THÔNG TIN TÀI LIỆU
A NEW SYSTEM OF GENERALIZED NONLINEAR RELAXED COCOERCIVE VARIATIONAL INEQUALITIES KE DING, WEN-YONG YAN, AND NAN-JING HUANG Received 21 November 2004; Revised 13 April 2005; Accepted 28 June 2005 We introduce and study a new system of generalized nonlinear relaxed cocoercive in- equality problems and construct an iterative algorithm for approximating the solutions of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces. We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality problems and the convergence of iterative sequences generated by thealgorithm.Wealsostudytheconvergenceandstabilityofanewperturbediterative algorithm for approximating the solution. Copyright © 2006 Ke Ding et al. This is an open access article distributed under the Cre- ative Commons Att ribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Variational inequality problems have various applications in mechanics and physics, opti- mization and control, linear and nonlinear programming, economics and finance, trans- portation equilibrium and eng ineering science, and so forth. Consequently considerable attention has been devoted to the study of the theory and efficient numerical methods for variational inequality problems (see, e.g., [2–17] and the references therein). In [15], Verma introduced a new system of nonlinear strongly monotone variational inequalities and studied the approximate of this system based on the projection method, and in [16], Verma discussed the approximate solvability of a system of nonlinear relaxed cocoercive variational inequalities in Hilbert spaces. Recently, Kim and Kim [14] introduced and studied a system of nonlinear mixed variational inequalities in Hilbert spaces, and ob- tained some approximate solvability results. In the recent paper [6], Cho et al. introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities. They also constructed an iterative algorithm for approximating the solution of the system of nonlinear variational inequalities. Some related works, we refer to [2, 3, 5, 7–10, 12, 13]. Motivated and inspired by these works, in this paper, we introduce and study a new system of generalized nonlinear relaxed cocoercive variational Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 40591, Pages 1–14 DOI 10.1155/JIA/2006/40591 2 Nonlinear relaxed cocoercive variational inequalities inequality problems and construct an iterative algorithm for approximating the solutions of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces. We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality problems and the convergence of iterative sequences generated by thealgorithm.Wealsostudytheconvergenceandstabilityofanewperturbediterative algorithm for approximating the solution. The results presented in this paper improve and extend the previously known results in this area. 2. Preliminaries Let H be a Hilbert space endowed with a norm ·and inner product (·,·), respectively. Let CB(H) be the family of all nonempty subsets of H and K 1 , K 2 be two convex and closed subsets of H.Letg 1 , g 2 , m 1 , m 2 : H → H and F, G : H × H → H be mappings. We consider the following system of generalized nonlinear variational inequality problems: find x, y ∈ H such that g i (x) ∈ K i (x)fori = 1, 2, and F(x, y),z − g 1 (x) ≥ 0, ∀z ∈ K 1 (x), G(x, y),z − g 2 (y) ≥ 0, ∀z ∈ K 2 (y), (2.1) where K i (x) = m i (x)+K i for i = 1, 2. When K 1 and K 2 are both convex cones of H, it is easy to see that problem (2.1)is equivalent to the following system of generalized nonlinear co-complementarity prob- lems: find x, y ∈ H such that g i (x) ∈ K i (x)fori = 1, 2, and F(x, y) ∈ K 1 (x) − g 1 (x) ∗ , G(x, y) ∈ K 2 (y) − g 2 (y) ∗ , (2.2) where K i (x) = m i (x)+K i and (K i (x) − g i (x)) ∗ is the dual of K i (x) − g i (x)fori = 1, 2, that is, K i (x) − g i (x) ∗ = u ∈ H | (u,v) ≥ 0, ∀v ∈ K i (x) − g i (x) . (2.3) Some examples of problems (2.1)and(2.2)areasfollows. (I) If G = 0andF(x, y) = Tx +Ax for all x, y ∈ X,whereT, A : H → H are two map- pings, then problem (2.2) reduces to finding x ∈ H such that Tx+ Ax ∈ K 1 (x) − g 1 (x) ∗ , (2.4) which is called the generalized complementarity problem. The problem (2.4)wasex- tended and studied by Jou and Yao [11] in Hilbert spaces, and by Chen et al. [5]inthe setting of Banach spaces. (II) Let T : H × H → H be a mapping. If F(x, y) = ρT(y,x)+x − y, G(x, y) = ηT(x, y) + y − x for all x, y ∈ H, m 1 = m 2 = 0, K 1 = K 2 = K,andg 1 = g 2 = I,whereI is an identity Ke Ding et al. 3 mapping and ρ>0, η>0, then problem (2.1) reduces to finding x, y ∈ K such that ρT(y, x)+x − y,z − x ≥ 0, ∀z ∈ K, ηT(x, y)+y − x,z − y ≥ 0, ∀z ∈ K, (2.5) which is called the system of nonlinear variational inequality problems considered by Ver m a [ 16]. The special case of problem (2.5) was studied by Verma [15]. The problem (2.5) was extended and studied by Agarwal et al. [1], Kim and Kim [14], and Cho et al. [6]. (III) If m 1 = m 2 = 0, and g 1 = g 2 = I,thenproblem(2.1) reduces to finding x ∈ K 1 and y ∈ K 2 such that F(x, y),z − x ≥ 0, ∀z ∈ K 1 , G(x, y),z − y ≥ 0, ∀z ∈ K 2 , (2.6) which is just the problem considered in [12]withF, G being single-valued mappings. Definit ion 2.1. AmappingN : H × H → H is said to be (i) α-strongly monotone with respect to first argument if there exists some α>0such that N(x,·) − N(y,·),x − y ≥ αx − y 2 , ∀(x, y) ∈ H × H; (2.7) (ii) ξ-Lipschitz continuous with respect to the first argument, if there exists a constant ξ>0suchthat N(x,·) − N(y,·) ≤ ξ x − y , ∀(x, y) ∈ H × H. (2.8) Similarly, we can define the strong monotonicity and Lipschitzian continuity with re- spect to the second argument of N. Definit ion 2.2. AMappingN : H × H → H is said to be relaxed (a,b)-cocoercive with respect to the first argument if there exists constants a>0andb>0suchthat N(x,·) − N(y,·),x − y ≥ (−a)x − y 2 + bx − y 2 , ∀(x, y) ∈ H × H. (2.9) If a = 0, then N is b-strongly monotone. Similarly, we can define the relaxed (a,b)-co- coercivity with respect to the second argument of N. Lemma 2.3 [4]. If K ⊂ H is a closed convex subset and z ∈ H is a given point, then there exists x ∈ K such that (x − z, y − x) ≥ 0, ∀y ∈ K (2.10) if and only if x = P K z,whereP K is the projection of H onto K. Lemma 2.4 [4]. The projection P K is nonexpansive, that is, P K u − P K v ≤ u − v, ∀u,v ∈ H. (2.11) 4 Nonlinear relaxed cocoercive variational inequalities Lemma 2.5 [18]. Let {K n } beasequenceofclosedconvexsubsetsofH such that H(K n ,K) → 0 as n →∞,whereH(·,·) is the Hausdorff metric, that is, for any A,B ∈ CB(H), H(A,B) = max sup a∈A inf b∈B a − b,sup b∈B inf a∈A a − b . (2.12) Then P K n v − P K v −→ 0(n −→ ∞ ), ∀v ∈ H. (2.13) Lemma 2.6 [4]. If K(u) = m(u)+K for all u ∈ H, then P K(u) v = m(u)+P K v − m(u) . (2.14) From Lemmas 2.3 and 2.6, we have the following lemma. Lemma 2.7. If K 1 ,K 2 ⊂ H are t wo closed convex cones, and K i (·) = m(·)+K i (i = 1,2), then x, y ∈ H solve problem (2.1)ifandonlyifx, y ∈ H such that x = x − g 1 (x)+m 1 (x)+P K 1 g 1 (x) − ρF(x, y) − m 1 (x) , y = y − g 2 (y)+m 2 (y)+P K 2 g 2 (y) − ρG(x, y) − m 2 (y) , (2.15) where ρ>0 is a constant. Lemma 2.8 [17]. Let {μ n } be a real seque nce of nonnegative numbers and {ν n } beareal sequence of numbers in [0,1] with ∞ n=0 ν n =∞. If there exists a constant n 1 such that μ n+1 ≤ 1 − ν n μ n + ν n δ n , ∀n ≥ n 1 , (2.16) where δ n ≥ 0 for all n ≥ 0,andδ n → 0 (n →∞), then lim n→∞ μ n = 0. 3. Existence and convergence In this section, we construct an iterative algorithm to approximate the solution of prob- lem (2.1) and study the convergence of the sequence generated by the algorithm. Algorithm 3.1. For any given x 0 , y 0 ∈ H,wecompute x n+1 = x n − g 1 x n + m 1 x n + P K 1 g 1 x n − ρF x n , y n − m 1 x n , y n+1 = y n − g 2 y n + m 2 y n + P K 2 g 2 y n − ρG x n , y n − m 2 y n . (3.1) Theorem 3.2. Let g i : H → H be η i -strongly m onotone and ζ i -Lipschitz continuous and m i : H → H be γ i -Lipschitz continuous (i = 1,2). Let F : H × H → H be l 1 , l 2 -Lipschitz con- tinuous with respect to the first, second arguments, respectively, and relaxed (a,b)-cocoercive with respect to the first argument. Let G : H × H → H be n 1 , n 2 -Lipschitz continuous with respect to the first, second arguments, respectively, and relaxed (c,d)-cocoercive with respect Ke Ding et al. 5 to the second argument. If 2 1+ζ 2 1 − 2η 1 +2γ 1 + 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb + ρn 1 < 1, 2 1+ζ 2 2 − 2η 2 +2γ 2 + 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd + ρl 2 < 1. (3.2) then there exist x ∗ , y ∗ ∈ H, which solve problem (2.1). Moreover, the iterative sequences {x n } and {y n } generated by Algorithm 3.1 converge to x ∗ and y ∗ ,respectively. Proof. From (3.1)andLemma 2.6,wehave x n+1 − x n = x n − g 1 x n + m 1 x n + P K 1 g 1 x n − ρF x n , y n − m 1 x n − x n−1 − g 1 x n−1 + m 1 x n−1 + P K 1 g 1 x n−1 − ρF x n−1 , y n−1 − m 1 x n−1 ≤ x n − x n−1 − g 1 x n − g 1 x n−1 + m 1 x n − m 1 x n−1 + P K 1 g 1 x n − ρF x n , y n − m 1 x n − P K 1 g 1 x n−1 − ρF x n−1 , y n−1 − m 1 (x n−1 ) . (3.3) Since g 1 is ζ 1 -Lipschitz continuous and η 1 -strongly monotone, x n − x n−1 − g 1 x n − g 1 x n−1 2 ≤ 1+ζ 2 1 − 2η 1 x n − x n−1 2 . (3.4) From t he γ 1 -Lipschitzian continuity of m 1 ,wehave m 1 x n − m 1 x n−1 ≤ γ 1 x n − x n−1 . (3.5) Lemma 2.4 implies that P K 1 is nonexpansive and it follows from the strong monotonicity of g 1 that P K 1 g 1 x n − ρF x n , y n − m 1 x n − P K 1 g 1 x n−1 − ρF x n−1 , y n−1 − m 1 x n−1 ≤ g 1 x n − ρF x n , y n − m 1 x n − g 1 x n−1 − ρF x n−1 , y n−1 − m 1 x n−1 ≤ x n − x n−1 − g 1 x n − g 1 x n−1 + m 1 x n − m 1 x n−1 + x n − x n−1 − ρ F x n , y n − F x n−1 , y n + ρ F x n−1 , y n − F x n−1 , y n−1 . (3.6) 6 Nonlinear relaxed cocoercive variational inequalities Since F is relaxed (a,b)-cocoercive and l 1 -Lipschitz continuous with respect to the first argument, x n − x n−1 − ρ F x n , y n − F x n−1 , y n 2 = x n − x n−1 2 + ρ 2 F x n , y n − F x n−1 , y n 2 − 2 x n − x n−1 ,ρ F x n , y n − F x n−1 , y n ≤ x n − x n−1 2 + ρ 2 F x n , y n − F x n−1 , y n 2 +2ρa F x n , y n − F x n−1 , y n 2 − 2ρb x n − x n−1 2 = 1+l 2 1 ρ 2 +2ρal 2 1 − 2ρb x n − x n−1 2 . (3.7) Since F is l 2 -Lipschitz continuous with respect to the second argument, F x n−1 , y n − F x n−1 , y n−1 ≤ l 2 y n − y n−1 . (3.8) It follows from (3.3)–(3.8)that x n+1 − x n ≤ 2 1+ζ 2 1 − 2η 1 +2γ 1 + 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb x n − x n−1 + ρl 2 y n − y n−1 . (3.9) Similarly, we have y n+1 − y n ≤ 2 1+ζ 2 2 − 2η 2 +2γ 2 + 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd y n − y n−1 + ρn 1 x n − x n−1 . (3.10) Now (3.9)and(3.10)imply x n+1 − x n + y n+1 − y n ≤ 2 1+ζ 2 1 − 2η 1 +2γ 1 + 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb + ρn 1 x n − x n−1 + 2 1+ζ 2 2 − 2η 2 +2γ 2 + 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd + ρl 2 , y n − y n−1 ≤ ω x n − x n−1 + y n − y n−1 , (3.11) where ω = max 2 1+ζ 2 1 − 2η 1 +2γ 1 + 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb + ρn 1 , 2 1+ζ 2 2 − 2η 2 +2γ 2 + 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd + ρl 2 . (3.12) Ke Ding et al. 7 It f ollows from (3.2)thatω<1. Thus (3.11) implies that {x n } and {y n } are both Cauchy sequences in H,and {x n } converges to x ∗ ∈ H, {y n } converges to y ∗ ∈ H.Sincem 1 , m 2 , g 1 , g 2 , P K 1 , P K 2 , F, G are all continuous, we have x ∗ = x ∗ − g 1 x ∗ + m 1 x ∗ + P K g 1 (x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ , y ∗ = y ∗ − g 2 y ∗ + m 2 y ∗ + P K g 2 y ∗ − ρG x ∗ , y ∗ − m 2 y ∗ , (3.13) The result follows then from Lemma 2.7. This completes the proof. Remark 3.3. Let ρ>0 be a number satisfying the conditions. ρ − b − al 2 1 − 1 − e 1 n 1 l 2 1 − n 2 1 < 1 − e 1 2 − 1+ b − al 2 1 − 1 − e 1 n 1 2 / l 2 1 − n 2 1 l 2 1 − n 2 1 , ρn 1 < 1 − e 1 , n 1 <l 1 , ρ − d − cn 2 2 − 1 − e 2 l 2 n 2 2 − l 2 2 < 1 − e 2 2 − 1+ d − cn 2 2 − 1 − e 2 l 2 2 / n 2 2 − l 2 2 n 2 2 − l 2 2 , ρl 2 < 1 − e 2 , l 2 <n 2 , (3.14) where e 1 = 2 1+ζ 2 1 − 2η 1 +2γ 1 and e 2 = 2 1+ζ 2 2 − 2η 2 +2γ 2 .Then(3.2)holds. 4. Perturbed algorithm and stability In this section, we construct a new perturbed iterative algorithm for solving problem (2.1) and prove the convergence and stability of the iterative sequence generated by the algorithm. Definit ion 4.1. Let T be a self-map of H, x 0 ∈ H and let x n+1 = f (T, x n ) define an iteration procedure which yields a sequence of points {x n } ∞ n=0 in H. Suppose that {x ∈ H : Tx = x} =∅and {x n } ∞ n=0 converge to a fixed point x ∗ of T.Let{u n }⊂H and let n =u n+1 − f (T,u n ).Iflim n = 0 implies that limu n = x ∗ , then the iteration procedure defined by x n+1 = f (T,x n )issaidtobeT-stable or stable with respect to T. Some results for the stability of various iterative processes, we refer to [1, 10] and the references therein. Let {K 1 n } and {K 2 n } be two sequences of closed convex subsets of H such that H(K 1 n ,K) → 0, H(K 2 n ,K) → 0, when n →∞. Now we consider the following perturbed algorithm for solving problem (2.1). 8 Nonlinear relaxed cocoercive variational inequalities Algorithm 4.2. For any given x 0 , y 0 ∈ H,wecompute x n+1 = 1−t n x n + t n x n − g 1 (x n + m 1 x n + P K 1 n g 1 x n − ρF x n , y n − m 1 x n + t n e n , y n+1 = 1−t n y n + t n y n − g 2 y n + m 2 y n + P K 2 n g 2 y n − ρG x n , y n − m 2 y n + t n j n , (4.1) for all n = 0,1,2, ,where{e n } and { j n } are two sequences of the elements of H, and the sequence {t n } satisfies the following conditions 0 ≤ t n ≤ 1, ∀n ≥ 0, ∞ n=0 t n =∞. (4.2) Let {u n } and {v n } be any sequences in H and define n = 1 n + 2 n by 1 n = u n+1 − 1 − t n u n + t n u n − g 1 u n + m 1 u n + P K 1 g 1 u n − ρF u n ,v n + m 1 u n + t n e n 2 n = v n+1 − 1 − t n v n + t n v n − g 2 v n + m 2 v n + P K 2 g 2 v n − ρG u n ,v n + m 2 v n + t n j n . (4.3) Theorem 4.3. Let g i : X → X be η i -strongly monotone and ζ i -Lipschitz continuous, and m i : X → X be τ i -Lipschitz continuous for i = 1,2.LetF : X × X → X be l 1 , l 2 -Lipschitz continuous with respect to the first and s econd arguments, respectively, and relaxed (a,b)- cocoercive with respect to the first argument. Let G : X × X → X be n 1 , n 2 -Lipschitz continu- ous with respect to the first and second arguments, respectively, and relaxed (c,d)-cocoercive with respect to the second argument. Suppose H(K n ,K) → 0(n →∞) and ρ − b − al 2 1 − 1 − e 1 n 1 l 2 1 − n 2 1 < 1 − e 1 2 − 1+ b − al 2 1 − 1 − e 1 n 1 2 / l 2 1 − n 2 1 l 2 1 − n 2 1 , ρn 1 < 1 − e 1 , n 1 <l 1 , ρ − d − cn 2 2 − 1 − e 2 l 2 n 2 2 − l 2 2 < 1 − e 2 2 − 1+ d − cn 2 2 − 1 − e 2 l 2 2 / n 2 2 − l 2 2 n 2 2 − l 2 2 , ρl 2 < 1 − e 2 , l 2 <n 2 , (4.4) where e i = 2 1+ζ 2 i − 2η i +2γ i for i = 1,2.Iflim n→∞ e n =0 and lim n→∞ j n =0, then we have the following conclusions. (I) The iterative sequences generated by Algorithm 4.2 converge to the unique solution of (2.1). (II) Moreover, if 0 <t ≤ t n , then limu n = x ∗ , limv n = y ∗ if and only if lim( 1 n + 2 n ) = 0, where 1 n and 2 n are defined by (4.3). Ke Ding et al. 9 Proof. By Theorem 3.2,problem(2.1) admits a solution (x ∗ , y ∗ ). It is easy to prove that (x ∗ , y ∗ ) is the unique solution of (4.1). From Lemma 2.7,wehave x ∗ = 1 − t n x ∗ + t n x ∗ − g 1 x ∗ + m 1 x ∗ + P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ , y ∗ = 1 − t n y ∗ + t n y ∗ − g 2 y ∗ + m 2 y ∗ + P K 2 g 2 y ∗ − ρG x ∗ , y ∗ − m 2 y ∗ , (4.5) Since P K is nonexpansive and it follows f rom (4.1)and(4.5)that x n+1 − x ∗ = 1−t n x n +t n x n −g 1 x n +m 1 x n +P K 1 n g 1 x n − ρF x n , y n − m 1 x n +t n e n − 1−t n x ∗ −t n x ∗ −g 1 x ∗ +m 1 x ∗ +P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ ≤ 1−t n x n −x ∗ +t n x n −x ∗ +g 1 x n − g 1 x ∗ +t n m 1 (x)−m 1 x ∗ +t n e n +t n P K 1 n g 1 x n − ρF x n , y n − m 1 x n − P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ ≤ 1−t n x n −x ∗ +t n x n −x ∗ +g 1 x n − g 1 x ∗ +t n m 1 (x)−m 1 x ∗ +t n e n +t n P K 1 n g 1 x n − ρF x n , y n − m 1 x n − P K 1 n g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ +t n P K 1 n g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ − P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ ≤ 1−t n x n −x ∗ +t n x n −x ∗ +g 1 x n − g 1 x ∗ +t n m 1 (x)−m 1 x ∗ +t n e n + t n x n − x ∗ − g 1 x n − g 1 x ∗ + t n m 1 x n − m 1 x ∗ + t n x n − x ∗ − ρ F x n , y n − F x ∗ , y n + ρt n F x ∗ , y n − F x ∗ , y ∗ +t n P K 1 n g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ − P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ . (4.6) Since F is l 2 -Lipschitz continuous with respect to the second argument, F x ∗ , y n − F x ∗ , y ∗ ≤ l 2 y n − y ∗ . (4.7) From the strong monotonicity and Lipschitzian continuity of g 1 ,weobtain x n − x ∗ − g 1 x n − g 1 x ∗ 2 ≤ 1+ζ 2 1 − 2η 1 x n − x ∗ 2 . (4.8) The Lipschitzian continuity of m 1 implies m 1 x n − m 1 x ∗ ≤ γ 1 x n − x ∗ . (4.9) 10 Nonlinear relaxed cocoercive variational inequalities Since F is relaxed (a,b)-cocoercive and l 1 -Lipschitz continuous with respect to the first argument, x n − x ∗ − ρ F x n , y n − F x ∗ , y n ≤ 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb x n − x ∗ . (4.10) It follows from (4.6)–(4.10)that x n+1 − x ∗ ≤ 2t n 1+ζ 2 1 − 2η 1 +2t n γ 1 + t n 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb +1− t n x n − x ∗ + t n ρl 2 y n − y ∗ + t n b n + t n e n , (4.11) where b n = P K 1 n g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ − P K 1 g 1 x ∗ − ρF x ∗ , y ∗ − m 1 x ∗ . (4.12) From t he fact of H(K 1 n ,K 1 ) → 0andLemma 2.5,weknowthatb n → 0. Similarly, we have y n+1 − y ∗ ≤ 2t n 1+ζ 2 2 − 2η 2 +2t n γ 2 + t n 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd +1− t n y n − y ∗ + t n ρn 1 x n − x ∗ + t n c n + t n j n , (4.13) where c n = P K 2 n g 2 y ∗ − ρG x ∗ , y ∗ − m 2 y ∗ − P K 2 g 2 y ∗ − ρF x ∗ , y ∗ − m 2 y ∗ , (4.14) and c n → 0. Now (4.11)and(4.13)imply x n+1 − x ∗ + y n+1 − y ∗ ≤ 2t n 1+ζ 2 1 − 2η 1 +2t n γ 1 +1− t n + t n 1+ρ 2 l 2 1 +2ρal 2 1 − 2ρb + t n ρn 1 x n − x ∗ + 2t n 1+ζ 2 2 − 2η 2 +2t n γ 2 +1− t n + t n 1+ρ 2 n 2 2 +2ρcn 2 2 − 2ρd + t n ρl 2 y n − y ∗ + t n c n + t n b n + t n e n + t n j n . 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Applied Mathematics Letters 17 (2004), no 3, 345– 352 14 Nonlinear relaxed cocoercive variational inequalities [4] C Baiocchi and A Capelo, Variational and Quasivariational Inequalities, Application to Free Boundary Problems, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1984 [5] J Y Chen, N C Wong, and J C Yao, Algorithm for generalized co-complementarity problems in Banach spaces,... proof of Conclusion I Next we prove Conclusion II By using (4.1), we obtain un+1 − x∗ ≤ un+1 − 1 − tn un +tn un − g1 un +m1 un +PK1 g1 un − ρF un ,vn +m1 un + ≤ 1 − tn un +tn un − g1 un +m1 un +PK1 g1 un − ρF un ,vn +m1 un 1 − tn un +tn un − g1 un +m1 un +PK1 g1 un − ρF un ,vn +m1 un +tn en +tn en − x∗ 1 +tn en − x∗ + n (4.21) 12 Nonlinear relaxed cocoercive variational inequalities As the proof of. .. completes the proof Acknowledgments The authors thank the referees for their valuable suggestions This work was supported by the National Natural Science Foundation of China and the Educational Science Foundation of Chongqing (KJ051307) References [1] R P Agarwal, Y J Cho, J Li, and N J Huang, Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive... pseudo-contractive mapping, Proceedings of the American Mathematical Society 113 (1991), no 3, 727–731 [18] S Z Zhou, Perturbation for elliptic variational inequalities, Science in China (Scientia Sinica) Series A 34 (1991), no 6, 650–659 Ke Ding: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China E-mail address: keding@yahoo.com Wen-Yong Yan: Department of Mathematics, Sichuan University,... (4.23) Similarly, we have 2 vn+1 − y ∗ ≤ 2tn 1 + ζ2 − 2η2 + 2tn γ2 + tn 1 + ρ2 n2 + 2ρcn2 − 2ρd + 1 − tn 2 2 + tn ρn1 un − x∗ + tn cn + tn jn + vn − y ∗ 2 n, (4.24) where cn is defined by (4.14) As the proof of inequality (4.17), and since 0 < t ≤ tn , (4.23) and (4.24) yield un+1 − x∗ + vn+1 − y ∗ ≤ 1 − (1 − h)tn un − x∗ + vn − y ∗ + tn (bn + cn + en + jn ) + 1 2 n+ n ≤ 1 − (1 − h)tn un − x∗ + vn − y ∗ . study a new system of generalized nonlinear relaxed cocoercive in- equality problems and construct an iterative algorithm for approximating the solutions of the system of generalized relaxed cocoercive. introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities 10.1155/JIA/2006/40591 2 Nonlinear relaxed cocoercive variational inequalities inequality problems and construct an iterative algorithm for approximating the solutions of the system of generalized relaxed cocoercive
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