Iterative Schemes for Fixed Point Computation of Nonexpansive Mappings

and Applied Analysis 3 Moreover, PC is characterized by the following properties: 〈 x − PCx, y − PCx 〉 ≤ 0, ∥ ∥x − y∥∥2 ≥ ‖x − PCx‖ ∥ ∥y − PCx ∥ ∥ 2 , 2.4 for all x ∈ H and y ∈ C. We also need other sorts of nonlinear operators which are introduced below. Let T : H → H be a nonlinear operator. a T is nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ H. b T is firmly nonexpansive if 2T − I is nonexpansive. Equivalently, T I S /2, where S : H → H is nonexpansive. Alternatively, T is firmly nonexpansive if and only if ∥ ∥Tx − Ty∥∥2 ≤ 〈Tx − Ty, x − y〉, x, y ∈ H. 2.5 c T is averaged if T 1− τ I τS, where τ ∈ 0, 1 and S : H → H is nonexpansive. In this case, we also say that T is τ-averaged. A firmly nonexpansive mapping is 1/2-averaged. It is well known that both PC and I − PC are firmly nonexpansive. We will need to use the following notation: i Fix T stands for the set of fixed points of T ; ii xn ⇀ x stands for the weak convergence of {xn} to x; iii xn → x stands for the strong convergence of {xn} to x. Let T1, T2, . . . be infinite mappings of C into itself, and let ξ1, ξ2, . . . be real numbers such that 0 ≤ ξi ≤ 1 for every i ∈ N. For any n ∈ N, define a mappingWn of C into itself as follows: Un,n 1 I, Un,n ξnTnUn,n 1 1 − ξn I, Un,n−1 ξn−1Tn−1Un,n 1 − ξn−1 I, .. Un,k ξkTkUn,k 1 1 − ξk I, Un,k−1 ξk−1Tk−1Un,k 1 − ξk−1 I, .. Un,2 ξ2T2Un,3 1 − ξ2 I, Wn Un,1 ξ1T1Un,2 1 − ξ1 I. 2.6 4 Abstract and Applied Analysis Such Wn is called the W-mapping generated by Tn, Tn−1, . . . , T2, T1 and ξn, ξn−1, . . . , ξ2, ξ1. For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to 21 . We have the following crucial lemmas concerning Wn which can be found in 22 . Lemma 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1, T2, . . . be nonexpansive mappings of C into itself such that ⋂∞ n 1 F Tn is nonempty, and let ξ1, ξ2, . . . be real numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then, for every x ∈ C and k ∈ N, the limit limn→∞Un,kx exists. Lemma 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1, T2, . . . be nonexpansive mappings of C into itself such that ⋂∞ n 1 F Tn is nonempty, and let ξ1, ξ2, . . . be real numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then, F W ⋂∞ n 1 F Tn . The following remark 23 is important to prove our main results. Remark 2.3. Using Lemma 2.1, one can define a mapping W of C into itself as Wx limn→∞Wnx limn→∞Un,1x, for every x ∈ C. If {xn} is a bounded sequence in C, then one has lim n→∞ ‖Wxn −Wnxn‖ 0. 2.7 Throughout this paper, we will assume that 0 < ξi ≤ b < 1 for every i ∈ N. Lemma 2.4 see 24 . Let K be a nonempty closed convex subset of a real Hilbert space H. Let T : K → K be a nonexpansive mapping with Fix T / ∅. Then T is demiclosed on K, that is, if xn ⇀ x ∈ K weakly and xn − Txn → 0, then x Tx. Lemma 2.5 see 25 . Let {xn} and {zn} be bounded sequences in a Banach spaceX, and let {βn} be a sequence in 0, 1 with 0 < lim infn→∞βn ≤ lim supn→∞βn < 1. Suppose xn 1 1−βn zn βnxn for all integers n ≥ 0 and lim supn→∞ ‖zn 1 − zn‖ − ‖xn 1 −xn‖ ≤ 0. Then, limn→∞‖zn −xn‖ 0. Lemma 2.6 see 26 . Assume {an} is a sequence of nonnegative real numbers such that an 1 ≤ 1 − γn an δn where {γn} is a sequence in 0, 1 and {δn} is a sequence such that 1 ∑∞ n 1 γn ∞; 2 lim supn→∞δn/γn ≤ 0 or ∑∞ n 1 |δn| < ∞. Then limn→∞an 0. 3. Main Result In this section, we introduce our algorithm and prove its strong convergence. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n 1 be a sequence of nonexpansive mappings from C to C such that the common fixed point set F : ⋂∞ n 1 F Tn / ∅. Let f : C → H be a κ-contraction and B : H → H be a self-adjoint, strongly Abstract and Applied Analysis 5 positive bounded linear operator with coefficient α > 0. Let σ be a constant such that 0 < σκ < α. For an arbitrary initial point x0 belonging to C, one defines a sequence {xn}n≥0 iterativelyand Applied Analysis 5 positive bounded linear operator with coefficient α > 0. Let σ be a constant such that 0 < σκ < α. For an arbitrary initial point x0 belonging to C, one defines a sequence {xn}n≥0 iteratively xn 1 PC [ αnσf xn I − αnB Wnxn ] , ∀n ≥ 0, 3.1 where {αn} is a real sequence in 0, 1 . Assume the sequence {αn} satisfies the following conditions: C1 limn→∞αn 0; C2 ∑∞ n 0 αn ∞. Then the sequence {xn} generated by 3.1 converges in norm to the unique solution x∗ which solves the following variational inequality: x∗ ∈ F such that 〈σf x∗ − Bx∗, x̃ − x∗〉 ≤ 0, ∀x̃ ∈ F. 3.2 Proof. Let x̃ ∈ F. From 3.1 , we have ‖xn 1 − x̃‖ ∥ PC [ αnσf xn I − αnB Wnxn ] − x̃∥∥ ≤ ∥∥αnσf xn I − αnB Wnxn − x̃ ∥ ∥ ≤ αnσ ∥ ∥f xn − f x̃ ∥ ∥ ‖I − αnB‖‖Wnxn − x̃‖ αn ∥ ∥σf x̃ − Bx̃∥∥ ≤ αnσκ‖xn − x̃‖ 1 − αnα ‖xn − x̃‖ αn ∥ ∥σf x̃ − Bx̃∥∥ 1 − α − σκ αn ‖xn − x̃‖ α − σκ αn ∥ ∥f x̃ − Bx̃∥∥ α − σκ . 3.3 It follows by induction that ‖xn 1 − x̃‖ ≤ max { ‖xn − x̃‖, ∥ ∥f x̃ − Bx̃∥∥ α − σκ } ≤ max { ‖x0 − x̃‖, ∥ ∥f x̃ − Bx̃∥∥ α − σκ }