Skip to main content

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.

References

  1. Azam A, Mehmood N, Ahmad J, Radenovic S: Multivalued fixed point theorems in cone b -metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 582

    Google Scholar 

  2. Shukla S, Radojevic S, Veljkovic Z, Radenovic S: Some coincidence and common fixed point theorems for ordered Presic-Reich type contractions. J. Inequal. Appl. 2013., 2013: Article ID 520

    Google Scholar 

  3. Shukla, S, Sen, R, Radenovic, S: Set-valued Presic type contraction in metric spaces. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. (2012, in press)

  4. Long W, Shukla S, Radenovic S, Radojevic S: Some coupled coincidence and common fixed point results for hybrid pair of mappings in 0-complete partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 145

    Google Scholar 

  5. Kadelburg Z, Radenovic S: Some results on set-valued contractions in abstract metric spaces. Comput. Math. Appl. 2012, 62: 342–350.

    Article  MathSciNet  MATH  Google Scholar 

  6. Khojasteh F, Karapınar E, Radenovic S: θ -Metric spaces: a generalization. Math. Probl. Eng. 2013., 2013: Article ID 504609

    Google Scholar 

  7. Shatanawi W, Rajic VC, Radenovic S, Rawashdeh AA: Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 106

    Google Scholar 

  8. Ali, MU: Mizoguchi-Takahashi’s type common fixed point theorem. J. Egypt. Math. Soc. (in press)

  9. Ali MU, Kamran T: Hybrid generalized contractions. Math. Sci. 2013., 7: Article ID 29

    Google Scholar 

  10. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28

    Google Scholar 

  11. Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60

    Google Scholar 

  12. Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870

    Google Scholar 

  13. Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152

    Google Scholar 

  14. Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong-type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054

    Google Scholar 

  15. Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271

    Google Scholar 

  16. Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275

    Google Scholar 

  17. Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized α -contractive mappings. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013,75(2):3–10.

    MathSciNet  MATH  Google Scholar 

  18. Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized ϕ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971

    Google Scholar 

  19. Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54

    Google Scholar 

  20. 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.2011.10.014

    Article  MathSciNet  MATH  Google Scholar 

  21. Karapınar E, Samet B: Generalized α - ψ -contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486

    Google Scholar 

  22. Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212

    Google Scholar 

  23. Ali MU, Kamran T: On ( α ,ψ)-contractive multi-valued mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 137

    Google Scholar 

  24. Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24

    Google Scholar 

  25. Amiri P, Rezapour S, Shahzad N: Fixed points of generalized α - ψ -contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2013. 10.1007/s13398-013-0123-9

    Google Scholar 

  26. Minak G, Altun I: Some new generalizations of Mizoguchi-Takahashi type fixed point theorem. J. Inequal. Appl. 2013., 2013: Article ID 493

    Google Scholar 

  27. Ali MU, Kamran T, Karapınar E: (α,ψ,ξ)-Contractive multi-valued mappings. Fixed Point Theory Appl. 2014., 2014: Article ID 7

    Google Scholar 

  28. Ali MU, Kamran T, Sintunavarat W, Katchang P: Mizoguchi-Takahashi’s fixed point theorem with α , η functions. Abstr. Appl. Anal. 2013., 2013: Article ID 418798

    Google Scholar 

  29. Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca; 2001.

    MATH  Google Scholar 

  30. Bianchini RM, Grandolfi M: Transformazioni di tipo contracttivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. 1968, 45: 212–216.

    MathSciNet  MATH  Google Scholar 

  31. Proinov PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal. TMA 2007, 67: 2361–2369. 10.1016/j.na.2006.09.008

    Article  MathSciNet  MATH  Google Scholar 

  32. Proinov PD: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 2010, 26: 3–42. 10.1016/j.jco.2009.05.001

    Article  MathSciNet  MATH  Google Scholar 

  33. Karapınar E: Discussion on α - ψ -contractions in generalized metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 962784

    Google Scholar 

  34. Berzig, M, Karapınar, E: On modified α-ψ-contractive mappings with application. Thai J. Math. 12 (2014)

  35. Jleli M, Karapınar E, Samet B: Best proximity points for generalized α - ψ -proximal contractive type mappings. J. Appl. Math. 2013., 2013: Article ID 534127

    Google Scholar 

  36. Jleli M, Karapınar E, Samet B: Fixed point results for α - ψ λ -contractions on gauge spaces and applications. Abstr. Appl. Anal. 2013., 2013: Article ID 730825

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Erdal Karapınar.

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.

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

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

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords