Research Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

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1 Abstract and Alied Analysis Volume 2012, Article ID , 11 ages doi: /2012/ Research Article An iterative Algorithm for Hemicontractive Maings in Banach Saces Youli Yu, 1 Zhitao Wu, 2 and Pei-Xia Yang 2 1 School of Mathematics and Information Engineering, Taizhou University, Linhai , China 2 Deartment of Mathematics, Tianjin Polytechnic University, Tianjin , China Corresondence should be addressed to Youli Yu, yuyouli@tzc.edu.cn Received 26 Aril 2012; Acceted 11 May 2012 Academic Editor: Yonghong Yao Coyright q 2012 Youli Yu et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. We First introduce a three-ste iterative algorithm for aroximating the fixed oints of the hemicontractive maings in Banach saces. Consequently, we rove the strong convergence of the roosed algorithm under some assumtions. Since three-ste iterations include Ishikawa iterations as secial cases, our result continue to hold for these roblems. Our main results can be viewed as an imortant refinement of the reviously known results. 1. Introduction In recent years, several convergence results have been roved on iterative methods for aroximating fixed oints of seudocontractive maings see, e.g., 1 10 and references therein. It is worth mentioning that such iterative tye methods are known as Mann iterations and Ishikawa iterations. It is clear that a Lischitz seudocontractive maing with a unique fixed oint for which the Mann iteration sequence failed to converge, but it does converge for the sequence obtained by the Ishikawa iterations, see 11. In 2000, Noor 12 suggested and analyzed three-ste iterative methods for finding the aroximate solution of a continuous maing in the Hilbert sace using the technique of udating the solution. Three-ste iterations are also known as Noor iteration. It is well known 13 that three-ste iterative schemes include one-ste Mann and two-ste Ishikawa iterations as secial cases. It raises an interesting question. Is there exist any Lischitz seudo-contractive maing with a unique fixed oint for which Ishikawa iteration sequence fail to convergence, but Does convergence for the sequence obtained from Noor iteration? This is an oen and challenging roblem. To the best of our knowledge, main result of Ishikawa 14, see Theorem IS has never been extended to more general Banach saces. Motivated and insired by the

2 2 Abstract and Alied Analysis recent research activities in this filed, we suggest and analyze a three-ste iterative scheme associated with hemi-contractive maings in Banach saces. We also rove the strong convergence of the sequence generated by the three-ste iterations under mild conditions. Since three-ste iterations include Ishikawa iterations as secial cases, our results continue to hold for these roblems. It is worth mentioning that our results may be considered as very significant, interesting and imortant extensions of the reviously known results concerning seudo-contractive maings. Let E be a real Banach sace and E be its dual sace. The normalized duality maing from E to 2 E is defined by J x {x E : x, x x x, x x }, x E, 1.1 where, denotes the generalized duality airing. Let C be a nonemty subset of E, a maing T : C C is called seudo-contractive if there exists j x y J x y such that Tx Ty,j ( x y ) x y 2, 1.2 for all x, y C.LetF T : {x C : Tx x}. A maing T : C C is called hemicontractive if F T / and Tx x,j x x x x 2, x C, x F T. 1.3 It is easy to see that the class of seudo-contractive maings with fixed oints is a subclass of the class of hemicontractions. A maing T : C C is called Lischitzian if there exists a constant L 0 such that Tx Ty L x y for each x, y C. In 1974, Ishikawa 14 roved the following result for the seudo-contractive maings. Theorem 1.1 see 14. If C is a comact convex subset of a Hilbert sace H, T : C C is a Lischitzian seudo-contractive maing. For x 0 C, define the sequence {x n } iteratively by x n 1 1 α n x n α n Ty n, y n ( 1 β n ) xn β n Tx n, n 0, 1.4 where {α n }, {β n } are sequences of ositive numbers satisfying the conditions i 0 α n β n < 1; ii lim n β n 0; iii n 1 α n β n. Then the sequence {x n } defined by 1.4 converges strongly to a fixed oint of T. Since its ublication in 1974, Theorem IS, as far as we know, has never been extended to more general Banach saces. In this aer, we suggest and analyze a three-ste iteration below Algorithm 1.2 associated with hemi-contractive maings having a strong convergence in the setting of Banach saces under some aroriate conditions.

3 Abstract and Alied Analysis 3 Algorithm 1.2. For arbitrary x 1 C, let the sequence {x n } be generated by z n ( 1 γ n ) xn γ n Tx n, y n ( 1 β n ) xn β n Tz n, 1.5 x n 1 1 α n x n α n Ty n, n 1. It is clear that Algorithm 1.2 includes Mann one-ste and Ishikawa two-ste iterations as secial cases. For some related works, lease refere to Preliminaries Let E be a Banach sace, the modulus of convexity of E is the function δ E : 0, 2 0, 1 defined by { δ E ɛ inf 1 1 x y : x 1, y 1, x y ɛ } A Banach sace E is called uniformly convex if and only if δ E ɛ > 0 for all ɛ 0, 2. For > 1, the generalized duality maing J : E 2 E is defined as J x : {x E : x, x x, x x 1 }. In articular, J J 2 is the normalized duality maing on E. It is known that J x x 2 J x, x / 0. A Banach sace E is called -uniformly convex if there exists a constant c>0 such that δ E ɛ cɛ,0<ɛ 2. It is known see e.g., 12 that L is 2 uniformly convex, if 1 < 2, uniformly convex, if For roving our main results, we shall need the following lemmas. Lemma 2.1 see 20. Let >1be a given real number. Then the following statements about a Banach sace E are equivalent: i E is -uniformly convex; ii there is a constant c > 0 such that for every x, y E, j x J x, the following inequality holds: x y x y, j x c y. 2.3 Remark 2.2. Relacing x by x y, y by y in inequality 2.3 and using the Cauchy- Schwarz inequality, we can obtain x y x y x y

4 4 Abstract and Alied Analysis Lemma 2.3 see 20. Let >1 be a given real number. Let E be a -uniformly convex Banach sace. Then, there exists a constant d>0 such that λx 1 λ y λ x 1 λ y W λ d x y, 2.5 for all λ 0, 1 and x, y E,whereW λ λ 1 λ λ 1 λ. Lemma 2.4 see 4. Let {ρ n } : {σ n } be two nonnegative sequences and for all integers n N 0 (for some fixed N 0 ), ρ n 1 ρ n σ n. i if n 1 σ n <,thenlim n ρ n exists; ii if n 1 σ n < and {ρ n } has a sequence converging to zero, then lim n ρ n Main Results In the sequel, c and d will denote the constants aearing in inequalities 2.3 and 2.5, resectively. For the rest of this aer, we shall assume that E be a real -uniformly convex Banach sace such that 2 2 d > 1 c 1 and 1 c. For L saces with 1 < 2, the following inequalities hold see 12, ages : x y 2 x 2 2 y, J x c y 2, λx 1 λ y 2 λ x 2 1 λ y 2 W2 λ ( 1 ) x y 2, 3.1 for λ 0, 1 and for all x, y E, where c 1 t 1 1 t 1,andfor0<t < 1, t is the unique solution of the equation g t 2 t 1 1 t Remark 3.1. We observe that the function h : 0, 1 0, defined by h x 1 x 1 / 1 x 1 is increasing on 0, 1 h x 1 x 2 1 x 2 1 / 1 x 2 2 0, hence for L 1 < 2, we have c 1andd 1. Therefore, the conditions 2 2 d > 1 c 1 and 1 c are satisfied. Lemma 3.2. Let E be a real -uniformly convex Banach sace, / C E nonemty closed convex and bounded, and T : C C a hemi-contractive maing with F T /. Then, for each x C and for each integer n 1, the following inequality holds: c Tx x ( 1 ) x x x Tx, x F T. 3.2 Proof. Relacing x by 1/2 x x and y by 1/2 Tx x in inequality 2.3, we can get ( ) 1 x Tx x x 2 Tx 1 x,j 2 x x c Tx x 3.3 x x x x c Tx x.

5 Abstract and Alied Analysis 5 Since j 1/2 x x J 1/2 x x 2 1 x x 2 J x x so that c Tx x ( 1 ) x x x Tx. 3.4 This comletes the roof. Remark 3.3. We note that the function f : 0,, defined by f x L x d 1 x c 1 is strictly increasing on 0,. Hence, it has at most one zero on 0,, rovided that f 0 1 c 1 d2 2 < 0. In this case, since f 1 L 1 c 1 > 0, it follows that the zero t 0, 1. Lemma 3.4. Let E be a real -uniformly convex Banach sace such that 2 2 d > 1 c 1 and 1 c.letc be a nonemty closed convex and bounded subset of E,letT : C C be a Lischitz hemi-contractive maing with Lischitz constant L 0 and F T /. Let{α n }, {β n } and {γ n } be three real sequences in 0, 1 satisfying the following conditions: ɛ 1 dc 1 α n 2 2 β n b, γ n <. n for all integers n 1, someɛ>0 and b 0,t,wheret is the unique solution of the equation: L x d 1 x 2 2 ( 1 ) c 1 0, 3.6 on 0,. For arbitrary x 1 C, let the sequence {x n } be generated by z n ( 1 γ n ) xn γ n Tx n, y n ( 1 β n ) xn β n Tz n, 3.7 x n 1 1 α n x n α n Ty n, n 1. Then, lim n x n Tx n 0. Proof. We shall use M to denote the ossible different constants aearing in the following reasoning. Let x F T. Using inequality 2.5, we have x n 1 x 1 αn x n x α n ( Tyn x ) 1 α n x n x α n Tyn x 3.8 W α n d xn Ty n.

6 6 Abstract and Alied Analysis From 3.2, we have c Tx n x ( 1 ) x n x x n Tx n, c Tyn x ( 1 ) yn x yn Ty n Moreover, we also have y n x ( 1 β n ) xn x β n Tz n x ( 1 β n ) xn x β n Tz n x 3.11 ( ) W βn d xn Tz n, y n Ty n ( )( ) ( ) 1 β n xn Ty n βn Tzn Ty n ( 1 β n )xn Ty n βn Tzn Ty n 3.12 W ( βn ) d xn Tz n. At the same time, alying 2.4, we can obtain the following estimates: Tz n x Tx n x Tz n Tx n Tx n x Tz n Tx n Tz n x 1 Tx n x L z n x n Tz n x Tx n x Lγ n Tx n x n Tz n x 1 Tx n x Mγ n, x n Tx n Tz n Tx n x n Tz n x n Tz n Tz n Tx n x n Tx n x n Tz n Mγ n, Tzn Ty n Txn Ty n Tz n Tx n Txn Ty n Tzn Tx n Tzn Ty n Tx n Ty n Mγ n. Substitute 3.13 and 3.14 into 3.11 to get yn x ( 1 β n ) xn x β n Tx n x W ( βn ) d xn Tx n Mγ n, 3.16

7 Abstract and Alied Analysis 7 this together with 3.9 imlies that yn x ( 1 β n ) xn x β n c 1 {( ) 1 xn x x n Tx n } W ( βn ) d xn Tx n Mγ n [ 1 β n c 1 ( ) ] 1 c x n x [ β n c 1 ( ) W βn d ] x n Tx n Mγ n Set t n β n c 1 1 c, r n β n c 1 W β n d. Then, yn x 1 t n x n x r n x n Tx n Mγ n From 3.12, 3.14, and 3.15, we have yn Ty n ( 1 β n )xn Ty n β n Txn Ty n W ( βn ) d xn Tx n Mγ n Substitution of 3.18 and 3.19 into 3.10 yields c Tyn x ( 1 ) 1 t n x n x ( 1 ) r n x n Tx n ( 1 β n ) x n Ty n β n Tx n Ty n W ( βn ) d xn Tx n Mγ n ( 1 ) 1 t n x n x [( 1 ) r n W ( βn ) d ] 3.20 x n Tx n ( 1 β n ) x n Ty n β n Txn Ty n Mγn.

8 8 Abstract and Alied Analysis Substitution of this inequality into 3.8 now gives x n 1 x 1 α n x n x α n c 1 {( ) 1 1 tn x n x [( 1 ) ( ) ] r n W βn d x n Tx n ( ) 1 β xn n Ty n β Txn n Ty n } W α n d xn Ty n Mγn [ 1 α n α n c 1 ( ) 1 1 tn ] x n x [ α n c 1 ( ) 1 βn W α n d]xn Ty n 3.21 α n c 1 [( ) ( ) ] 1 rn W βn d xn Tx n α n β n c 1 Txn Ty n Mγn, that is, x n 1 x { [ ( ) ]} 1 α n 1 c 1 1 t n 1 x n x [ W α n d c 1 ( ) ] α n 1 βn xn Ty n c 1 [ ( ) ( ) ] α n W βn d 1 rn xn Tx n α n β n c 1 Tx n Ty n Mγ n Observe that c t n 1 c 2 1 c 1 β n c and that by condition 3.5, since W α n α n 1 α n 2 2,wegetW α n d c 1 α n 1 β n 0. so that { x n 1 x 1 α n c 2 ( )[( ) ] } 1 c 1 βn c x n x α n c 1 [ ( ) ( ) ] W βn d 1 rn xn Tx n α n β n c 1 Tx n Ty n Mγ n Since T is Lischitzian, we have Txn Ty n L xn y n L βn x n Tz n L β n x n Tz n L βn x n Tx n Tx n Tz n L β n [ x n Tx n Tx n Tz n x n Tz n 1] 3.24 L β n x n Tx n Mγ n.

9 Abstract and Alied Analysis 9 By the assumtion 1 c, hence [ x n 1 x x n x α n β n c 1 d ( ) 1 β n 2 2 ( 1 ) c 1 βnl ] x n Tx n Mγ n Since b 0,t, it follows that δ d 1 b c 1 b L > 0. We can choose some ɛ such that ɛ 1 1 ɛ 2 2 c d>0. Then, condition 3.5 imlies α n ɛ > 0. Furthermore, inequality 3.25 now yields the following estimates: x n 1 x x n x ɛɛ c 1 δ x n Tx n Mγ n x n x Mγ n Since n 0 γ n <, it follows from Lemma 2.3 that lim n x n x exists. Let lim n x n x r. Inequality 3.26 also yields 0 <ɛɛ c 1 δ x n Tx n x n x x n 1 x Mγ n Hence, lim n x n Tx n 0. This comletes the roof. Remark 3.5. The interest and imortance of Lemma 3.2 lie in the fact that strong convergence of the sequence {x n } is achieved under certain mild comactness assumtions either on T or on its domain. Now, we give a strong convergence theorem as follows. Theorem 3.6. Let E be a real -uniformly convex Banach sace such that 2 2 d > 1 c 1 and 1 c.letc be a nonemty closed convex and bounded subset of E, T : C C be a comletely continuous Lischitz hemi-contractive maing with Lischitz constant L 0 and F T /.Let{α n }, {β n }, and {γ n } be three real sequences in 0, 1 satisfying the following conditions: ɛ 1 dc 1 α n 2 2 β n b, γ n <, n for all integers n 1, someɛ>0 and b 0,t,wheret is the unique solution of the equation: L x d 1 x 2 2 ( 1 ) c 1 0, 3.29 on 0,. For arbitrary x 1 C, let the sequence {x n } be defined by 3.7. Then, {x n } converges strongly to a fixed oint of T.

10 10 Abstract and Alied Analysis Proof. By Lemma 3.2 lim n x n Tx n 0. Since T is comletely continuous, there exists a subsequence {Tx ni } of {Tx n } such that Tx ni y. This imlies, by Lemma 3.2 that x ni y By the continuity of T and Lemma 3.2, weobtainty y,thatis,y is a fixed oint of T. Relacing the x by y in inequality 3.26, weobtainthat xn 1 y xn y ɛɛ c 1 δ x n Tx n xn y Mγ n From 3.30, we know that { x n y } has a sequence converging to zero. We note that the condition n 0 γ n <. Hence from inequality 3.31 and Lemma 2.3, we can conclude that x n y as n,thatis,{x n } converges to a fixed oint of T. This comletes the roof. From Theorem 3.6, we can obtain the following result. Corollary 3.7. Let E be a real -uniformly convex Banach sace such that 2 2 d > 1 c 1 and 1 c.letc be a nonemty closed convex and bounded subset of E, T : C C be a comletely continuous Lischitz hemi-contractive maing with Lischitz constant L 0 and F T /.Let{α n } and {β n } be two real sequences in 0, 1 satisfying the following condition: ɛ 1 dc 1 α n 2 2 β n b, 3.32 for all integers n 1, someɛ>0 and b 0,t,wheret is the unique solution of the equation: L x d 1 x 2 2 ( 1 ) c on 0,. For arbitrary x 1 C, let the sequence {x n } be defined by 1.4. Then, {x n } converges strongly to a fixed oints of T. Acknowledgment This research was artially suorted by the Youth Foundation of Taizhou University 2011QN11. References 1 H. Zhou, Convergence theorems of common fixed oints for a finite family of Lischitz seudocontractions in Banach saces, Nonlinear Analysis: Theory, Methods & Alications, vol. 68, no. 10, , Y. Yao, Y.-C. Liou, and R. Chen, Strong convergence of an iterative algorithm for seudocontractive maing in Banach saces, Nonlinear Analysis: Theory, Methods & Alications, vol. 67, no. 12, , 2007.

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