D such that α( x 0 , x 1 )1;

  3. (iii)

    G is continuous.

Then G has a fixed point.

Proof By hypothesis, there exist x 0 D and x 1 G x 0 D 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 D, then x 1 is a fixed point. Let x 1 G x 1 D. From (2.1) we have

0 < α ( x 0 , x 1 ) H ( G x 0 D , G x 1 D ) ψ ( max { d ( x 0 , x 1 ) , d ( x 0 , G x 0 ) + d ( x 1 , G x 1 ) 2 , d ( x 0 , G x 1 ) + d ( x 1 , G x 0 ) 2 } ) ψ ( max { d ( x 0 , x 1 ) , d ( x 1 , G x 1 ) } )
(2.2)

since d ( x 0 , G x 1 ) 2 max{d( x 0 , x 1 ),d( x 1 ,G x 1 )} and d ( x 0 , G x 0 ) + d ( x 1 , G x 1 ) 2 max{d( x 0 , x 1 ),d( x 1 ,G x 1 )}. Assume that max{d( x 0 , x 1 ),d( x 1 ,G x 1 )}=d( x 1 ,G x 1 ). Then from (2.2) we have

0 < d ( x 1 , G x 1 D ) α ( x 0 , x 1 ) H ( G x 0 D , G x 1 D ) ψ ( d ( x 1 , G x 1 ) ) < d ( x 1 , G x 1 ) ,
(2.3)

a contradiction to our assumption. Thus max{d( x 0 , x 1 ),d( x 1 ,G x 1 )}=d( x 0 , x 1 ). Then from (2.2) we have

0<d( x 1 ,G x 1 D)ψ ( d ( x 0 , x 1 ) ) .
(2.4)

For q>1 by Lemma 1.1, there exists x 2 G x 1 D such that

0<d( x 1 , x 2 )<qd( x 1 ,G x 1 D)qψ ( d ( x 0 , x 1 ) ) .
(2.5)

Applying ψ in (2.5), we have

0<ψ ( d ( x 1 , x 2 ) ) <ψ ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.6)

Put q 1 = ψ ( q ψ ( d ( x 0 , x 1 ) ) ) ψ ( d ( x 1 , x 2 ) ) . Then q 1 >1. Since G is an α-admissible mapping, α( x 1 , x 2 )1. If x 2 G x 2 D, then x 2 is a fixed point. Let x 2 G x 2 D. From (2.1) we have

0 < α ( x 1 , x 2 ) H ( G x 1 D , G x 2 D ) ψ ( max { d ( x 1 , x 2 ) , d ( x 1 , G x 1 ) + d ( x 2 , G x 2 ) 2 , d ( x 1 , G x 2 ) + d ( x 2 , G x 1 ) 2 } ) ψ ( max { d ( x 1 , x 2 ) , d ( x 2 , G x 2 ) } )
(2.7)

since d ( x 1 , G x 2 ) 2 max{d( x 1 , x 2 ),d( x 2 ,G x 2 )} and d ( x 1 , G x 1 ) + d ( x 2 , G x 2 ) 2 max{d( x 1 , x 2 ),d( x 2 ,G x 2 )}. Assume that max{d( x 1 , x 2 ),d( x 2 ,G x 2 )}=d( x 2 ,G x 2 ). Then from (2.7) we have

0 < d ( x 2 , G x 2 D ) α ( x 1 , x 2 ) H ( G x 1 D , G x 2 D ) ψ ( d ( x 2 , G x 2 ) ) < d ( x 2 , G x 2 ) ,
(2.8)

a contradiction to our assumption. Thus max{d( x 1 , x 2 ),d( x 2 ,G x 2 )}=d( x 1 , x 2 ). Then from (2.7) we have

0<d( x 2 ,G x 2 D)ψ ( d ( x 1 , x 2 ) ) .
(2.9)

For q 1 >1 by Lemma 1.1, there exists x 3 G x 2 D such that

0<d( x 2 , x 3 )< q 1 d( x 2 ,G x 2 D) q 1 ψ ( d ( x 1 , x 2 ) ) =ψ ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.10)

Applying ψ in (2.10), we have

0<ψ ( d ( x 2 , x 3 ) ) < ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.11)

Put q 2 = ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) ψ ( d ( x 2 , x 3 ) ) . Then q 2 >1. Since G is an α-admissible mapping, α( x 2 , x 3 )1. If x 3 G x 3 D, then x 3 is a fixed point. Let x 3 G x 3 D. From (2.1) we have

0 < α ( x 2 , x 3 ) H ( G x 2 D , G x 3 D ) ψ ( max { d ( x 2 , x 3 ) , d ( x 2 , G x 2 ) + d ( x 3 , G x 3 ) 2 , d ( x 2 , G x 3 ) + d ( x 3 , G x 2 ) 2 } ) ψ ( max { d ( x 2 , x 3 ) , d ( x 3 , G x 3 ) } )
(2.12)

since d ( x 2 , G x 3 ) 2 max{d( x 2 , x 3 ),d( x 3 ,G x 3 )} and d ( x 2 , G x 2 ) + d ( x 3 , G x 3 ) 2 max{d( x 2 , x 3 ),d( x 3 ,G x 3 )}. Assume that max{d( x 2 , x 3 ),d( x 3 ,G x 3 )}=d( x 3 ,G x 3 ). Then from (2.12) we have

0 < d ( x 3 , G x 3 D ) α ( x 2 , x 3 ) H ( G x 2 D , G x 3 D ) ψ ( d ( x 3 , G x 3 ) ) < d ( x 3 , G x 3 ) ,
(2.13)

a contradiction to our assumption. Thus max{d( x 2 , x 3 ),d( x 3 ,G x 3 )}=d( x 2 , x 3 ). Then from (2.12) we have

0<d( x 3 ,G x 3 D)ψ ( d ( x 2 , x 3 ) ) .
(2.14)

For q 2 >1 by Lemma 1.1, there exists x 4 G x 3 D such that

0<d( x 3 , x 4 )< q 2 d( x 3 ,G x 3 D) q 2 ψ ( d ( x 2 , x 3 ) ) = ψ 2 ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.15)

Applying ψ in (2.15), we have

0<ψ ( d ( x 3 , x 4 ) ) < ψ 3 ( q ψ ( d ( x 0 , x 1 ) ) ) .
(2.16)

Continuing the same process, we get a sequence { x n } in D such that x n + 1 G x n D, x n + 1 x n , α( x n , x n + 1 )1, and

d( x n + 1 , x n + 2 )< ψ n ( q ψ ( d ( x 0 , x 1 ) ) ) for each nN{0}.
(2.17)

For m,nN, we have

d( x n , x n + m ) i = n n + m 1 d( x i , x i + 1 )< i = n n + m 1 ψ i 1 ( d ( x 0 , x 1 ) ) .

Since ψΨ, it follows that { x n } is a Cauchy sequence in D. Since D is complete, there exists x D such that x n x as n. By the continuity of G, we have

d ( x , G x ) lim n H ( G x n , G x ) =0.

 □

Theorem 2.4 Let (X,d) be a metric space, D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly (α,ψ)-contractive type mapping on D. Assume that the following conditions hold:

  1. (i)

    G is an α-admissible map;

  2. (ii)

    there exist x 0 D and x 1 G x 0 D such that α( x 0 , x 1 )1;

  3. (iii)

    either

    1. (a)

      for any sequence { x n } in D such that x n x as n and α( x n , x n + 1 )1 for each nN{0}, lim n α( x n ,x)1,

      or

    2. (b)

      for any sequence { x n } in D such that x n x as n and α( x n , x n + 1 )1 for each nN{0}, α( x n ,x)1 for each nN{0}.

Then G has a fixed point.

Proof Following the proof of Theorem 2.3, there exists a Cauchy sequence { x n } in D with x n x as n and α( x n , x n + 1 )1 for each nN{0}. Suppose that d( x ,G x )0. From (2.1) we have

α ( x n , x ) d ( x n + 1 , G x D ) α ( x n , x ) H ( G x n D , G x D ) ψ ( max { d ( x n , x ) , d ( x n , G x n ) + d ( x , G x ) 2 , d ( x n , G x ) + d ( x , G x n ) 2 } ) < max { d ( x n , x ) , d ( x n , G x n ) + d ( x , G x ) 2 , d ( x n , G x ) + d ( x , G x n ) 2 } .
(2.18)

Letting n in (2.18), we have

lim n α ( x n , x ) d ( x , G x D ) d ( x , G x ) 2 .
(2.19)

Since lim n α( x n , x )1, by condition (iii)(a), we have

d ( x , G x D ) lim n α ( x n , x ) d ( x , G x D ) d ( x , G x ) 2 .
(2.20)

Further, it is clear that d( x ,G x )d( x ,G x D). Then from (2.20) we have

d ( x , G x ) d ( x , G x ) 2 ,

which is impossible. Thus d( x ,G x )=0. If we use (iii)(b), then from (2.1) we have

d ( x n + 1 , G x D ) α ( x n , x ) H ( G x n D , G x D ) ψ ( max { d ( x n , x ) , d ( x n , G x n ) + d ( x , G x ) 2 , d ( x n , G x ) + d ( x , G x n ) 2 } ) < max { d ( x n , x ) , d ( x n , G x n ) + d ( x , G x ) 2 , d ( x n , G x ) + d ( x , G x n ) 2 } .
(2.21)

Letting n in (2.21), we have

d ( x , G x ) d ( x , G x D ) d ( x , G x ) 2 ,

which is impossible. Thus d( x ,G x )=0. □

Example 2.5 Let X=(,8)[0,) be endowed with the usual metric d, and let D=[0,). Define G:DCL(X) by

Gx={ [ 0 , x 4 ] if  0 x < 4 , { 0 } if  x = 4 , ( , 3 x ] [ x , x 2 ] if  x > 4

and α:D×D[0,) by

α(x,y)={ 1 if  x , y [ 0 , 4 ] , 0 otherwise .

Clearly, GxD for each xD. Let ψ(t)= t 2 for each t0. To see that G is a strictly (α,ψ)-contractive type mapping on D, we consider the following cases.

Case (i) When x,y[0,4), we have

α(x,y)H(GxD,GyD)= | x 4 y 4 | | x y | 2 =ψ ( d ( x , y ) ) ψ ( M ( x , y ) ) .

Case (ii) When x[0,4) and y=4, we have

α(x,y)H(GxD,GyD)= | x 4 | ψ ( d ( x , G x ) + d ( y , G y ) 2 ) ψ ( M ( x , y ) ) .

Case (iii) Otherwise, we have

α(x,y)H(GxD,GyD)=0ψ ( M ( x , y ) ) ,

where M(x,y)=max{d(x,y), d ( x , G x ) + d ( y , G y ) 2 , d ( x , G y ) + d ( y , G x ) 2 }.

Thus, G is a strictly (α,ψ)-contractive type mapping on D. For α(x,y)1, we have x,y[0,4], then GxD,GyD[0,1], thus α(u,v)=1 for each uGxD and vGyD. Further, for any sequence { x n } in D such that x n x as n and α( x n , x n + 1 )=1 for each nN{0}, lim n α( x n ,x)=1. Therefore, all the conditions of Theorem 2.4 hold and G has a fixed point.

Corollary 2.6 Let (X,,d) be an ordered metric space, let (D,) be a nonempty subset of X which is complete with respect to the metric induced by d. Let G:DCL(X) be a mapping such that GxD for each xD and for each x,yD with xy, we have

H(GxD,GyD)ψ ( M ( x , y ) ) ,

where M(x,y)=max{d(x,y), d ( x , G x ) + d ( y , G y ) 2 , d ( x , G y ) + d ( y , G x ) 2 } and ψ is an increasing function in  Ψ. Also, assume that the following conditions hold:

  1. (i)

    there exist x 0 D and x 1 G x 0 D such that x 0 x 1 ;

  2. (ii)

    if xy then GxD r GyD;

  3. (iii)

    either

    1. (a)

      G is continuous,

      or

    2. (b)

      for any sequence { x n } in D such that x n x as n and x n x n + 1 for each nN{0}, x n x as n,

      or

    3. (c)

      for any sequence { x n } in D such that x n x as n and x n x n + 1 for each nN{0}, x n x for each nN{0}.

Then G has a fixed point.

Proof Define α:D×D[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 that xy implies GxD r GyD; by using the definitions of α and r , we have that α(x,y)=1 implies α(u,v)=1 for each uGxD and vGyD. Moreover, it is easy to check that G is a strictly (α,ψ)-contractive type mapping on D. Therefore, all the conditions of Theorem 2.3 (or Theorem 2.4 for (iii)(b), (iii)(c)) hold, hence G has a fixed point. □

Remark 2.7 Condition (a), in the statement of Theorem 2.4, was introduced by Samet et al. [20]. In Theorem 2.4 we introduce a new condition (b). The following examples show that (a) and (b) are independent conditions.

Example 2.8 Let X={ 1 n :nN}{0}. Consider x n = 1 n + 1 for each nN{0}, then x n 0= x as n. Define α:X×X[0,) by

α(x,y)={ max { 1 x , 1 y } if  x 0  and  y 0 , 1 x + y if either  x = 0  or  y = 0 , 1 if  x = y = 0 .

Now, we have α( x n , x n + 1 )=α( 1 n + 1 , 1 n + 2 )=n+2>1 for each nN{0} and α( x n , x )=α( 1 n + 1 ,0)=n+11 for each nN{0}. Thus condition (a) holds but lim n α( x n , x )= lim n (n+1)=. Thus condition (b) does not hold.

Example 2.9 Let X={ 1 n :nN}{0}. Consider x n = 1 n + 1 for each nN{0}, then x n 0= x as n. Define α:X×X[0,) by

α(x,y)={ max { 1 x , 1 y } if  x 0  and  y 0 , 1 1 + ( x + y ) / 2 if either  x = 0  or  y = 0 , 1 if  x = y = 0 .

Now, we have α( x n , x n + 1 )=α( 1 n + 1 , 1 n + 2 )=n+2>1 for each nN{0} and α( x n , x )=α( 1 n + 1 ,0)= 2 n + 2 2 n + 3 . Then lim n α( x n , x )= lim n 2 n + 2 2 n + 3 =1. Thus condition (b) holds but for n=0, we have α( x n , x )= 2 3 ; for n=1, we have α( x n , x )= 4 5 ; for n=2, we have α( x n , x )= 6 7 , which implies that α( x n ,x)1 for each nN{0}. Thus condition (a) does not hold.

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The authors are grateful to the reviewers for their careful reviews and useful comments.

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Correspondence to Erdal Karapınar.

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Ali, M.U., Kamran, T. & Karapınar, E. A new approach to (α,ψ)-contractive nonself multivalued mappings. J Inequal Appl 2014, 71 (2014). https://doi.org/10.1186/1029-242X-2014-71

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  • DOI: https://doi.org/10.1186/1029-242X-2014-71

Keywords

  • α-admissible maps
  • α-ψ-contractive type condition
  • nonself α-admissible maps
  • nonself (α,ψ)-contractive type condition