Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 354202, 19 pages doi:10.1155/2010/354202 Research Article Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of λ i -Strict Pseudocontractions in q-Uniformly Smooth Banach Spaces Yanlai Song and Changsong Hu Department of Mathematics, Hubei Normal University, Huangshi 435002, China Correspondence should be addressed to Yanlai Song, songyanlai2009@163.com Received 9 August 2010; Revised 2 October 2010; Accepted 14 November 2010 Academic Editor: Mohamed Amine Khamsi Copyright q 2010 Y. Song and C. Hu. 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. We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions in q-uniformly smooth Banach spaces. We also discuss the strong convergence theorems for the new iterative scheme in q-uniformly smooth Banach space. Our results improve and extend the corresponding results announced by many others. 1. Introduction Throughout this paper, we denote by E and E ∗ a real Banach space and the dual space of E, respectively. Let C be a subset of E,andlrtT be a non-self-mapping of C.WeuseFT to denote the set of fixed points of T. The norm of a Banach space E is said to be G ˆ ateaux differentiable if the limit lim t → 0   x  ty   −  x  t 1.1 exists for all x, y on the unit sphere SE{x ∈ E : x  1}. If, for each y ∈ SE,the limit 1.1 is uniformly attained for x ∈ SE, then the norm of E is said to be uniformly G ˆ ateaux differentiable. The norm of E is said to be Fr ´ echet differentiable if, for each x ∈ SE, the limit 1.1 is attained uniformly for y ∈ SE. The norm of E is said to be uniformly Fr ´ echet differentiable or uniformly smooth if the limit 1.1 is attained uniformly for x, y ∈ SE × SE. 2 Fixed Point Theory and Applications Let ρ E : 0, 1 → 0, 1 be the modulus of smoothness of E defined by ρ E  t   sup  1 2    x  y      x − y    − 1:x ∈ S  E  ,   y   ≤ t  . 1.2 A Banach space E is said to be uniformly smooth if ρ E t/t → 0ast → 0. Let q>1. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c>0 such that ρ E t ≤ ct q . It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fr ´ echet differentiable. If E is q-uniformly smooth, then q ≤ 2andE is uniformly smooth, and hence the norm of E is uniformly Fr ´ echet differentiable, in particular, the norm of E is Fr ´ echet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L p , where p>1. More precisely, L p is min{p, 2}-uniformly smooth for every p>1. By a gauge we mean a continuous strictly increasing function ϕ defined R  :0, ∞ such that ϕ00 and lim r →∞ ϕr∞. We associate with a gauge ϕ a generally multivalued duality map J ϕ : E → E ∗ defined by J ϕ  x    x ∗ ∈ E ∗ :  x, x ∗    x  ϕ   x   ,  x ∗   ϕ   x    . 1.3 In particular, the duality mapping with gauge function ϕtt q−1 denoted by J q , is referred to the generalized duality mapping. The duality mapping with gauge function ϕtt denoted by J, is referred to the normalized duality mapping. Browder 1 initiated the study J ϕ .Setfort ≥ 0 Φ  t    t 0 ϕ  r  dr. 1.4 Then it is known that J ϕ x is the subdifferential of the convex function Φ· at x. It is well known that if E is smooth, then J q is single valued, which is denoted by j q . The duality mapping J q is said to be weakly sequentially continuous if the duality mapping J q is single valued and for any {x n }∈E with x n x, J q x n  ∗ J q x. Every l p 1 <p<∞ space has a weakly sequentially continuous duality map with the gauge ϕtt p−1 . Gossez and Lami Dozo 2 proved that a space with a weakly continuous duality mapping satisfies Opial’s condition. Conversely, if a space satisfies Opial’s condition and has a uniformly G ˆ ateaux differentiable norm, then it has a weakly continuous duality mapping. We already know t hat in q-uniformly smooth Banach space, there exists a constant C q > 0 such that   x  y   q ≤  x  q  q  y, J q  x    C q   y   q , 1.5 for all x, y ∈ E. Recall that a mapping T is said to be nonexpansive, if   Tx − Ty   ≤   x − y   ∀x, y ∈ C. 1.6 Fixed Point Theory and Applications 3 T is said to be a λ-strict pseudocontraction in the terminology of Browder and Petryshyn 3, if there exists a constant λ>0 such that  Tx − Ty,j q  x − y  ≤   x − y   q − λ   I − Tx − I − Ty   q , 1.7 for every x, y,andC for some j q x − y ∈ J q x − y. It is clear that 1.7 is equivalent to the following:   I − T  x −  I − T  y, j q  x − y  ≥ λ   I − Tx − I − Ty   q . 1.8 The following famous theorem is referred to as the Banach contraction principle. Theorem 1.1 Banach 4. Let X, d be a complete metric space and let f be a contraction on X, that is, there exists r ∈ 0, 1 such that dfx,fy ≤ rdx, y for all x, y ∈ X.Thenf has a unique fixed point. Theorem 1.2 Meir and Keeler 5. Let X, d be a complete metric space and let φ be a Meir-Keeler contraction (MKC, for short) on X, that is, for every ε>0, there exists δ>0 such that dx, y <ε δ implies dφx,φy <εfor all x, y ∈ X.Thenφ has a unique fixed point. This theorem is one of generalizations of Theorem 1.1, because contractions are Meir- Keeler contractions. In a smooth Banach space, we define an operator A is strongly positive if there exists a constant γ>0 with the property  Ax, J  x   ≥ γ  x  2 ,  aI − bA   sup  x  ≤1 {|  aI − bA  x, J  x  | : a ∈  0, 1  ,b∈  0, 1  } , 1.9 where I is the identity mapping and J is the normalized duality mapping. Attempts to modify the normal Mann’s iteration method for nonexpansive mappings and λ-strictly pseudocontractions so that strong convergence is guaranteed have recently been made; see, for example, 6–11 and the references therein. Kim and Xu 6 introduced the following iteration process: x 1  x ∈ C, y n  β n x n   1 − β n  Tx n , x n1  α n u   1 − α n  y n ,n≥ 0, 1.10 where T is a nonexpansive mapping of C into itself u ∈ C is a given point. They proved the sequence {x n } defined by 1.10 converges strongly to a fixed point of T, provided the control sequences {α n } and {β n } satisfy appropriate conditions. 4 Fixed Point Theory and Applications Hu and Cai 12 introduced the following iteration process: x 1  x ∈ C, y n  P C  β n x n   1 − β n  N  i1 η n i T i x n  , x n1  α n γf  x n   γ n x n   1 − γ n  I − α n A  y n ,n≥ 1. 1.11 where T i is non-self-λ i -strictly pseudocontraction, f is a contraction and A is a strong positive linear bounded operator in Banach space. They have proved, under certain appropriate assumptions on the sequences {α n }, {γ n },and{β n },that{x n } defined by 1.11 converges strongly to a common fixed point of a finite family of λ i -strictly pseudocontractions, which solves some variational inequality. Question 1. Can Theorem 3.1 of Zhou 8, T heorem 2.2 of Hu and Cai 12 and so on be extended from finite λ i -strictly pseudocontraction to infinite λ i -strictly pseudocontraction? Question 2. We know that the Meir-Keeler contraction MKC, for short is more general than the contraction. What happens if the contraction is replaced by the Meir-Keeler contraction? The purpose of this paper is to give the affirmative answers to these questions mentioned above. In this paper we study a general iterative scheme as follows: x 1  x ∈ C, y n  P C  β n x n   1 − β n  ∞  i1 η n i T i x n  , x n1  α n γφ  x n   γ n x n   1 − γ n  I − α n A  y n ,n≥ 1, 1.12 where T n is non-self λ n -strictly pseudocontraction, φ is a MKC contraction and A is a strong positive linear bounded operator in Banach space. Under certain appropriate assumptions on the sequences {α n }, {β n }, {γ n },and{μ n i },that{x n } defined by 1.12 converges strongly to a common fixed point of an infinite family of λ i -strictly pseudocontractions, which solves some variational inequality. 2. Preliminaries In order to prove our main results, we need the following lemmas. Lemma 2.1 see 13. Let {x n }, {z n } be bounded sequences in a Banach space E and {β n } be a sequence in 0, 1 which satisfies the following condition: 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1. Suppose that x n1 1 − β n x n  β n z n for all n ≥ 0 and lim sup n →∞ z n1 − z n −x n1 − x n  ≤ 0. Then, lim n →∞ z n − x n   0. Fixed Point Theory and Applications 5 Lemma 2.2 see Xu 14. Assume that {α n } is a sequence of nonnegative real numbers such that α n1 ≤ 1 − γ n α n  δ n ,whereγ n is a sequence in (0, 1) and {δ n } is a sequence in R such that i  ∞ n1 γ n  ∞, ii lim sup n →∞ δ n /γ n  ≤ 0 or  ∞ n1 |δ n | < ∞. Then lim n →∞ α n  0. Lemma 2.3 see 15 demiclosedness principle. Let C be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial’s condition, and suppose T : C → E is nonexpansive. Then the mapping I − T is demiclosed at zero, that is, x n x, x n − Tx n → 0 implies x  Tx. Lemma 2.4 see 16, Lemmas 3.1, 3.3. Let E be real smooth and strictly convex Banach space, and C be a nonempty closed convex subset of E which is also a sunny nonexpansive retraction of E. Assume that T : C → E is a nonexpansive mapping and P is a sunny nonexpansive retraction of E onto C,thenFTFPT. Lemma 2.5 see 17, Lemma 2.2. Let C be a nonempty convex subset of a real q-uniformly smooth Banach space E and T : C → C be a λ-strict pseudocontraction. For α ∈ 0, 1, we define T α x  1 − αx  αTx. Then, as α ∈ 0,μ, μ  min{1, {qλ/C q } 1/q−1 }, T α : C → C is nonexpansive such that FT α FT. Lemma 2.6 see 12, Remark 2.6. When T is non-self-mapping, the Lemma 2.5 also holds. Lemma 2.7 see 12, Lemma 2.8. Assume that A is a strongly positive linear bounded operator on a smooth Banach space E with coefficient γ>0 and 0 <ρ≤A −1 . Then,   I − ρA   ≤ 1 − ρ γ. 2.1 Lemma 2.8 see 18, Lemma 2.3. Let φ be an MKC on a convex subset C of a Banach space E. Then for each ε>0,thereexistsr ∈ 0, 1 such that   x − y   ≥ ε implies   φx − φy   ≤ r   x − y   ∀x, y ∈ C. 2.2 Lemma 2.9. Let C be a closed convex subset of a reflexive Banach space E which admits a weakly sequentially continuous duality mapping J q from E to E ∗ .LetT : C → C be a nonexpansive mapping with FT /  ∅ and φ : C → C be a MKC, A is strongly positive linear bounded operator with coefficient γ>0. Assume that 0 <γ<γ. Then the sequence {x t } define by x t  tγφx t 1 −tATx t converges strongly as t → 0 to a fixed point x of T which solves the variational inequality:  A − γφ  x, J q  x − z   ≤ 0,z∈ F  T  . 2.3 Proof. The definition of {x t } is well definition. Indeed, from the definition of MKC, we can see MKC is also a nonexpansive mapping. Consider a mapping S t on C defined by S t x  tγφ  x    I − tA  Tx, x ∈ C. 2.4 6 Fixed Point Theory and Applications It is easy to see that S t is a contraction. Indeed, by Lemma 2.8, we have   S t x − S t y   ≤ tγ   φ  x  − φ  y        I − tA   Tx − Ty    ≤ tγ   φ  x  − φ  y      1 − t γ    x − y   ≤ tγ   x − y     1 − t γ    x − y   ≤  1 − t  γ − γ    x − y   . 2.5 Hence, S t has a unique fixed point, denoted by x t , which uniquely solves the fixed point equation x t  tγφ  x t    I − tA  Tx t . 2.6 We next show the uniqueness of a solution of the variational inequality 2.3. Suppose both x ∈ FT and x ∈ FT are solutions to 2.3, not lost generality, we may assume there is a number ε such that  x − x≥ε. Then by Lemma 2.8, there is a number r such that φ x − φ x≤r x − x.From2.3,weknow  A − γφ  x, J q  x − x   ≤ 0,  A − γφ  x, J q  x − x   ≤ 0. 2.7 Adding up 2.7 gets  A − γφ  x −  A − γφ  x, J q  x − x   ≤ 0. 2.8 Noticing that  A − γφ  x −  A − γφ  x, J q  x − x    A  x − x  ,J q  x − x  −γ  φ x − φ x, J q  x − x   ≥ γ  x − x  q − γ   φ x − φ x    x − x  q−1 ≥ γ  x − x  q − γr  x − x  q ≥  γ − γr   x − x  q ≥  γ − γr  ε q > 0. 2.9 Therefore x  x and the uniqueness is proved. Below, we use x to denote the unique solution of 2.3. We observe that {x t } is bounded. Indeed, we may assume, with no loss of generality, t<A −1 , for all p ∈ FT, fixed ε 1 , for each t ∈ 0, 1. Case 1 x t − p <ε 1 . In this case, we can see easily that {x t } is bounded. Fixed Point Theory and Applications 7 Case 2 x t − p≥ε 1 . In this case, by Lemmas 2.7 and 2.8, there is a number r 1 such that   φ  x t  − φ  p    <r 1   x t − p   ,   x t − p      tγφ  x t    I − tA  Tx t − p      t  γφ  x t  − Ap    I − tA   Tx t − p    ≤ t   γφ  x t  − Ap     1 − t γ     x t − p    ≤ t   γφ  x t  − γφ  p       γφ  p  − Ap     1 − t γ    x t − p   ≤ tγr 1   x t − p    t   γφ  p  − Ap     1 − t γ    x t − p   , 2.10 therefore, x t − p≤γφp − Ap/γ − γr 1 . This implies the {x t } is bounded. To prove that x t → x  x ∈ FT as t → 0. Since {x t } is bounded and E is reflexive, there exists a subsequence {x t n } of {x t } such that x t n x ∗ .Byx t − Tx t  tγφx t  − ATx t . We have x t n − Tx t n → 0, as t n → 0. Since E satisfies Opial’s condition, it follows from Lemma 2.3 that x ∗ ∈ FT. We claim  x t n − x ∗  −→ 0. 2.11 By contradiction, there is a number ε 0 and a subsequence {x t m } of {x t n } such that x t m − x ∗ ≥ ε 0 .FromLemma 2.8, there is a number r ε 0 > 0 such that φx t m  − φx ∗ ≤r ε 0 x t m − x ∗ ,we write x t m − x ∗  t m  γφ  x t m  − Ax ∗    I − t m A  Tx t m − x ∗  , 2.12 to derive that  x t m − x ∗  q  t m γφ  x t m  − Ax ∗ ,J q  x t m − x ∗      I − t m A  Tx t m − x ∗  ,J q  x t m − x ∗   ≤ t m  γφ  x t m  − Ax ∗ ,J q  x t m − x ∗     1 − t m γ   x t m − x ∗  q . 2.13 It follows that  x t m − x ∗  q ≤ 1 γ  γφ  x t m  − Ax ∗ ,J q  x t m − x ∗    1 γ  γφ  x t m  − γφ  x ∗  ,J q  x t m − x ∗     γφ  x ∗  − Ax ∗ ,J q  x t m − x ∗   ≤ 1 γ  γr ε 0  x t m − x ∗  q   γφ  x ∗  − Ax ∗ ,J q  x t m − x ∗   . 2.14 Therefore,  x t m − x ∗  q ≤  γφ  x ∗  − Ax ∗ ,J q  x t m − x ∗   γ − γr ε 0 . 2.15 8 Fixed Point Theory and Applications Using that the duality map J q is single valued and weakly sequentially continuous from E to E ∗ ,by2.15,wegetthatx t m → x ∗ . It is a contradiction. Hence, we have x t n → x ∗ . We next prove that x ∗ solves the variational inequality 2.3. Since x t  tγφ  x t    I − tA  Tx t , 2.16 we derive that  A − γφ  x t  − 1 t  I − tA  I − T  x t . 2.17 Notice   I − T  x t −  I − T  z, J q  x t − z   ≥  x t − z  q −  Tx t − Tz  x t − z  q−1 ≥  x t − z  q −  x t − z  q  0. 2.18 It follows that, for z ∈ FT,  A − γφ  x t ,J q  x t − z    − 1 t   I − tA  I − T  x t ,J q  x t − z    − 1 t   I − T  x t −  I − T  z, J q  x t − z     A  I − T  x t ,J q  x t − z   ≤  A  I − T  x t ,J q  x t − z   . 2.19 Now replacing t in 2.19 with t n and letting n →∞, noticing I−Tx t n → I−Tx ∗  0 for x ∗ ∈ FT,weobtainA − γφx ∗ ,J q x ∗ − z≤0. That is, x ∗ ∈ FT is a solution of 2.3; Hence x  x ∗ by uniqueness. In a summary, we have shown that each cluster point of {x t } at t → 0 equals x, therefore, x t → x as t → 0. Lemma 2.10 see, e.g., Mitrinovi ´ c 19, page 63. Let q>1. Then the following inequality holds: ab ≤ 1 q a q  q − 1 q b q/q−1 , 2.20 for arbitrary positive real numbers a, b. Lemma 2.11. Let E be a q-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping J q from E to E ∗ and C be a nonempty convex subset of E. Assume that T i : C → E is a countable family of λ i -strict pseudocontraction for some 0 <λ i < 1 and inf{λ i : i ∈ N} > 0 such that F   ∞ i1 FT i  /  ∅. Assume that {η i } ∞ i1 is a positive sequence such that  ∞ i1 η i  1.Then  ∞ i1 η i T i : C → E is a λ-strict pseudocontraction with λ  inf{λ i : i ∈ N} and F  ∞ i1 η i T i F. Fixed Point Theory and Applications 9 Proof. Let G n x  η 1 T 1 x  η 2 T 2 x  ··· η n T n x 2.21 and  n i1 η i  1. Then, G n : C → E is a λ i -strict pseudocontraction with λ  min{λ i :1≤ i ≤ n}. Indeed, we can firstly see the case of n  2.   I − G 2  x −  I − G 2  y, J q  x − y   η 1  I − T 1  x  η 2  I − T 2  x − η 1  I − T 1  y − η 2  I − T 2  y, J q  x − y    η 1   I − T 1  x −  I − T 1  y, J q  x − y   η 2   I − T 2  x −  I − T 2  y, J q  x − y  ≥ η 1 λ 1   I − T 1 x − I − T 1 y   q  η 2 λ 2   I − T 2 x − I − T 2 y   q ≥ λ  η 1    I − T 1  x −  I − T 1  y   q  η 2    I − T 2  x −  I − T 2  y   q  ≥ λ   I − G 2 x − I − G 2 y   q , 2.22 which shows that G 2 : C → E is a λ-strict pseudocontraction with λ  min{λ i : i  1, 2}.By the same way, our proof method easily carries over to the general finite case. Next, we prove the infinite case. From the definition of λ-strict pseudocontraction, we know   I − T n  x −  I − T n  y, J q  x − y  ≥ λ   I − T n x − I − T n y   q . 2.23 Hence, we can get    I − T n  x −  I − T n  y   ≤  1 λ  1/q−1   x − y   . 2.24 Taking p ∈ FT n ,from2.24, we have   I − T n  x      I − T n  x −  I − T n  p   ≤  1 λ  1/q−1   x − p   . 2.25 Consquently, for all x ∈ E,ifF   ∞ i1 FT i  /  ∅, η i > 0 i ∈ N and  ∞ i1 η i  1, then  ∞ i1 η i T i strongly converges. Let Tx  ∞  i1 η i T i x, 2.26 we have Tx  ∞  i1 η i T i x  lim n →∞ n  i1 η i T i x  lim n →∞ 1  n i1 η i n  i1 η i T i x. 2.27 10 Fixed Point Theory and Applications Hence,   I − T  x −  I − T  y, J q  x − y   lim n →∞  I − 1  n i1 η i n  i1 η i T i  x   I − 1  n i1 η i n  i1 η i T i  y, J q  x − y    lim n →∞ 1  n i1 η i n  i1 η i   I − T i  x −  I − T i  y, J q  x − y  ≥ lim n →∞ 1  n i1 η i n  i1 η i λ   I − T i x − I − T i y   q ≥ λ lim n →∞       I − 1  n i1 η i n  i1 η i T i  x −  I − 1  n i1 η i n  i1 η i T i  y      q  λ   I − Tx − I − Ty   q . 2.28 So, we get T is λ-strict pseudocontraction. Finally, we show F  ∞ i1 η i T i F. Suppose that x   ∞ i1 η i T i x,itissufficient to show that x ∈ F. Indeed, for p ∈ F, we have   x − p   q   x − p, J q  x − p    ∞  i1 η i T i x − p, J q  x − p    ∞  i1 η i  T i x − p, J q  x − p  ≤   x − p   q − λ ∞  i1 η i  x − T i x  q , 2.29 where λ  inf{λ i : i ∈ N}. Hence, x  T i x for each i ∈ N, this means that x ∈ F. 3. Main Results Lemma 3.1. Let E be a real q-uniformly smooth, strictly convex Banach space and C be a closed convex subset of E such that C ± C ⊂ C.LetC be also a sunny nonexpansive retraction of E.Let φ : C → C be a MKC. Let A : C → C be a strongly positive linear bounded operator with the coefficient γ>0 such that 0 <γ<γ and T i : C → E be λ i -strictly pseudo-contractive non-self- mapping such that F   ∞ i1 FT i  /  ∅.Letλ  inf {λ i : i ∈ N} > 0.Let{x n } be a sequence of C generated by 1.12 with the sequences {α n },{β n } and {γ n } in 0, 1, assume for each n, {η n i } be an infinity sequence of positive number such that  ∞ i1 η n i  1 for all n and η n i > 0. 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General Iterative Process with Meir-Keeler Contractions for a Countable Family of λ i -Strict Pseudocontractions in q-Uniformly Smooth Banach Spaces Yanlai Song and Changsong Hu Department of Mathematics,. G. Cai and C. S. Hu, Strong convergence theorems of a general iterative process for a finite family of λ i -strict pseudo -contractions in q-uniformly smooth Banach spaces,” Computers & Mathematics. 2008. Fixed Point Theory and Applications 19 9 X. Qin and Y. Su, “Approximation of a zero point of accretive operator in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 329,