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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007, Article ID 28619, 8 pages doi:10.1155/2007/28619 Research Article An Iteration Method for Nonexpansive Mappings in Hilbert Spaces Lin Wang Received 22 August 2006; Revised 2 November 2006; Accepted 2 November 2006 Recommended by Nan-Jing Huang In real Hilbert space H, from an arbitrary initial point x 0 ∈ H, an explicit iteration scheme is defined as follows: x n+1 = α n x n +(1− α n )T λ n+1 x n ,n ≥ 0, where T λ n+1 x n = Tx n − λ n+1 μF(Tx n ), T : H → H is a nonexpansive mapping such that F(T) ={x ∈ K : Tx= x} is nonempty, F : H → H is a η-strongly monotone and k-Lipschitzian mapping, {α n }⊂ (0,1), and {λ n }⊂[0,1). Under some suitable conditions, the sequence {x n } is shown to converge strongly to a fixed point of T and the necessary and sufficient conditions that {x n } converges strongly to a fixed point of T are obtained. Copyright © 2007 Lin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro- duction in a ny medium, provided the original work is properly cited. 1. Introduction Let H be a Hilbert space with inner product ·,· and norm ·.AmappingT : H → H is said to be nonexpansive if Tx− Ty≤x − y for any x, y ∈ H.AmappingF : H → H is said to be η-strongly monotone i f there exists constant η>0suchthat Fx− Fy,x − y≥ ηx − y 2 for any x, y ∈ H. F : H → H is said to be k-Lipschitzian if there exists constant k>0suchthat Fx− Fy≤kx − y for any x, y ∈ H. The interest and importance of construction of fixed points of nonexpansive map- pings stem mainly from the fact that it may be applied in many areas, such as imagine recover y and signal processing (see, e.g., [1–3]). Iterative techniques for approximat- ing fixed points of nonexpansive mappings have been studied by various authors (see, e.g., [1, 4–10], etc.), using famous Mann iteration method, Ishikawa iteration method, and many other iteration methods such as, viscosity approximation method [6]andCQ method [7]. Let F : H → H be a nonlinear mapping and K nonempty closed convex subset of H. The variational inequality problem is formulated as finding a point u ∗ ∈ K such 2 Fixed Point Theory and Applications that  VI(F,K)  F  u ∗  ,v − u ∗  ≥ 0, ∀v ∈ K. (1.1) The variational inequalities were initially studied by Kinderlehrer and Stampacchia [11], and ever since have been widely studied. It is well known that the VI(F,K)isequivalent to the fixed point equation u ∗ = P K  u ∗ − μF  u ∗  , (1.2) where P K is the projection from H onto K and μ is an arbitrarily fixed constant. In fact, when F is an η-strongly monotone and Lipschitzian mapping on K and μ>0small enough, then the mapping defined by the right-hand side of (1.2) is a contraction. For reducing the complexity of computation caused by the projection P K ,Yamada[12] proposed an iteration method to solve the variational inequalities VI(F,K). For arbitrary u 0 ∈ H, u n+1 = Tu n − λ n+1 μF  T  u n  , n ≥ 0, (1.3) where T is a nonexpansive mapping from H into itself, K is the fixed point set of T, F is an η-strongly monotone and k-Lipschitzian mapping on K, {λ n } is a real sequence in [0,1), and 0 <μ<2η/k 2 .ThenYamada[12]provedthat{u n } converges strongly to the unique solution of the VI(F,K)as {λ n } satisfies the following conditions: (1) lim n→∞ λ n = 0, (2)  ∞ n=0 λ n =∞, (3) lim n→∞ (λ n − λ n+1 )/λ 2 n+1 = 0. Motivated by the above work, we propose a new explicit iteration scheme with map- ping F to approximate the fixed point of nonexpansive mapping T in Hilbert space. The strong and weak convergence theorems to a fixed point of T are obtained. The necessary and sufficient conditions for strong convergence of this iteration scheme a re obtained, too. 2. Preliminaries Let T be a nonexpansive mapping from H into itself, F : H → H an η-strongly mono- tone and k-Lipschitzian mapping, {λ n }⊂(0,1), {λ n }⊂[0,1), and μ afixedconstantin (0,2η/k 2 ). Starting with an initial point x 0 ∈ H, the explicit iteration scheme with map- ping F is defined as follows: x n+1 = α n x n +  1 − α n  Tx n − λ n+1 μF  Tx n  , n ≥ 0. (2.1) For simplicity, we define a mapping T λ : H → H by T λ x = Tx− λμF(Tx), ∀x ∈ H. (2.2) Then (2.1)maybewrittenasfollows: x n+1 = α n x n +  1 − α n  T λ n+1 x n , n ≥ 0. (2.3) Lin Wang 3 In fact, as λ n = 0, n ≥ 1, then the iteration scheme (2.3) reduces to the famous Mann iteration scheme. ABanachspaceE is said to satisfy Opial’s condition if for any sequence {x n } in E, x n  x implies that limsup n→∞ x n − x < limsup n→∞ x n − y for all y ∈ E with y = x, where x n  x denotes that {x n } converges weakly to x. It is well known that e very Hilbert space satisfies Opial’s condition. AmappingT : K → E is said to be semicompact if, for any sequence {x n } in K such that x n − Tx n →0(n →∞), there exists subsequence {x n j } of {x n } such that {x n j } converges strongly to x ∗ ∈ K. AmappingT with domain D(T) and range R(T)inE is said to be demiclosed at p;if whenever {x n } is a sequence in D(T)suchthat{x n } converges weakly to x ∗ ∈ D(T)and {Tx n } converges strongly to p,thenTx ∗ = p. Lemma 2.1 [13]. Let {α n } and {t n } be two nonnegative sequences satisfying α n+1 ≤  1+a n  α n + b n , ∀n ≥ 1. (2.4) If  ∞ n=1 a n < ∞ and  ∞ n=1 b n < ∞, then lim n→∞ α n exists. Lemma 2.2 [12]. Let T λ x = Tx − λμF(Tx),whereT : H → H is a nonexpansive mapping from H into itself and F is an η-strongly monotone and k-Lipschitzian mapping from H into itself. If 0 ≤ λ<1 and 0 <μ<2η/k 2 , then T λ is a contraction and satisfies   T λ x − T λ y   ≤ (1 − λτ)x − y, ∀x, y ∈ H, (2.5) where τ = 1 −  1 − μ(2η − μk 2 ). Lemma 2.3 [14]. Let K beanonemptyclosedconvexsubsetofarealHilbertspaceH and T a nonexpansive mapping from K into itself. If T has a fixed point, then I − T is demiclosed at zero, where I is the identity mapping of H,thatis,whenever {x n } isasequenceinK weakly converging to some x ∈ K and the sequence {(I − T)x n } strongly converges to some y, it follows that (I − T)x = y. 3. Main results Lemma 3.1. Let H be a Hilbert space, T : H → H a nonexpansive mapping with F(T) = φ, and F : H → H an η-strongly monotone and k-Lipschitzian mapping. For any given x 0 ∈ H, {x n } is defined by x n+1 = α n x n +  1 − α n  T λ n+1 x n , n ≥ 0, (3.1) where {α n } and {λ n }⊂[0,1) satisfy the following conditions: (1) α ≤ α n ≤ β for some α,β ∈ (0,1); (2)  ∞ n=1 λ n < ∞; (3) 0 <μ<2η/k 2 . Then, (1) lim n→∞ x n − q ex ists for each q ∈ F(T); (2) lim n→∞ x n − Tx n =0. 4 Fixed Point Theory and Applications Proof. (1) For any q ∈ F(T), we have   x n+1 − q   2 =   α n  x n − q  +  1 − α n  T λ n+1 x n − q    2 = α n   x n − q   2 +  1 − α n    T λ n+1 x n − q 2 − α n  1 − α n    x n − T λ n+1 x n   2 , (3.2) where (by Lemma 2.2)   T λ n+1 x n − q   =   T λ n+1 x n − T λ n+1 q + T λ n+1 q − q   ≤   T λ n+1 x n − T λ n+1 q   +   T λ n+1 q − q   ≤  1 − λ n+1 τ    x n − q   + λ n+1 μ   F(q)   . (3.3) Furthermore,   T λ n+1 x n − q   2 ≤  1 − λ n+1 τ    x n − q   2 + λ n+1 μ 2 τ   F(q)   2 . (3.4) Thus,   x n+1 − q   2 ≤ α n   x n − q   2 +  1 − α n  1 − λ n+1 τ    x n − q   2 +  1 − α n  λ n+1 μ 2 τ   F(q)   2 − α n  1 − α n    x n − T λ n+1 x n   2 ≤ α n   x n − q   2 +  1 − α n  1 − λ n+1 τ    x n − q   2 +  1 − α n  λ n+1 μ 2 τ   F(q)   2 − α n   x n+1 − x n   2 ≤   x n − q   2 + λ n+1 μ 2 τ   F(q)   2 − α n   x n+1 − x n   2 . (3.5) Since  ∞ n=1 λ n < ∞,itfollowsfromLemma 2.1 that lim n→∞ x n − q exists for each q ∈ F(T). It also implies that {x n } is bounded. (2) From (3.5), we have α   x n+1 − x n   2 ≤ α n   x n+1 − x n   2 ≤   x n − q   2 −   x n+1 − q   2 + λ n+1 μ 2 τ   F(q)   2 . (3.6) Therefore, lim n→∞ x n+1 − x n =0. In addition, (1 − β)   x n − T λ n+1 x n   ≤  1 − α n    x n − T λ n+1 x n   =   x n+1 − x n   . (3.7) Hence, lim n→∞ x n − T λ n+1 =0. Thus,   x n − Tx n   =   x n − T λ n+1 x n + T λ n+1 x n − Tx n   ≤   x n − T λ n+1 x n   + λ n+1 μ   F  Tx n    . (3.8) Since {x n } is bounded, then {Tx n } and {F(Tx n )} are bounded, as well. Therefore, lim n→∞ x n − Tx n =0. The proof is completed.  Lin Wang 5 Theorem 3.2. Let H be a Hilbert space, T : H → H a nonexpansive mapping with F(T) = φ,andF : H → H an η-strongly monotone and k-Lipschitzian mapping. For any given x 0 ∈ H, {x n } is defined by x n+1 = α n x n +  1 − α n  T λ n+1 x n , n ≥ 0, (3.9) where {α n } and {λ n }⊂[0,1) satisfy the following conditions: (1) α ≤ α n ≤ β for some α,β ∈ (0,1); (2)  ∞ n=1 λ n < ∞; (3) 0 <μ<2η/k 2 . Then, (1) {x n } converges w eakly to a fixed point of T; (2) {x n } converges strong ly to a fixed point of T if and only if liminf n→∞ d(x n ,F(T)) = 0. Proof. (1) It follows from Lemma 3.1 that {x n } is bounded. Thus, let q 1 and q 2 be weak limits of subsequences {x n k } and {x n j } of {x n }, respectively. It follows from Lemmas 2.3 and 3.1 that q 1 ,q 2 ∈ F(T). Assume q 1 = q 2 , then by Opial’s condition, we obtain lim n→∞   x n − q 1   = lim k→∞   x n k − q 1   < lim k→∞   x n k − q 2   = lim j→∞   x n j − q 2   < lim k→∞   x n k − q 1   = lim n→∞   x n − q 1   , (3.10) which is a contradiction; hence, q 1 = q 2 .Then,{x n } converges weakly to a common fix ed point of T. (2) Suppose that {x n } converges strongly to a fix ed point q of T, then lim n→∞ x n − q=0. Since 0 ≤ d(x n ,F(T)) ≤x n − q, we have liminf n→∞ d(x n ,F(T)) = 0. Conversely, suppose that liminf n→∞ d(x n ,F(T)) = 0. For any p ∈ F(T), F(p)≤  F(p) − F(x n ) + F(x n )≤kx n − p + F(x n ).Since{x n } and {F(x n )} are bounded, F(p) is bounded for any p ∈ F(T), that is, there exists constant M>0 such that F(p)≤M for all p ∈ F(T). In addition, it follows from (3.5)that   x n+1 − p   2 ≤   x n − p   2 + λ n+1 μ 2 τ   F(p)   2 . (3.11) So,   x n+1 − p   2 ≤   x n − p   2 + λ n+1 μ 2 τ  2k 2   x n − p   2 +2   F  x n    2  =  1+2k 2 λ n+1 μ 2 τ    x n − p   2 +2 λ n+1 μ 2 τ   F  x n    2 . (3.12) Thus,  d  x n+1 ,F(T)  2 ≤  1+2k 2 λ n+1 μ 2 τ   d  x n ,F(T)  2 +2 λ n+1 μ 2 τ   F  x n    2 . (3.13) In addition, we obtain that  ∞ n=1 2k 2 (λ n+1 μ 2 /τ) < ∞ and  ∞ n=1 2(λ n+1 μ 2 /τ)F(x n ) 2 < ∞ since  ∞ n=1 λ n < ∞ and {F(x n )} is bounded. It follows from Lemma 2.1 that 6 Fixed Point Theory and Applications lim n→∞ d(x n ,F(T)) exists. Furthermore, since liminf n→∞ d(x n ,F(T)) = 0, we have lim n→∞ d(x n ,F(T)) = 0. We now prove that {x n } is a Cauchy sequence. Tak ing M 1 = max{2e (2μ 2 k 2 /τ)  ∞ i=1 λ i ,4(μ 2 M 2 /τ)e (2μ 2 k 2 /τ)  ∞ i=1 λ i },forany > 0, there exists positive integer N such that d(x n ,F(T)) <  /4M 1 and  ∞ i=n λ i < /4M 1 as n ≥ N.Taking q ∈ F(T), for any n,m ≥ N,itfollowsfrom(3.12)that   x n − x m   2 2 ≤   x n − q   2 +   x m − q   2 ≤  1+2k 2 λ n μ 2 τ    x n−1 − q   2 +2 λ n μ 2 τ   F  x n−1    2 +  1+2k 2 λ m μ 2 τ    x m−1 − q   2 +2 λ m μ 2 τ   F  x m−1    2 ≤  1+2k 2 λ n μ 2 τ    x n−1 − q   2 +2 λ n μ 2 τ M 2 +  1+2k 2 λ m μ 2 τ    x m−1 − q   2 +2 λ m μ 2 τ M 2 ≤ n  i=N+1  1+2k 2 λ i μ 2 τ    x N − q   2 + n−1  i=N+1 2 λ i μ 2 τ M 2 n  j=i+1  1+2k 2 λ j μ 2 τ  +2 λ n μ 2 τ M 2 + m  i=N+1  1+2k 2 λ i μ 2 τ    x N − q   2 + m−1  i=N+1 2 λ i μ 2 τ M 2 m  j=i+1  1+2k 2 λ j μ 2 τ  +2 λ m μ 2 τ M 2 ≤ 2e (2μ 2 k 2 /τ)  ∞ i=N+1 λ i   x N − q   2 +4 μ 2 M 2 τ e (2μ 2 k 2 /τ)  ∞ i=N+1 λ i ∞  i=N+1 λ i . (3.14) Thus,   x n − x m   2 ≤ 2M 1   x N − q   2 +2M 1 ∞  i=N+1 λ i . (3.15) Taking the infimum for all q ∈ F(T), we have   x n − x m   2 ≤ 2M 1  d  x N ,F(T)  2 +2M 1 ∞  i=N+1 λ i < . (3.16) This implies that {x n } is a Cauchy sequence. Therefore, there exists p ∈ H such that {x n } converges strongly to p.ItfollowsfromLemma 3.1 that p − Tp≤   p − x n   +   x n − Tx n   −→ 0, as n −→ ∞ . (3.17) Hence, p ∈ F(T). The proof is completed.  Lin Wang 7 Corollary 3.3. Under the conditions of Lemma 3.1 ,ifT is completely continuous, the n {x n } convergesstronglytoafixedpointofT. Proof. By Lemma 3.1, {x n } is bounded and lim n→∞ x n − Tx n =0, then {Tx n } is also bounded. Since T is completely continuous, there exists subsequence {Tx n j } of {Tx n } such that Tx n j → p as j →∞.ItfollowsfromLemma 3.1 that lim j→∞ x n j − Tx n j =0. So by the continuity of T and Lemma 2.3,wehavelim j→∞ x n j − p=0andp ∈ F(T). Furthermore, by Lemma 3.1,wegetthatlim n→∞ x n − p exists. Thus, lim n→∞ x n − p= 0. The proof is completed.  Corollary 3.4. Under the conditions of Lemma 3.1,ifT is demicompact, then {x n } con- verges strongly to a fixed point of T. Proof. Since T is demicompact, {x n } is bounded and lim n→∞ x n − Tx n =0, then there exists subsequence {x n j } of {x n } such that {x n j } converges strongly to q ∈ H.Itfollows from Lemma 2.3 that q ∈ F(T). Thus, lim n→∞ x n − q exists by Lemma 3.1.Sincethe subsequence {x n j } of {x n } such that {x n j } converges strongly to q,then{x n } converges strongly to the common fixed point q ∈ F(T). The proof is completed.  For studying the strong convergence of fixed points of a nonexpansive mapping, Sen- ter and Dotson [9] introduced Condition (A). Later on, Maiti and Ghosh [5]wellasTan and Xu [10] studied Condition (A) and pointed out that Condition (A)isweakerthan the requirement of demicompactness for nonexpansive mappings. A mapping T : K → K with F(T) ={x ∈ K : Tx = x} = φ is said to satisfy condition (A)ifthereexistsanon- decreasing function f :[0, ∞) → [0,∞)with f (0) = 0and f (t) > 0forallt ∈ (0, ∞)such that x − Tx≥ f (d(x,F(T))) for all x ∈ K,whered(x,F(T)) = inf{x − q : q ∈ F(T)}. Theorem 3.5. Under the conditions of Lemma 3.1,ifT satisfies condition (A), then {x n } convergesstronglytoafixedpointofT. Proof. Since T satisfies condition (A), then f (d(x n ,F(T))) ≤x n − Tx n .Itfollowsfrom Lemma 3.1 that liminf n→∞ d(x n ,F(T)) = 0. Thus, it follows from Theorem 3.2 that {x n } convergesstronglytoafixedpointofT.Theproofiscompleted.  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, no. 2, pp. 197–228, 1967. [2] C. 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