Someresultsonfixedpointsofα-ψ-Ciric generalized multifunctions

Transcription

1 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 R E S E A R C H Open Access Someresultsonfixedpointsofα-ψ-Ciric generalized multifunctions B Mohammadi 1,SRezapour and N Shahzad 3* * Correspondence: nshahzad@kau.edu.sa 3 Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 1859, Saudi Arabia Full list of author information is available at the end of the article Abstract In 01, Samet, Vetro and Vetro introduced α-ψ-contractive mappings and gave some results on a fixed point of the mappings (Samet et al. in Nonlinear Anal. 75: , 01). In fact, their technique generalized some ordered fixed point results (see (Alikhani et al. in Filomat, 01, to appear) and (Samet et al. in Nonlinear Anal. 75: , 01)). By using the main idea of (Samet et al. in Nonlinear Anal. 75: , 01), we give some new results for α-ψ-ciric generalized multifunctions and some related self-maps. Also, we give an affirmative answer to a recent open problem which was raised by Haghi, Rezapour and Shahzad in 01. Keywords: α-ψ-ciric generalized multifunction; fixed point; quasi-contractive multifunction 1 Introduction In 01, Samet, Vetro and Vetro introduced α-ψ-contractive mappings and gave fixed point results for such mappings [1]. Their results generalized some ordered fixed point results (see [1]). Immediately, using their idea, some authors presented fixed point results in the field (see, for example, []). Denote by the family of nondecreasing functions ψ :[0,+ ) [0, + ) suchthat + n=1 ψ n (t) <+ for each t >0.Itiswellknownthat ψ(t) <t for all t >0.Let(X, d) beametricspace,β : X X [0, + ) be a mapping and ψ. AmultivaluedoperatorT : X X is said to be β-ψ-contractive whenever β(tx, Ty)H(Tx, Ty) ψ(d(x, y)) for all x, y X,whereH is the Hausdorff distance (see []). Alikhani, Rezapour and Shahzad proved fixed point results for β-ψ-contractive multifunctions []. Let (X, d)beametricspace,α : X X [0, + ) be a mapping and ψ. We say that T : X X is an α-ψ-ciric generalized multifunction if α(x, y)h(tx, Ty) ψ max d(x, y), d(x, Tx), d(y, Ty), for all x, y X. One can find an idea of this notion in [3]. Also, we say that the self-map F on X is α-admissible whenever α(x, y) 1impliesα(Fx, Fy) 1[1]. In this paper, we give fixedpointresultsfor α-ψ-ciric generalized multifunctions. In 01, Haghi, Rezapour and Shahzad proved that some fixed point generalizations are not real ones [4]. Here, by presenting a result and an example, we are going to show that obtained results in this new field are real generalizations in respect to old similar results in the literature (compare the next result and the example with main results of [5] and[6]). The following result has been proved in [7]. 013 Mohammadi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page of 10 Lemma 1.1 Let (X, d) be a complete metric space, α : X X [0, ) be a function, ψ and T be a self-map on X such that α(x, y)d(tx, Ty) ψ max d(x, y), d(x, Tx), d(y, Ty), 1 [ ] for all x, y X. Suppose that T is α-admissible and there exists x 0 Xsuchthatα(x 0, Tx 0 ) 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 T has a fixed point. Here, we give the following example which shows that Lemma 1.1 holds while we cannot use some similar old results, for example, Theorem 1 of [5]. Example 1.1 Let M 1 = { m n : m =0,1,3,9,... andn =3k +1(k 0)}, d(x, y)= x y, M = { m n : m =1,3,9,7,... andn =3k +(k 0)} and M 3 = {k : k 1}. Now,putM = M 1 M M 3 and define the self-map T : M M by Tx = 3x 11 whenever x M 1, Tx = x 8 whenever x M and Tx =x whenever x M 3.PutM(x, y) =max{d(x, y), d(x, Tx), d(y, Ty), d(x,ty)+d(y,tx) } for all x, y M. Ifx M 1 and y M,thend(Tx, Ty)= 3x 11 y 8 = 3 11y x Now, we consider two cases. If x > 11 4 y,thenwehave 3 11 x 11y 4 = 3 ( x 11y ) 3 ( x y ) = 3 d(x, Ty) Hence, d(tx, Ty) 6 d(x,ty) 6 11y M(x, y). If x < ,then 3 11 x 11y 4 = 3 ( ) 11y 11 4 x 3(y x) = d(x, y) 6 M(x, y). 11 Hence, d(tx, Ty) 6 11 M(x, y). If x, y M 1,then d(tx, Ty) = 3x 11 3y 11 = 3 11 d(x, y) 6 M(x, y). 11 If x, y M,thend(Tx, Ty)= x 8 y 8 = 1 8 d(x, y) 6 M(x, y). Define α(x, y) =1whenever 11 x, y M 1 M and α(x, y)=1otherwise.also,putψ(t)= 6t for all t 0. Then it is easy 11 to see that α(x, y)d(tx, Ty) ψ(m(x, y)) for all x, y M. Also,α(1, T1) = α(1, 3/11) 1. If α(x, y) 1, then x, y M 1 M and so α(tx, Ty) 1. Thus, T is α-admissible. It is easy to show that if {x n } is a sequence in M 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. Now,byusingTheorem1.1, T has a fixed point. Now, we show that Theorem 1 in [5] does not apply here. Put x =1andy =.Thend(T1, T) = = 11 11, d(1, ) = 1, d(1, T1) = d(1, 3 11 )= 8, d(, T) = d(, 4) =, d(1, T) = d(1, 4) = 11 3, d(, T1) = d(, )=.Thus,N(1, ) = 3 and so d(t1, T) = >3=N(1, ), where N(x, y) =max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}. An affirmative answer for an open problem After providing some results on fixed points of quasi-contractions on normal cone metric spaces by Ilic and Rakocevic in 009 [8], Kadelburg, Radenovic and Rakocevic generalized the results by considering an additional assumption and deleting the assumption on

3 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 3 of 10 normality [9]. In 011, Haghi, Rezapour and Shahzad proved same results without the additional assumption and for λ (0,1) [10]. Then Amini-Harandi proved a result on the existence of fixed points of set-valued quasi-contraction maps in metric spaces by using the technique of [10] (see[11]). But similar to [9], he could prove it only for λ (0, 1 ) [11]. In 01, Haghi, Rezapour and Shahzad proved the main result of [11]byusingasimple method [1]. Also, they introduced quasi-contraction type multifunctions and showed that the main result of [11] holds for quasi-contraction type multifunctions. They raised an open problem about the difference between quasi-contraction and quasi-contraction type multifunctions [1]. In this section, we give a positive answer to the question. Let (X, d) be a metric space. Recall that the multifunction T : X X is called quasi-contraction whenever there exists λ (0, 1) such that H(Tx, Ty) λ max { d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) } for all x, y X [11]. Also, a multifunction T : X X is called quasi-contraction type whenever there exists λ (0, 1) such that H(Tx, Ty) λ max { d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx) } for all x, y X [1]. It is clear that each quasi-contraction type multifunction is a quasicontractive multifunction. In 01, Haghi, Rezapour and Shahzad raised this question that (see [1]): Is there any quasi-contractive multifunction which is not quasi-contraction type? By providing the following example, we give a positive answer to the problem. Example.1 Let X =[0,]andd(x, y) = x y. DefineT : X CB(X) bytx =[1, 5 4 ] whenever x [0, 3/4], Tx =[ 7 8, 9 8 ]wheneverx ( 3 4, 5 4 )andtx =[3 4,1]whenever x [ 5 4,]. We show that T is a quasi-contraction while it is not a quasi-contractive type multifunction. Put N(x, y)=max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)} for all x, y X. First,we show that T is a quasi-contraction. It is easy to check that H(Tx, Ty) 1 N(x, y)whenever x, y [0, 3 4 ]orx, y ( 3 4, 5 4 )orx, y [ 5 4,].Ifx [0, 3 4 ]andy [ 5 4,],thenH(Tx, Ty)= 1 4 and d(x, y) = 1.Hence,H(Tx, Ty)= d(x, y) 1 N(x, y). If x [0, 3 4 ]andy ( 3 4, 5 4 ), then H(Tx, Ty)= 1 8 and d(x, Tx) 1 4.Hence, H(Tx, Ty)= d(x, Tx) 1 N(x, y). If x [ 5 4,] and y ( 3 4, 5 4 ), then H(Tx, Ty)=H([ 3 4,1],[7 8, 9 8 ]) = 1 8 and d(x, Tx) 1 4.Hence, H(Tx, Ty) = d(x, Tx) 1 N(x, y). Therefore, T is a quasi-contraction with λ = 1. Now, put x = 3 4 and y = 5 4.Thenwehaved( 3 4, T 3 4 )=d( 3 4,[1, 5 4 ]) = 1 4, d( 5 4, T 5 4 )=d( 5 4,[3 4,1])= 1 4, d( 3 4, T 5 4 )=d( 3 4,[3 4,1]) = 0 and d( 5 4, T 3 4 )=d( 5 4,[1, 5 4 ]) = 0. Hence, N 1( 3 4, 5 4 )=max{d( 3 4, T 3 4 ), d( 5 4, T 5 4 ), d( 3 4, T 5 4 ), d( 5 4, T 3 4 )} = max{ 1 4,0} = 1 4.Also,wehaveH(T 3 4, T 5 4 )=H([1, 5 4 ], [ 3 4,1])= 1 4.Thus,H(T 3 4, T 5 4 )>λn 1( 3 4, 5 ) for all λ (0, 1). This shows that T is not a quasicontractive type 4 multifunction. 3 Main results Now, we are ready to state and prove our main results. Let (X, d) beametricspace, α : X X [0, ) be a mapping and T : X CB(X) be a multifunction. We say that X satisfies the condition (C α )wheneverforeachsequence{x n } in X with α(x n, x n+1 ) 1 for all n

4 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 4 of 10 and x n x,thereexistsasubsequence{x nk } of {x n } such that α(x nk, x) 1 for all k (see [13] for the idea of this notion). Recall that T is continuous whenever H(Tx n, Tx) 0 for all sequence {x n } in X with x n x.also,wesaythatt is α-admissible whenever for each x X and y Tx with α(x, y) 1, we have α(y, z) 1 for all z Ty. Note that this notion is different from the notion of α -admissible multifunctions which has been provided in [7]. But, by providing a similar proof to that of Theorem.1 in [7], we can prove the following result. Theorem 3.1 Let (X, d) be a complete metric space, α : X X [0, ) be a function, ψ be a strictly increasing map and T : X CB(X) be an α-admissible multifunction such that α(x, y)h(tx, Ty) ψ(d(x, y)) for all x, y Xandthereexistx 0 Xandx 1 Tx 0 with α(x 0, x 1 ) 1. If T is continuous or X satisfies the condition (C α ), then T has a fixed point. Corollary 3. Let (X, d) be a complete metric space, ψ be a strictly increasing map, x XandT: X CB(X) be a multifunction such that H(Tx, Ty) ψ(d(x, y)) for all x, y Xwithx Tx Ty. Suppose that there exist x 0 Xandx 1 Tx 0 such that x Tx 0 Tx 1. Assume that for each x Xandy Tx with x Tx Ty, we have x Ty Tz for all z Ty. If T is continuous or X satisfies the condition (C α ), then T has a fixed point. Proof Define α : X X [0, + ) byα(x, y) =1wheneverx Tx Ty and α(x, y) =0 otherwise. Then, by using Theorem 3.1, T has a fixed point. Let (X, ) beanorderedsetanda, B X. WesaythatA B whenever for each a A there exists b B such that a b. Corollary 3.3 Let (X,, d) be a complete ordered metric space, ψ be a strictly increasing map and T : X CB(X) be a multifunction such that H(Tx, Ty) ψ(d(x, y)) for all x, y XwithTx Ty or Ty Tx. Suppose that there exist x 0 Xandx 1 Tx 0 such that Tx 0 Tx 1 or Tx 1 Tx 0. Assume that for each x Xandy Tx with Tx Ty or Ty Tx, we have Ty Tz or Tz Ty for all z Ty. If T is continuous or X satisfies the condition (C α ), then T has a fixed point. Now, we give the following result. Theorem 3.4 Let (X, d) be a complete metric space, α : X X [0, ) be a function, ψ be a strictly increasing map and T : X CB(X) be an α-admissible α-ψ-ciric generalized multifunction, and let there exist x 0 Xandx 1 Tx 0 with α(x 0, x 1 ) 1. If T is continuous, then T has a fixed point. Proof If x 1 = x 0,thenwehavenothingtoprove.Letx 1 x 0.Thenwehave d(x 1, Tx 1 ) α(x 0, x 1 )H(Tx 0, Tx 1 ) ψ max d(x 0, x 1 ), d(x 0, Tx 0 ), d(x 1, Tx 1 ), d(x 0, Tx 1 )+d(x 1, Tx 0 ) = ψ max d( 0, x 1 ), d(x 1, Tx 1 ), d(x 0, Tx 1 ) ψ max d(x 0, x 1 ), d(x 1, Tx 1 ), d(x 0, x 1 )+d(x 1, Tx 1 ) = ψ ( max { d(x 0, x 1 ), d(x 1, Tx 1 ).

5 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 5 of 10 If max{d(x 0, x 1 ), d(x 1, Tx 1 )} = d(x 1, Tx 1 ), then d(x 1, Tx 1 ) ψ(d(x 1, Tx 1 )) and so we get d(x 1, Tx 1 )=0.Thus,d(x 0, x 1 ) = 0, which is a contradiction. Hence, we obtain max{d(x 0, x 1 ), d(x 1, Tx 1 )} = d(x 0, x 1 )andsod(x 1, Tx 1 ) ψ(d(x 0, x 1 )). If x 1 Tx 1,thenx 1 is a fixed point of T.Letx 1 / Tx 1 and q >1.Then 0<d(x 1, Tx 1 )<qψ ( d(x 0, x 1 ) ). Put t 0 = d(x 0, x 1 ). Then t 0 >0andd(x 1, Tx 1 )<qψ(t 0 ). Hence, there exists x Tx 1 such that d(x 1, x )<qψ(t 0 )andsoψ(d(x 1, x )) < ψ(qψ(t 0 )). It is clear that x x 1.Putq 1 = ψ(qψ(t 0)) ψ(d(x 1,x )). Then q 1 >1andwehave d(x, Tx ) α(x 1, x )H(Tx 1, Tx ) ψ max d(x 1, x ), d(x 1, Tx 1 ), d(x, Tx ), d(x 1, Tx )+d(x, Tx 1 ) = ψ max d(x 1, x ), d(x, Tx ), d(x 1, Tx ) ψ ( max { d(x 1, x ), d(x, Tx ). Similarly, we should have max{d(x 1, x ), d(x, Tx )} = d(x, x )andsoweget d(x, Tx ) ψ ( d(x 1, x ) ). If x Tx,thenx is a fixed point of T.Letx / Tx.Then0<d(x, Tx )<q 1 ψ(d(x 1, x )). Hence, there exists x 3 Tx such that d(x, x 3 )<q 1 ψ(d(x 1, x )) = ψ(qψ(t 0 )). It is clear that x 3 x and ψ(d(x, x 3 )) < ψ (qψ(t 0 )). Put q = ψ (qψ(t 0 )) ψ(d(x,x 3 )).Thenq >1.Also,wehave d(x 3, Tx 3 ) α(x, x 3 )H(Tx, Tx 3 ) ψ max d(x, x 3 ), d(x, Tx ), d(x 3, Tx 3 ), d(x, Tx 3 )+d(x 3, Tx ) = ψ max d(x, x 3 ), d(x 3, Tx 3 ), d(x, Tx 3 ) ψ ( max { d(x, x 3 ), d(x 3, Tx 3 ). By continuing this process, we obtain a sequence {x n } in X such that x n Tx n 1, x n x n 1 and d(x n, x n+1 ) ψ n 1 (qψ(t 0 )) for all n.letm > n.then m 1 m 1 d(x n, x m ) d(x i, x i+1 ) ψ i 1( qψ(t 0 ) ) i=n i=n and so {x n } is a Cauchy sequence in X.Hence,thereexistsx X such that x n x.ift is continuous, then d ( x, Tx ) = lim n d( x n+1, Tx ) lim n H( Tx n, Tx ) =0 and so x Tx.

6 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 6 of 10 Corollary 3.5 Let (X, d) be a complete metric space, ψ be a strictly increasing map, x XandT: X CB(X) be a multifunction such that H(Tx, Ty) ψ max d(x, y), d(x, Tx), d(y, Ty), for all x, y Xwithx Tx Ty. Suppose that there exist x 0 Xandx 1 Tx 0 such that x Tx 0 Tx 1. Assume that for each x Xandy Tx with x 0 Tx Ty, we have x 0 Ty Tz for all z Ty. If T is continuous, then T has a fixed point. Corollary 3.6 Let (X,, d) be a complete ordered metric space, ψ be a strictly increasing map and T : X CB(X) be a multifunction such that H(Tx, Ty) ψ max d(x, y), d(x, Tx), d(y, Ty), for all x, y XwithTx Ty or Ty Tx. Suppose that there exist x 0 Xandx 1 Tx 0 such that Tx 0 Tx 1 or Tx 1 Tx 0. Assume that for each x Xandy Tx with Tx Ty or Ty Tx, we have Ty Tz or Tz Ty for all z Ty. If T is continuous, then T has a fixed point. Now, we give the following result about a fixed point of self-maps on complete metric spaces. Theorem 3.7 Let (X, d) be a complete metric space, α : X X [0, ) be a mapping, φ :[0, ) [0, ) be a continuous and nondecreasing map such that φ(t) <t for all t >0 and T be a self-map on X such that α(x, y)d(tx, Ty) φ max d(x, y), d(x, Tx), d(y, Ty), for all x, y X. Assume that there exists x 0 Xsuchthatα(T i x 0, T j x 0 ) 1 for all i, j 0 with i j. Suppose that T is continuous or α(t i x 0, x) 1 for all i 0 whenever T i x 0 x. Then T has a fixed point. Proof It is easy to check that lim n φ n (t) = 0 for all t >0.Letx n = Tx n 1 for all n 1. If x n = x n 1 for some n,thenx n 1 is a fixed point of T.Supposethatx n x n 1 for all n.then we have d(x n+1, x n )=d(tx n, Tx n 1 ) α(x n, x n 1 )d(tx n, Tx n 1 ) φ max d(x n, x n 1 ), d(x n, Tx n ), d(x n 1, Tx n 1 ), d(x n 1, Tx n )+d(x n, Tx n 1 ) = φ max d(x n 1, x n ), d(x n, x n+1 ), d(x n 1, x n+1 ) φ max d(x n 1, x n ), d(x n, x n+1 ), d(x n 1, x n )+d(x n, x n+1 ) = φ ( max { d(x n 1, x n ), d(x n, x n+1 ).

7 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 7 of 10 Since d(x n+1, x n ) φ(d(x n+1, x n )) does not hold, we get d(x n+1, x n ) φ(d(x n, x n 1 )). Since φ is nondecreasing, we have d(x n+1, x n ) φ ( d(x n, x n 1 ) ) φ ( d(x n 1, x n ) ) φ n( d(x 1, x 0 ) ) for all n. Hence,d(x n+1, x n ) 0. If {x n } is not a Cauchy sequence, then there exists ε >0 and subsequences {x ni } and {x mi } of {x n } with n i < m i such that d(x ni, x mi )>ε for all i.for each n i,putk i = min{m i d(x ni, x mi )>ε}.thenwehave ε < d(x ni, x ki ) d(x ni, x ki 1)+d(x ki 1, x ki ) ε + d(x ki 1, x ki ) and so d(x ni, x ki ) ε.butwehave d(x ni, x ki ) d(x ni, x ni +1) d(x ki, x ki +1) d(x ni +1, x ki +1) d(x ni, x ki )+d(x ni, x ni +1)+d(x ki, x ki +1) and so d(x ni +1, x ki +1) ε.thus,weget { max d(x ni, x ki ), d(x ni +1, x ni ), d(x ki +1, x ki ), 1 [ d(xni +1, x ki )+d(x ki +1, x ni ) ]} { max d(x ni, x ki ), d(x ni +1, x ni ), d(x ki +1, x ki ), 1 [ d(xni +1, x ki +1)+d(x ki +1, x ki )+d(x ki +1, x ki )+d(x ki, x ni ) ]} { max d(x ni, x ki ), d(x ni +1, x ni ), d(x ki +1, x ki ), 1 [ d(xni, x ki )+3d(x ki +1, x ki )+d(x ni +1, x ni ) ]} d(x ni, x ki )+ 3 d(x k i +1, x ki )+d(x ni +1, x ni ) and so M(x ni, x ki ) ε. On the other hand, we have d(x ni +1, x ki +1)=d(Tx ni, Tx ki ) α(x ni, x ki )d(tx ni, Tx ki ) φ ( M(x ni, x ki ) ) and so ε φ(ε). This contradiction shows that {x n } is a Cauchy sequence. Since X is complete, there exists x X such that x n x.ift is continuous, then we get d(x, Tx) = lim n d(x n+1, Tx)= lim n d(tx n, Tx)=0. Now, suppose that α(t i x 0, x) 1 for all i 0wheneverT i x 0 x.then d(x, Tx) = lim d(x n+1, Tx)= lim d(tx n, Tx) lim α(x n, x)d(tx n, Tx) n n n lim φ( max { d(x n, x), d(x n, x n+1 ), d(x, Tx), ( d(x n, Tx)+d(x, x n+1 ) ) / n = φ ( d(x, Tx) ). Thus, d(x, Tx)=0.

8 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 8 of 10 Here, we give the following example to show that there are discontinuous mappings satisfying the conditions of Theorem 3.7. Example 3.1 Let X =[0, )andd(x, y)= x y.definet : X X by Tx = x + whenever x [0, 1], Tx = 3 whenever x (1, ) and Tx = x3 +x + 1 whenever x [, ). Also, define the mappings φ :[0, ) [0, )andα : X X [0, )byφ(t)= t, α(x, y)=1whenever x, y (1, ) and α(x, y) = 0 otherwise. An easy calculation shows us that α(x, y)d(tx, Ty) φ max d(x, y), d(x, Tx), d(y, Ty), for all x, y X. Putx 0 = 3.SinceTi x 0 = x 0 for all i 1, α(t i x 0, T j x 0 ) 1 for all i, j 0 with i j and α(t i x 0, x) 1 for all i 0wheneverT i x 0 x.thus,themapt satisfies the conditions of Theorem 3.7.Notethatx 0 = 3 is a fixed point of T. Corollary 3.8 Let (X,, d) be a complete ordered metric space, φ :[0, ) [0, ) be a continuous and nondecreasing map such that φ(t)<tforallt>0and T be a self-map on Xsuchthat d(tx, Ty) φ max d(x, y), d(x, Tx), d(y, Ty), for all comparable elements x, y X. Assume that there exists x 0 XsuchthatT i x 0 and T j x 0 are comparable for all i, j 0 with i j. Suppose that T is continuous or T i x 0 and x are comparable for all i 0 whenever T i x 0 x. Then T has a fixed point. Proof Define the mapping α : X X [0, + ) byα(x, y)=1wheneverx and y are comparable and α(x, y) = 0 otherwise. Then, by using Theorem 3.7, T has a fixed point. Corollary 3.9 Let (X,, d) be a complete ordered metric space, z X, φ :[0, ) [0, ) be a continuous and nondecreasing map such that φ(t)<tforallt>0and T be a self-map on X such that d(tx, Ty) φ max d(x, y), d(x, Tx), d(y, Ty), for all x, y Xwhicharecomparablewithz. Assume that there exists x 0 XsuchthatT i x 0 and T j x 0 are comparable with z for all i, j 0 with i j. Suppose that T is continuous or T i x 0 and x are comparable with z for all i 0 whenever T i x 0 x. Then T has a fixed point. Proof Define the mapping α : X X [0, + ) byα(x, y)=1wheneverx and y are comparable with z and α(x, y) = 0 otherwise. Then, by using Theorem 3.7, T has a fixed point. Finally, by using [14], we can find also some equivalent conditions for some presented results. We give two following results in this way. This shows us the importance of the main results of [14]. Also, Example 3.1 leads us to the fact that there are discontinuous mappings satisfying the conditions of the following results.

9 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 9 of 10 Proposition 3.10 Let (X, d) be a complete metric space, α : X X [0, ) be a mapping, ψ :[0, ) [0, ) be a lower semi-continuous function, η :[0, ) [0, ) be a map such that η 1 ({0={0} and lim inf t η(t)>0for all t >0and T be a self-map on X such that ψ ( α(x, y)d(tx, Ty) ) ψ ( M(x, y) ) η ( M(x, y) ) for all x, y X. Assume that there exists x 0 Xsuchthatα(T i x 0, T j x 0 ) 1 for all i, j 0 with i j. Suppose that T is continuous or α(t i x 0, x) 1 for all i 0 whenever T i x 0 x. Then T has a fixed point. Proposition 3.11 Let (X, d) be a complete metric space, α : X X [0, ) be a mapping, ψ be a nondecreasing map which is continuous from right at each point, φ : [0, ) [0, ) be a map such that φ(t)<tforallt>0and T be a self-map on X such that ψ(α(x, y)d(tx, Ty)) φ(ψ(m(x, y))) for all x, y X. Assume that there exists x 0 X such that α(t i x 0, T j x 0 ) 1 for all i, j 0 with i j. Suppose that T is continuous or α(t i x 0, x) 1 for all i 0 whenever T i x 0 x. Then T has a fixed point. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Azarshahr, Iran. 3 Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah, 1859, Saudi Arabia. Acknowledgements The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper. Received: 6 July 01 Accepted: 13 January 013 Published: 6 February 013 References 1. Samet, B, Vetro, C, Vetro, P: Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Anal. 75, (01). Alikhani, H, Rezapour, S, Shahzad, N: Fixed points of a new type contractive mappings and multifunctions. Filomat (013, to appear) 3. Ciric, LB: Fixed points for generalized multivalued mappings. Mat. Vesn. 9(4),65-7 (197) 4. Haghi, RH, Rezapour, S, Shahzad, N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 74, (011) 5. Ciric, LB: A generalization of Banach contraction principle. Proc. Am. Math. Soc. 45, (1974) 6. Ciric, LB: Generalized contractions and fixed point theorems. Publ. Inst. Math. (Belgr.) 1(6),19-6 (1971) 7. Asl, JH, Rezapour, S, Shahzad, N: On fixed points of α-ψ-contractive multifunctions. Fixed Point Theory Appl. 01, 1 (01). doi: / Ilic, D, Rakocevic, V: Quasi-contraction on a cone metric space. Appl. Math. Lett., (009) 9. Kadelburg, Z, Radenovic, S, Rakocevic, V: Remarks on quasi-contraction on a cone metric space. Appl. Math. Lett., (009) 10. Rezapour, S, Haghi, RH, Shahzad, N: Some notes on fixed points of quasi-contraction maps. Appl. Math. Lett. 3, (010) 11. Amini-Harandi, A: Fixed point theory for set-valued quasi-contraction maps in metric spaces. Appl. Math. Lett. 4, (011) 1. Haghi, RH, Rezapour, S, Shahzad, N: On fixed points of quasi-contraction type multifunctions. Appl. Math. Lett. 5, (01) 13. Aleomraninejad, SMA, Rezapour, S, Shahzad, N: Some fixed point results on a metric space with a graph. Topol. Appl. 159, (01) 14. Jachymski, J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, (011)

10 Mohammadi et al. Fixed Point Theory and Applications 013, 013:4 Page 10 of 10 doi: / Cite this article as: Mohammadi et al.: Some results on fixed points of α-ψ-ciric generalized multifunctions. Fixed Point Theory and Applications :4.