Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

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1 Journal of Applied Mathematics Volume 2012, Article ID , 15 pages doi: /2012/ Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space Bin-Chao Deng, Tong Chen, and Zhi-Fang Li School of Management, Tianjin University, Tianjin , China Correspondence should be addressed to Bin-Chao Deng, Received 17 April 2012; Revised 1 June 2012; Accepted 1 June 2012 Academic Editor: Hong-Kun Xu Copyright q 2012 Bin-Chao Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let {T i } N i 1 be N strictly pseudononspreading mappings defined on closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality γf μb x,v x 0, v N i 1 F ix T i. 1. Introduction Throughout this paper, we always assume that C is a nonempty, closed, and convex subset of a real Hilbert space H. Let B : C H be a nonlinear mapping. Recall the following definitions. Definition 1.1. B is said to be i monotone if Bx By, x y 0, x, y C, 1.1 ii strongly monotone if there exists a constant α>0 such that Bx By, x y α x y 2, x, y C, 1.2 for such a case, B is said to be α-strongly-monotone,

2 2 Journal of Applied Mathematics iii inverse-strongly monotone if there exists a constant α>0 such that Bx By, x y α Bx By 2, x, y C, 1.3 for such a case, B is said to be α-inverse-strongly monotone, iv k-lipschitz continuous if there exists a constant k 0 such that Bx By k x y x, y C. 1.4 Remark 1.2. Let F μb γf, where B is a k-lipschitz and η-strongly monotone operator on H with k>0andf is a Lipschitz mapping on H with coefficient L>0, 0 <γ μη/l. It is a simple matter to see that the operator F is μη γl -strongly monotone over H, that is: Fx Fy,x y ( μη γl )x y 2, ( x, y ) H H. 1.5 Following the terminology of Browder-Petryshyn 1, we say that a mapping T : D T H H is 1 k-strict pseudocontraction if there exists k 0, 1 such that Tx Ty 2 x y 2 k x Tx ( y Ty ) 2, x, y D T, k-strictly pseudononspreading if there exists k 0, 1 such that Tx Ty 2 x y 2 k x Tx ( y Ty ) 2 2 x Tx,y Ty, 1.7 for all x, y D T, 3 nonspreading in 2 if Tx Ty 2 Tx y 2 Ty x 2, x, y C. 1.8 It is shown in 3 that 1.8 is equivalent to Tx Ty 2 x y 2 2 x Tx,y Ty, x, y C. 1.9 Clearly every nonspreading mapping is 0-strictly pseudononspreading. Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see 2, 4 6. Let C be a closed convex subset of H,andlet{T i } N i 1 be nk i-strictly pseudocontractive mappings on C such that N i 1 F ix T i /.Letx 1 C and {α n } n 1 be a sequence in 0, 1.In 7, Acedo and Xu introduced an explicit iteration scheme called the followting cyclic algorithm

3 Journal of Applied Mathematics 3 for iterative approximation of common fixed points of {T i } N i 1 in Hilbert spaces. They define the sequence {x n } cyclically by x 1 α 0 x 0 1 α 0 T 0 x 0 ; x 2 α 1 x 1 1 α 1 T 1 x 1 ; x N α N 1 x N 1 1 α N 1 T N 1 x N 1 ; x N 1 α N x N 1 α N T 0 x N ; In a more compact form, they rewrite x n 1 as x N 1 α N x N 1 α N T N x n, 1.12 where T N T i,withi n mod N, 0 i N 1. Using the cyclic algorithm 1.12, Acedo and Xu 7 show that this cyclic algorithm 1.12 is weakly convergent if the sequence {α n } of parameters is appropriately chosen. Motivated and inspired by Acedo and Xu 7, we consider the following cyclic algorithm for finding a common element of the set of solutions of k i -strictly pseudononspreading mappings {T i } N i 1. The sequence {x n} i 1 generated from an arbitrary x 1 H as follows: x 1 α 0 γf x 0 ( I μα 0 B ) T ω0 x 0 ; x 2 α 1 γf x 1 ( I μα 1 B ) T ω1 x 1 ; x N α N 1 γf x N 1 ( I μα N 1 B ) T ωn 1 x N 1 ; x N 1 α N γf x N ( I μα N B ) T ω0 x N ; Indeed, the algorithm above can be rewritten as x n 1 α n γf x n ( I μα n B ) T ω n x n, 1.15 where T ω n I ω n I ω n T n, T n T n mod N ; namely, T N is one of T 1,T 2,...,T N circularly.

4 4 Journal of Applied Mathematics 2. Preliminaries Throughout this paper, we write x n xto indicate that the sequence {x n } converges weakly to x. x n x implies that {x n } converges strongly to x. The following definitions and lemmas are useful for main results. Definition 2.1. A mapping T is said to be demiclosed, if for any sequence {x n }which weakly converges to y, and if the sequence {Tx n } strongly converges to z, then T y z. Definition 2.2. T : H H is called demicontractive on H, if there exists a constant α<1 such that Tx q 2 x q 2 α x Tx 2, ( x, q ) H F ix T. 2.1 Remark 2.3. Every k-strictly pseudononspreading mapping with a nonempty fixed point set F ix T is demicontractive see 8, 9. Remark 2.4 See 10. LetT be a α-demicontractive mapping on H with F ix T / and T ω 1 ω I ωt for ω 0, : A1 Tα-demicontractive is equivalent to x Tx,x q α x Tx 2, ( x, q ) H F ix T, 2.2 A2 F ix T F ix T ω if ω / 0. Remark 2.5. According to I T ω ω I T with T being a k-strictly pseudononspreading mapping, we obtain x Tω x, x q ω 1 k x Tx 2, ( x, q ) H F ix T Proposition 2.6 see 2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C C be a k-strictly pseudononspreading mapping. If F ix T /, then it is closed and convex. Proposition 2.7 see 2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C C be a k-strictly pseudononspreading mapping. Then I T is demiclosed at 0. Lemma 2.8 see 11. Let {T n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {T nj } j 0 of {T n } which satisfies T nj < T nj 1 for all j 0. Also consider the sequence of integers {δ n } n n0 defined by δ n max{k n T k < T k 1 }. 2.4 Then {δ n } n n0 is a nondecreasing sequence verifying lim δ n, n n 0 ; it holds that T δ n < T δ n 1 and we have T n < T δ n

5 Journal of Applied Mathematics 5 Lemma 2.9. Let H be a real Hilbert space. The following expressions hold: i tx 1 t y 2 t x 2 1 t y 2 t 1 t x y 2, ii x y 2 x 2 2 y, x y, x, y H. x, y H, t 0, 1, Lemma 2.10 see 6. Let C be a closed convex subset of a Hilbert space H, and let T : H H be a k-strictly pseudononspreading mapping with a nonempty fixed point set. Let k ω<1 be fixed and define T ω C C by T ω x 1 ω x ωt x, x C. 2.6 Then F ix T ω F ix T. Lemma Assume C is a closed convex subset of a Hilbert space H. a Given an integer N 1, assume, T i : H H is a k i -strictly pseudononspreading mapping for some k i 0, 1, i 1, 2,...,N.Let{λ i } N i 1 be a positive sequence such that N i 1 λ i 1. Suppose that {T i } N i 1 has a common fixed point and N i 1 F ix T i /. Then, ( ) N F ix λ i T i i 1 N F ix T i. i b Assuming T i : H H is a k i -strictly pseudononspreading mapping for some k i 0, 1, i 1, 2,...,N,letT ωi 1 ω i I ω i T i, 1 i N.If N i 1 F ix T i /, then N F ix T ω1 T ω2 T ωn F ix T ωi. i Proof. To prove a, we can assume N 2. It suffices to prove that F ix F F ix T 1 F ix T 2, where F 1 λ T 1 λt 2 with 0 <λ<1. Let x F ix F and write V 1 I T 1 and V 2 I T 2. From Lemma 2.9 and taking z F ix T 1 F ix T 2 to deduce that z x 2 1 λ z T 1 x λ z T 2 x 2 1 λ z T 1 x 2 λ z T 2 x 2 λ 1 λ T 1 x T 2 x 2 ( 1 λ z x 2 k x T 1 x 2) λ ( z x 2 k x T 2 x 2) 2.9 λ 1 λ T 1 x T 2 x 2 [ z x 2 k 1 λ V 1 x 2 λ V 2 x 2] λ 1 λ V 1 x V 2 x 2, it follows that [ λ 1 λ V 1 x V 2 x 2 k 1 λ V 1 x 2 λ V 2 x 2]. 2.10

6 6 Journal of Applied Mathematics Since 1 λ V 1 x λv 2 x 0, we obtain 1 λ V 1 x 2 λ V 2 x 2 λ 1 λ V 1 x V 2 x This together with 2.10 implies that 1 k λ 1 λ V 1 x V 2 x Since 0 <λ<1andk<1, we get V 1 x V 2 x 0 which implies T 1 x T 2 x which in turn implies that T 1 x T 2 x x since 1 λ T 1 x λt 2 x x. Thus,x F ix T 1 F ix T 2. By induction, we also claim that F ix N i 1 λ it i N i 1 F ix T i with {λ i } N i 1 is a positive sequence such that N i 1 λ i 1, i 1, 2,...,N. To prove b, we can assume N 2. Set T ω1 1 ω 1 I ω 1 T 1 and T ω2 1 ω 2 I ω 1 T 2, 0 <k i <ω i < 1/2, i 1, 2. Obviously F ix T ω1 F ix T ω2 F ix T ω1 T ω Now we prove F ix T ω1 F ix T ω2 F ix T ω1 T ω2, 2.14 for all q F ix T ω1 T ω2 and T ω1 T ω2 q q. IfT ω2 q q, then T ω1 q q; the conclusion holds. From Lemma 2.10, we can know that F ix T ω1 F ix T ω2 F ix T 1 F ix T 2 /. Taking p F ix T ω1 F ix T ω2, then p q 2 p Tω1 T ω2 q 2 p [ 1 ω1 ( T ω2 q ) ω 1 T 1 T ω2 q ] 2 1 ω1 ( p T ω2 q ) ω 1 ( p T1 T ω2 q ) 2 1 ω 1 p T ω2 q 2 ω 1 p T 1 T ω2 q 2 ω 1 1 ω 1 T ω2 q T 1 T ω2 q 2 1 ω 1 p Tω2 q 2 ω 1 1 ω 1 Tω2 q T 1 T ω2 q 2 ω 1 [p Tω2 q 2 k 1 Tω2 q T 1 T ω2 q 2 2 p T 1 T ω2 p, T ω2 q T 1 T ω2 q ] ω 1 p Tω2 q 2 ω 1 1 ω 1 Tω2 q T 1 T ω2 q 2 [ ω 1 p Tω2 q 2 k Tω2 1 q T 1 T ω2 q 2] p Tω2 q 2 ω 1 1 ω 1 k 1 Tω2 q T 1 T ω2 q 2 p q 2 ω1 1 ω 1 k 1 Tω2 q T 1 T ω2 q 2. Since 0 <k 1 <ω 1 < 1/2, we obtain Tω2 q T 1 T ω2 q

7 Journal of Applied Mathematics 7 Namely, T ω2 q T 1 T ω2 q,thatis: T ω2 q F ix T 1 F ix T ω1, T ω2 q T ω1 T ω2 q By induction, we also claim that the Lemma 2.11 b holds. Lemma Let K be a closed convex subset of a real Hilbert space H, givenx H and y K. Then y P K x if and only if there holds the inequality x y, y z 0, z K Cyclic Algorithm In this section, we are concerned with the problem of finding a point p such that N N p F ix T ωi F ix T i, N 1, 3.1 i 1 i 1 where T ωi 1 ω i I ω i T i, {ω i } N i 1 0, 1/2 and {T i} N i 1 are k i-strictly pseudononspreading mappings with k i 0,ω i, i 1, 2,...,N, defined on a closed convex subset C in Hilbert space H. HereF ix T ωi {q C : T ωi q q} is the set of fixed points of T i,1 i N. Let H be a real Hilbert space, and let B : H H be η-strongly monotone and ρ- Lipschitzian on H with ρ>0, η>0. Let 0 <μ<2η/ρ 2,0<γ<μ η μρ 2 /2 /L τ/l.let N be a positive integer, and let T i : H H be a k i -strictly pseudononspreading mapping for some k i 0, 1, i 1, 2,...,N, such that N i 1 F ix T i /. We consider the problem of finding p N i 1 F ix T i such that ( ) N γf μb p, v p 0, v F ix T i. i Since N i 1 F ix T i is a nonempty closed convex subset of H, VI 3.2 has a unique solution. The variational inequality has been extensively studied in literature; see, for example, Remark 3.1. Let H be a real Hilbert space. Let B be a ρ-lipschitzian and η-strongly monotone operator on H with ρ > 0, η>0. Leting 0 < μ < 2η/ρ 2 and leting S I tμb and μ η μρ 2 /2 τ, then for x, y H and t 0, min{1, 1/τ}, S is a contraction. Proof. Consider Sx Sy 2 I tμb x I tμb y 2 ( I tμb ) x ( I tμb ) y, ( I tμb ) x ( I tμb ) y x y 2 t 2 μ 2Bx By 2 2tμ x y, Bx By

8 8 Journal of Applied Mathematics x y 2 t 2 μ 2 ρ 2 x y 2 2tμη x y 2 [ ( )] 1 2tμ η μρ2 x y tτ x y 2 1 tτ 2 x y It follows that Sx Sy 1 tτ x y. 3.4 So S is a contraction. Next, we consider the cyclic algorithm 1.15, respectively, for solving the variational inequality over the set of the common fixed points of finite strictly pseudononspreading mappings. Lemma 3.2. Assume that {x n } is defined by 1.15 ; ifp is solution of 3.2 with T : H H being strictly pseudononspreading mapping and demiclosed and {y n } H is a bounded sequence such that Ty n y n 0, then ( ) lim inf γf μb p, yn p Proof. By Ty n y n 0andT : H H demi-closed, we know that any weak cluster point of {y n } belongs to N i 1 F ix T i. Furthermore, we can also obtain that there exists y and a subsequence {y nj } of {y n } such that y nj y as j hence y N i 1 F ix T i and ( ) lim inf γf μb p, yn p (γf ) lim μb p, ynj p. j 3.6 From 3.2, we can derive that ( ) lim inf γf μb p, yn p ( γf μb ) p, y p It is the desired result. In addition, the variational inequality 3.7 can be written as ( ) N I μb γf p p, y p 0, y F ix T i. i 1 3.8

9 Journal of Applied Mathematics 9 So, by Lemma 2.12, it is equivalent to the fixed point equation ( ) P N i 1 F ix T i I μb γf p p. 3.9 Theorem 3.3. Let C be a nonempty closed convex subset of H and for 1 i N. LetT i : H H be k i -strictly pseudononspreading mappings for some k i 0,ω i, ω i 0, 1/2, i 1, 2,...,N, and k max{k i :1 i N}. Letf be L-Lipschitz mapping on H with coefficient L>0, and let B : H H be η-strongly monotone and ρ-lipschitzian on H with ρ>0, η>0. Let{α n } being a sequence in 0, min{1, 1/τ} satisfying the following conditions: c1 lim α n 0, c2 n 0 α n. Given x 0 C,let{x n } n 1 be the sequence generated by the cyclic algorithm 1.15.Then{x n} converges strongly to the unique element p in N i 1 F ix T i verifying ( ) p P N i 1 F ix T i I μb γf p, 3.10 which equivalently solves the variational inequality problem 3.2. Proof. Take a p N i 1 F ix T i.lett ω x 1 ω x ωtx and 0 <k<ω<1/2. Then x, y C, we have T ω x T ω y 2 ω x y 2 1 ω Tx Ty 2 ( ) ω 1 ω x Tx y Ty 2 ω [ x y 2 1 ω x y 2 k ( ) ] x Tx y Ty 2 2 x Tx,y Ty ω 1 ω x Tx ( y Ty ) 2 x y ω x Tx,y Ty 1 ω ω k x Tx y Ty 2 x y ω x Tx,y Ty x y ω ω 2 x Tω x, y T ω y From p F ix T and 3.11, we also have Tω x n p xn p Using 1.15 and 3.12, weobtain xn 1 p αn γ ( f x n f ( p )) ( ( ) ) ( α n γf p p I μαn B )( T ω n x n p ) α n γ f xn f ( p ) αn γf ( p ) p 1 αn τ xn p, 3.13

10 10 Journal of Applied Mathematics which combined with 3.12 and f x n f p L x n p amounts to x n 1 p ( 1 α n ( τ γl )) x n p α n γf ( p ) p Putting M 1 max{ x 0 p, γf p p }, we clearly obtain x n p M 1. Hence {x n } is bounded. We can also prove that the sequences {f x n } and {T ω n x n } are all bounded. From 1.15 we obtain that x n 1 x n α n ( μbxn γf x n ) ( I μα n B )( T ω n x n x n ), 3.15 hence xn 1 x n α n ( μbxn γf x n ),x n p 1 α n T ω n x n x n,x n p α n ( I μb )( Tω n x n x n ),xn p 1 α n T ω n x n x n,x n p α n ( I μb )( Tω n x n x n ) xn p 1 α n T ω n x n x n,x n p α n 1 τ Tω n x n x n xn p α n T ω n x n x n,x n p ω n α n 1 τ T n x n x n xn p. Moreover, by p N i 1 F ix T i and using Remark 2.5,weobtain xn T x ω n n,x n p 1 2 ω ( ) n 1 k n x n T n x n 2, 3.17 which combined with 3.16 entails xn 1 x n α n ( μb γf ) xn,x n p 1 2 ω n ( 1 k n ) 1 αn xn T n x n 2 ω n α n 1 τ T n x n x n xn p, 3.18 or equivalently x n x n 1,x n p ( ) α n μb γf xn,x n p 1 2 ω ( ) n 1 k n 1 αn xn T n x n ω n α n 1 τ T n x n x n xn p. Furthermore, using the following classical equality u, v 1 2 u u v v 2, u, v C, 3.20

11 Journal of Applied Mathematics 11 and setting T n 1/2 x n p 2, we have xn x n 1,x n p T n T n x n x n So 3.19 can be equivalently rewritten as T n 1 T n 1 2 x n x n 1 2 α n ( μb γf ) x n,x n p 1 2 ω ( ) n 1 k n 1 αn xn T n x n ω n α n 1 τ T n x n x n xn p. Now using 3.15 again, we have x n 1 x n 2 αn γf x n μbx n I μα n B T ω n x n x n Since B : H H is η-strongly monotone and k-lipschitzian on H, hence it is a classical matter to see that x n 1 x n 2 2α 2 n γf xn μbx n αn τ 2 Tω n x n x n 2, 3.24 which by T ω n x n x n ω n x n T n x n and 1 α n τ 2 1 α n τ yields 1 2 x n 1 x n 2 α 2 n γf xn μbx n 2 1 αn τ ω 2 n xn T n x n Then from 3.22 and 3.25, we have ( ) 1 ) xn T n 1 T n ω n 1 k n 1 αn ω n 1 α n τ T n x n 2 2( α n (α γf xn n μbx n 2 ( ) μb γf xn,x n p ω n 1 τ T n x n x n xn p ) The rest of the proof will be divided into two parts. Case 1. Suppose that there exists n 0 such that {T n } n n0 is nonincreasing. In this situation, {T n } is then convergent because it is also nonnegative hence it is bounded from below, so that lim T n 1 T n 0; hence, in light of 3.26 together with lim α n 0 and the boundedness of {x n },weobtain lim x n T n x n

12 12 Journal of Applied Mathematics It also follows from 3.26 that T n T n 1 α n ( α n γf xn μbx n 2 ( μb γf ) xn,x n p ω n 1 τ T n x n x n xn p ) Then, by n 0 α n, we obviously deduce that lim inf α n ( α γf xn n μbx n 2 ( ) μb γf xn,x n p ω n 1 τ T n x n x n xn p ) 0, 3.29 or equivalently as α n γf x n μbx n 2 0 ( ) lim inf μb γf xn,x n p Moreover, by Remark 1.2, we have 2 ( μη γl ) T n ( μb γf ) p, x n p ( μb γf ) x n,x n p, 3.31 which by 3.30 entails ( ) lim inf 2 μη γl Tn ( μb γf ) p, x n p 0, 3.32 hence, recalling that lim T n exists, we equivalently obtain 2 ( μη γl ) ( ) lim T n lim inf μb γf p, xn p 0, 3.33 namely 2 ( μη γl ) ( ) lim T n lim inf μb γf p, xn p From 3.27 and Lemma 3.2,weobtain ( ) lim inf μb γf p, xn p 0, 3.35 which yields lim T n 0, so that {x n } converges strongly to p.

13 Journal of Applied Mathematics 13 T nk 1 Case 2. Suppose there exists a subsequence {T nk } k 0 of {T n } n 0 such that T nk for all k 0. In this situation, we consider the sequence of indices {δ n } as defined in Lemma 2.8. It follows that T δ n 1 T δ n > 0, which by 3.26 amounts to ( 1 ( )( ) ( ω n 1 k n 1 αδ n ω n 1 αδ n τ )) xδ n T n x δ n 2 2 <α δ n ( α δ n γf ( x δ n ) μbxδ n 2 ( μb γf ) xδ n,x δ n p 3.36 ω n 1 τ T n x δ n x δ n xδ n p ). By the boundedness of {x n } and lim α n 0, we immediately obtain lim xδ n T n x δ n Using 1.15, we have x δ n 1 x δ n α δ n γf ( x δ n ) μbxδ n ( 1 α δ n τ )T ω n x δ n x δ n α δ n γf ( xδ n ) μbxδ n ω n ( 1 αδ n τ ) T n x δ n x δ n, 3.38 which together with 3.37 and lim α n 0 yields lim x δ n 1 x δ n Now by 3.36 we clearly have α δ n γf xδ n μbx δ n 2 ω n 1 τ T n x δ n x δ n xδ n p ( μb γf ) x δ n,x δ n p, 3.40 which in the light of 3.31 yields 2 ( μη γl ) T δ n ( μb γf ) p, x δ n p α δ n γf x δ n μbx δ n 2 ω n 1 τ T n x δ n x n x δ n p, 3.41 hence as lim α δ n γf x δ n μbx δ n 2 0and 3.37 it follows that 2 ( μη γl ) lim sup From 3.37 and Lemma 3.2,weobtain ( ) T δ n lim inf μb γf p, xδ n p lim ( μb γf ) p, x δ n p 0, 3.43

14 14 Journal of Applied Mathematics which by 3.42 yields lim sup T δ n 0, so that lim T δ n 0. Combining 3.39, we have lim T δ n 1 0. Then, recalling that T n < T δ n 1 by Lemma 2.8, weget lim T n 0, so that x n p strongly. Taking k i 0, we know that k i -strictly pseudononspreading mapping is nonspreading mapping and i n mod N, 0 i N 1. According to the proof Theorem 3.3, weobtain the following corollary. Corollary 3.4. Let C be a nonempty closed convex subset of H. LetT i : C C be nonspreading mappings and ω i 0, 1/2, i 1, 2,...,N. Letf be L-Lipschitz mapping on H with coefficient L>0 and let B : H H be η-strongly monotone and ρ-lipschitzian on H with ρ>0, η>0. Let {α n } be a sequence in 0, min{1, 1/τ} satisfying the following conditions: c1 lim α n 0, c2 n 0 α n. Given x 0 C,let{x n } n 1 be the sequence generated by the cyclic algorithm 1.15.Then{x n} converges strongly to the unique element p in N i 1 F ix T i verifying ( ) p P N i 1 F ix T i I μb γf p, 3.44 which equivalently solves the variational inequality problem 3.2. Acknowledgment This work is supported in part by China Postdoctoral Science Foundation Grant no References 1 F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, Journal of Mathematical Analysis and Applications, vol. 20, pp , M. O. Osilike and F. O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp , S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2082 e2089, K. Aoyama, Halperns iteration for a sequence of quasinonexpansive type mappings, Nonlinear Mathematics for Uncertainty and Its Applications, vol. 100, pp , K. Aoyama and F. Kohsaka, Fixed point and mean convergence theorems for a famliy of λ-hybrid mappings, Journal of Nonlinear Analysis and Optimization, vol. 2, no. 1, pp , N. Petrot and R. Wangkeeree, A general iterative scheme for strict pseudononspreading mapping related to optimization problem in Hilbert spaces, Journal of Nonlinear Analysis and Optimization, vol. 2, no. 2, pp , G. L. Acedo and H.-K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp , S. A. Naimpally and K. L. Singh, Extensions of some fixed point theorems of Rhoades, Journal of Mathematical Analysis and Applications, vol. 96, no. 2, pp , T. L. Hicks and J. D. Kubicek, On the Mann iteration process in a Hilbert space, Journal of Mathematical Analysis and Applications, vol. 59, no. 3, pp , 1977.

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