This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Graphical Approximation of Common Solutions to Generalized Nonlinear Relaxed Cocoercive Operator Equation Systems with (A,\eta)-accretive Mappings Fixed Point Theory and Applications 2012, 2012:14 doi:10.1186/1687-1812-2012-14 Fang Li (lifang1687@163.com) Heng-you Lan (hengyoulan@163.com) Yeol JE Cho (yjcho@gsnu.ac.kr) ISSN 1687-1812 Article type Research Submission date 24 April 2011 Acceptance date 15 February 2012 Publication date 15 February 2012 Article URL http://www.fixedpointtheoryandapplications.com/content/2012/1/14 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Fixed Point Theory and Applications manuscript No. (will be inserted by the editor) Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings Fang Li 1 , Heng-you Lan ∗1 and Yeol Je Cho 2 ∗ Corresponding author: hengyoulan@163.com FL: lifang1687@163.com YJC: yjcho@gsnu.ac.kr 1 Department of Mathematics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, People’s Republic of China 2 Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju 660-701, Korea Abstract In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of op- erator sequences with (A, η)-accretive mappings in Banach space. By using the generalized resolvent operator technique associated with (A, η)-accretive mappings, we also prove the existence of solutions for a class of generalized nonlinear relaxed co coercive operator equation systems and the variational convergence of the sequence generated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. The obtained results improve and gener- alize some well-known results in recent literatures. Keywords (A, η)-accretive mapping; Generalized resolvent operator technique; Gener- alized nonlinear relaxed cocoercive operator equation systems; New perturbed iterative algorithm with errors; Variational graphical convergence 2000 Mathematics Subject Classification: 47H05; 49J40 1 Introduction It is well known that standard Yosida regularizations/approximations have been tremendously effective to approximation solvability of general variational inclusion problems in the context of resolvent operators that turned out to be nonexpansive. This class of nonlinear Yosida approximations have been applied to approximation solvability of nonlinear inhomogeneous evolution inclusions of the form f(t) ∈ u (t) + M u(t) − ωu(t), u(0) = u 0 for almost all t ∈ [0, T ], where T ∈ (0, 1) is fixed, ω ∈ R (see [1]). For more gen- eral details on approximation solvability of general nonlinear inclusion prob- lems, we refer the reader to [2–18] and the references therein. On the other hand, it is well known that variational inequalities and vari- ational inclusions provide mathematical models to some problems arising in economics, mechanics, and engineering science and have been studied exten- sively. There are many methods to find solutions of variational inequality and variational inclusion problems. Among these methods, the resolvent opera- tor technique is very important. For some literature, we recommend to the following example, and the reader [2–15, 17, 18] and the references therein. Example 1.1. ([19]) Let V : R n → R be a local Lipschitz continuous function, and let K be a closed convex set in R n . If x ∗ is a solution to the following problem: min x∈K V (x), then 0 ∈ ∂V (x ∗ ) + N K (x ∗ ), where ∂V (x ∗ ) denotes the subdifferential of V at x ∗ and N K (x ∗ ) the normal cone of K at x ∗ . In 2006, Lan et al. [7] intro duced a new concept of (A, η)-accretive map- pings, which provides a unifying framework for maximal monotone operators, m-accretive operators, η-subdifferential operators, maximal η-monotone oper- ators, H-monotone operators, generalized m-accretive mappings, H-accretive operators, (H, η)-monotone operators, and A-monotone mappings. Recently, by using the concept of (A, η)-accretive mappings and the resolvent opera- tor technique associated with (A, η)-accretive mappings, Jin [5] introduced and studied a new class of nonlinear variational inclusion systems with (A, η)- accretive mappings in q-uniformly smooth Banach spaces and developed some new iterative algorithms to approximate the solutions of the mentioned nonlin- ear variational inclusion systems. Furthermore, by using the resolvent operator technique, Petrot [14] studied the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point prob- lems for Lipschitz mappings in Hilbert spaces, and Agarwal and Verma [2] in- troduced and studied a new system of nonlinear (set-valued) variational inclu- sions involving (A, η)-maximal relaxed monotone and relative (A, η)-maximal monotone mappings in Hilbert spaces and proved its approximation solvabil- ity based on the variational graphical convergence of operator sequences. For more literature, we recommend to the reader [9, 20] and the references therein. Motivated and inspired by the above works, the purpose of this paper is to consider and study the following generalized nonlinear operator equation system with (A, η)-accretive mappings in real Banach space B 1 × B 2 : Find (x, y) ∈ B 1 × B 2 and u ∈ S(x), v ∈ T (y) such that p(x) = R ρλ 1 ,A 1 η 1 ,M 1 (·,x) [(1 − λ 1 )A 1 (p(x)) + λ 1 (A 1 (f(y)) −ρN 1 (u, y) + a)], h(y) = R λ 2 ,A 2 η 2 ,M 2 (y,· ) [(1 − λ 2 )A 2 (h(y)) + λ 2 (A 2 (g(x)) −N 2 (x, v) + b)], (1.1) where for all (x, y) ∈ B 1 ×B 2 , R ρλ 1 ,A 1 η 1 ,M 1 (·,x) = (A 1 +ρλ 1 M 1 (·, x)) −1 and R λ 2 ,A 2 η 2 ,M 2 (y ,· ) = (A 2 + λ 2 M 2 (y, ·)) −1 are two resolvent operators and two constants ρ, > 0, N 1 : B 1 × B 2 → B 1 , N 2 : B 1 × B 2 → B 2 , p : B 1 → B 1 , h : B 2 → B 2 , f : B 2 → B 1 , g : B 1 → B 2 are single-valued operators, λ 1 , λ 2 > 0 are two constants, ( a, b) ∈ B 1 × B 2 is an any given element, and S : B 1 → 2 B 1 , T : B 2 → 2 B 2 , A i : B i → B i , η i : B i × B i → B i , M i : B i × B i → 2 B i (i = 1, 2) are any nonlinear op erators such that for all x ∈ B 1 , M 1 (·, x) : B 1 → 2 B 1 is an (A 1 , η 1 )-accretive mapping and M 2 (y, ·) : B 2 → 2 B 2 is an (A 2 , η 2 )-accretive mapping for all y ∈ B 2 , respectively. Based on the definition of the resolvent operators associated with (A, η)- accretive mappings, the Equation (1.1) can be written as a ∈ A 1 (p(x)) − A 1 (f(y)) + ρN 1 (u, y) + ρM 1 (p(x), x), b ∈ A 2 (h(y)) − A 2 (g(x)) + N 2 (x, v) + M 2 (y, h(y)) (1.2) Remark 1.1. For appropriate and suitable choices of B i , A i , η i , N i , M i (i = 1, 2), p, h, f, g, S, T , one can obtain a number (systems) of quasi-variational inclusions, generalized (random) quasi- variational inclusions, quasi-variational inequalities, and implicit quasi-variational inequalities as special cases of the Equation (1.1) (or problem (1.2)) include. Below are some special cases of problem. Example 1.2. If B i = B(i = 1, 2), p = f = h = g, N 1 (x, ·) = N 2 (·, y) = N(·) and M 1 (·, x) = M 1 (·), M 2 (y, ·) = M 2 (·) for all (x, y) ∈ B 1 × B 2 and a = b = 0, then the problem (1.2) collapses to the following nonlinear variational inclusion system with (A, η)-accretive mappings: 0 ∈ A 1 (g(x)) −A 1 (g(y)) + ρN(y) + ρM 1 (g(x)), 0 ∈ A 2 (g(y)) − A 2 (g(x)) + N(x) + M 2 (g(y)). (1.3) The system (1.3) was introduced and studied by Jin [5]. Further, when A i = A, M i = M(i = 1, 2) and y = x, the system (1.3) reduces to a nonlinear variational inclusion of find x ∈ B such that 0 ∈ N(x) + M(g(x)), which contains the variational inclusions with H-monotone operator, H-accretive mappings, or A-maximal (m)-relaxed monotone (AMRM) mappings in [2, 3] as special cases. Example 1.3. If B i = H(i = 1, 2) is a Hilbert space, a = b = 0, S : B 1 → B 1 and T : B 2 → B 2 are two single-valued mappings, p = f = h = g = S = T = I is the identity operator and M 1 (·, x) = M 2 (y, ·) = M (·) for all (x, y) ∈ B 1 × B 2 , then the problem (1.2) is equivalent to solve the following nonlinear variational inclusion system with (A, η)-monotone mappings: 0 ∈ A 1 (x) − A 1 (y) + ρN(y, x) + ρM(x), 0 ∈ A 2 (y) − A 2 (x) + N (x, y) + M (y), (1.4) The system (1.4) was introduced and studied by Wang and Wu [18] and con- tains the generalized system for mixed variational inequalities with maximal monotone operators in [14] as special cases. Moreover, taking y = x, then the system (1.4) reduces to finding an element x ∈ H such that 0 ∈ N(x, x) + M(x), which was considered by Verma [17]. Example 1.4. When B i = H, λ i = 1(i = 1 , 2), p = h, A 1 = A 2 = I, N 1 (x, ·) = N 2 (·, y) = N(·) and M 1 (·, x) = M 1 (·), N 2 (y, ·) = M 2 (·) for all (x, y) ∈ B 1 ×B 2 , the system (1.1) becomes to the following nonlinear operator equation systems: Finding (x, y) ∈ H × H such that h(x) = J ρ M 1 [f(y) − ρN(y)], h(y) = J M 2 [g(x) − N(x)], (1.5) where J ρ M 1 = (I + ρM 1 ) −1 and J M 2 = (I + M 2 ) −1 . Based on the definition of the resolvent operators, we know that the system (1.5) is equivalent to solve the following system of general variational inclusions: 0 ∈ h(x) − f (y) + ρN(y) + ρM 1 (h(x)), 0 ∈ h(y) −g(x) + N(x) + M 2 (h(y)), (1.6) which was studied by Noor et al. [12] when M i = M is maximal monotone for i = 1, 2. Moreover, some special cases of the problem (1.6) can be found in [4, 6] and the references therein. We also construct a new perturbed iterative algorithm framework with er- rors based on the variational graphical convergence of operator sequences with (A, η)-accretive mappings in Banach space for approximating the solutions of the nonlinear equation system (1.1) in smooth Banach spaces and prove the ex- istence of solutions and the variational convergence of the sequence generated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. The results present in this paper improve and generalize the corresponding results of [2, 3, 5, 12, 14, 17, 18] and many other recent works. 2 Preliminaries Let B be a real Banach space with dual space B ∗ , ·, ·be the dual pair between B and B ∗ , CB ( B) denote the family of all nonempty closed bounded subsets of B, and 2 B denote the family of all the nonempty subsets of B. The generalized duality mapping J q : B → 2 B ∗ is defined by J q (x) = f ∗ ∈ B ∗ : x, f ∗ = x q , f ∗ = x q − 1 , ∀x ∈ B, where q > 1 is a constant. In particular, J 2 is the usual normalized duality mapping. It is known that, in general, J q (x) = x q − 2 J 2 (x) for all x = 0, and J q is single-valued if B ∗ is strictly convex. In the sequel, we always suppose that B is a real Banach space such that J q is single-valued and H is a Hilbert space. If B = H, then J 2 becomes the identity mapping on H. The modulus of smoothness of B is the function χ B : [0, ∞) → [0, ∞) defined by χ B (t) = sup 1 2 (x + y + x − y) − 1 : x ≤ 1, y ≤ t . A Banach space B is called uniformly smooth if lim t→0 χ B (t) t = 0. B is called q-uniformly smooth if there exists a constant c > 0 such that χ B (t) ≤ ct q , q > 1. Remark that J q is single-valued if B is uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Ba- nach spaces, Xu [21] proved the following result: Lemma 2.1. Let B be a real uniformly smooth Banach space. Then, B is q-uniformly smooth if and only if there exists a constant c q > 0 such that for all x, y ∈ B, x + y q ≤ x q + qy, J q (x) + c q y q . In the sequel, we give some concept and lemmas needed later. Definition 2.1. Let B be a q-uniformly smooth Banach space and T, A : B → B be two single-valued mappings. T is said to be (i) accretive if T (x) − T (y), J q (x − y) ≥ 0, ∀x, y ∈ B; (ii) strictly accretive if T is accretive and T (x) − T (y), J q (x − y) = 0 if and only if x = y; (iii) r-strongly accretive if there exists a constant r > 0 such that T (x) − T (y), J q (x − y) ≥ rx −y q , ∀x, y ∈ B ; (iv) γ-strongly accretive with respect to A if there exists a constant γ > 0 such that T (x) − T (y), J q (A(x) − A(y)) ≥ γx − y q , ∀x, y ∈ B ; (v) m-relaxed cocoercive with respect to A if, there exists a constant m > 0 such that T (x) − T (y), J q (A(x) − A(y)) ≥ −mT (x) − T (y) q , ∀x, y ∈ B ; (vi) (π, ι)-relaxed cocoercive with respect to A if, there exist constants π, ι > 0 such that T (x) −T (y), J q (A(x) −A(y)) ≥ −πx −y q + ιT (x) −T (y) q , ∀x, y ∈ B ; (vii) s-Lipschitz continuous if there exists a constant s > 0 such that T (x) − T (y) ≤ sx − y, ∀x, y ∈ B. In a similar way, we can define (relaxed) cocoercivity and Lipschitz conti- nuity of the operator N(·, ·) : B × B → B in the first and second arguments. Remark 2.1. (1) The notion of the cocoercivity is applied in several direc- tions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods [16], while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoerciv- ity. Several classes of relaxed cocoercive variational inequalities and variational inclusions have been studied in [2, 5, 7–10, 12, 16–18]. (2) When B = H, (i)–(iv) of Definition 2.1 reduce to the definitions of monotonicity, strict monotonicity, strong monotonicity, and strong monotonic- ity with respect to A, respectively (see [3, 18]). Definition 2.2. A single-valued mapping η : B × B → B is said to be τ-Lipschitz continuous if there exists a constant τ > 0 such that η(x, y) ≤ τx − y, ∀x, y ∈ B. Definition 2.3. Let B be a q-uniformly smooth Banach space, η : B×B → B and A, H : B → B b e single-valued mappings. Then set-valued mapping M : B → 2 B is said to be (i) η-accretive if u − v, J q (η(x, y)) ≥ 0, ∀x, y ∈ B, u ∈ M(x), v ∈ M(y); (ii) r-strongly η-accretive if there exists a constant r > 0 such that u − v, J q (η(x, y)) ≥ rx − y q , ∀x, y ∈ B, u ∈ M(x), v ∈ M(y); (iii) m-relaxed η-accretive if there exists a constant m > 0 such that u − v, J q (η(x, y)) ≥ −mx − y q , ∀x, y ∈ B , u ∈ M(x), v ∈ M(y); (iv) ξ- ˆ H-Lipschitz continuous, if there exists a constant ξ > 0 such that ˆ H(M(x), M (y)) ≤ ξx − y, ∀x, y ∈ B, where ˆ H is the Hausdorff metric on CB(B); (v) (A, η)-accretive if M is m-relaxed η-accretive and (A + ρM )(B) = B for every ρ > 0. Remark 2.2. The (A, η)-accretivity generalizes the general (H, η)-accretivity, (I, η)-accretivity (so-called generalized m-accretivity), H-accretivity, classical m-accretivity, (A, η)-monotonicity, A-monotonicity, (H, η)-monotonicity, H- monotonicity, maximal η-monotonicity, and classical maximal monotonicity as special cases (see, for example, [1, 7, 8, 13] and the references therein.) Definition 2.4. Let A : B → B be a strictly η-accretive mapping and M : B → 2 B be an (A, η)-accretive mapping. The resolvent operator R ρ,A η ,M : B → B is defined by: R ρ,A η ,M (u) = (A + ρM ) −1 (u), ∀u ∈ B. Remark 2.3. The resolvent operators associated with (A, η)-accretive mappings include as special cases the corresponding resolvent operators asso- ciated with (H, η)-accretive mappings, (A, η)-monotone operators [8], (H , η)- monotone operators, H-accretive operators, generalized m-accretive operators, maximal η-monotone operators, H-monotone operators, A-monotone opera- tors, η-subdifferential operators, the classical m-accretive, and maximal mono- tone operators. See, for example, [1, 7, 8, 13] and the references therein. Lemma 2.2. ([7]) Let B be a q-uniformly smooth Banach space and η : B × B → B be τ-Lipschitz continuous, A : B → B be a r-strongly η-accretive mapping and M : B → 2 B be an (A, η)-accretive mapping. Then, the resolvent operator R ρ,A η ,M : B → B is τ q−1 r−ρm -Lipschitz continuous, i.e., R ρ,A η ,M (x) − R ρ,A η,M (y) ≤ τ q −1 r − ρm x − y, ∀x, y ∈ B, where ρ ∈ (0, r m ) is a constant. Definition 2.5. Let M n , M : B → 2 B be (A, η)-accretive mappings on B for n = 0, 1, 2, . . . . Let A : B → B be r-strongly η-monotone and β-Lipschitz continuous. The sequence M n is graph-convergent to M , denoted M n A−G −→ M , if for every (x, y) ∈ graph(M), there exists a sequence (x n , y n ) ∈ graph(M n ) such that x n → x, y n → y as n → ∞. Based on Definition 2.6 and Theorem 2.1 in [20], we have the following lemma. Lemma 2.3. Let M n , M : B → 2 B be (A, η)-accretive mappings on B for n = 0, 1, 2, . . Then, the sequence M n A−G −→ M if and only if R ρ,A η,M n (x) → R ρ,A η ,M (x), ∀x ∈ B, where R ρ,A η,M = (A + ρM n ) −1 , R ρ,A η ,M = (A + ρM) −1 , ρ > 0 is a constant, and A : B → B is r-strongly η-monotone and β-Lipschitz continuous. 3 Algorithms and graphical convergence In this section, by using resolvent operator technique associated with (A, η)- accretive mappings, we shall develop a new perturbed iterative algorithm framework with errors for solving the nonlinear operator equation system (1.1) with (A, η)-accretive mappings and relaxed cocoercive operators and prove the existence of solutions and the variational convergence of the sequence gen- erated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. Above all, we note that the equalities (1.1) can be written as p(x) = R ρλ 1 ,A 1 η 1 ,M 1 (·,x) (s), s = (1 − λ 1 )A 1 (p(x)) + λ 1 (A 1 (f(y)) − ρN 1 (u, y) + a), h(y) = R λ 2 ,A 2 η 2 ,M 2 (y, ·) (t), t = (1 − λ 2 )A 2 (h(y)) + λ 2 (A 2 (g(x)) − N 2 (x, v) + b), where ρ, λ > 0 are constants. This formulation allows us to construct the following perturbed iterative algorithm framework with errors. Algorithm 3.1. Step 1. For an arbitrary initial point (x 0 , y 0 ) ∈ B 1 × B 2 , take u 0 ∈ S(x 0 ) and v 0 ∈ T(y 0 ). Step 2. Choose sequences {d n } ⊂ B 1 and {e n } ⊂ B 2 are two error sequences to take into account a possible inexact computation of the op erator points, which satisfy the following conditions: lim n→∞ d n = lim n→∞ e n = 0, ∞ n=1 d n − d n−1 + e n − e n−1 < ∞. Step 3. Let the sequence {(s n , t n , x n , y n )} ⊂ B 1 × B 2 × B 1 × B 2 satisfy s n = (1 −λ 1 )A 1 (p(x n )) + λ 1 (A 1 (f(y n )) − ρN 1 (u n , y n ) + a), t n = (1 −λ 2 )A 2 (h(y n )) + λ 2 (A 2 (g(x n )) − N 2 (x n , v n ) + b), x n+1 = (1 −k)x n + k{x n − p(x n ) + R ρλ 1 ,A 1 η 1 ,M n 1 (·,x n ) (s n )} + d n , y n+1 = (1 −κ)y n + κ{y n − h(y n ) + R λ 2 ,A 2 η 2 ,M n 2 (y n ,·) (t n )} + e n , (3.1) where R ρλ 1 ,A 1 η 1 ,M n 1 (·,x) = (A 1 +ρλ 1 M n 1 (·, x)) −1 , R λ 2 ,A 2 η 2 ,M n 2 (y, ·) = (A 2 +λ 2 M n 2 (y, ·)) −1 , λ 1 , λ 2 , ρ, are nonnegative constants and k, κ ∈ (0, 1] are size constants. Step 4. Choose u n+1 ∈ S(x n+1 ) and v n+1 ∈ T(y n+1 ) such that (see [22]) u n − u n+1 ≤ (1 + 1 n+1 ) ˆ H(S(x n ), S(x n+1 )), v n − v n+1 ≤ (1 + 1 n+1 ) ˆ H(T (y n ), T (y n+1 )). (3.2) Step 5. If s n , t n , x n , y n , d n , and e n satisfy (3.1) and (3.2) to sufficient accuracy, stop; otherwise, set n := n + 1 and return to Step 2. Now, we prove the existence of a solution of problem (1.1) and the conver- gence of Algorithm 3.1. Theorem 3.1. For i = 1, 2, let B i be a q i -uniformly smooth Banach space with q i > 1, η i , A i , M i , N i (i = 1, 2) and p, h, f, g be the same as in the Equation (1.1). Also suppose that the following conditions hold: (H 1 ) η i is τ i -Lipschitz continuous, and A i is r i -strongly η i -accretive, and σ i -Lipschitz continuous for i = 1, 2, respectively; (H 2 ) p is δ 1 -strongly accretive and l p -Lipschitz continuous, h is δ 2 -strongly accretive and l h -Lipschitz continuous, f is l f -Lipschitz continuous and g is l g -Lipschitz continuous, S : B 1 → CB ( B 1 ) is ξ - ˆ H -Lipschitz continuous and T : B 2 → CB(B 2 ) is ζ- ˆ H-Lipschitz continuous; (H 3 ) N 1 is (π 1 , ι 1 )-relaxed cocoercive with respect to f 1 and 2 -Lipschitz continuous in the second argument, and N 2 is (π 2 , ι 2 )-relaxed cocoercive with respect to g 2 and 1 -Lipschitz continuous in the first argument, and N 1 is β 1 - Lipschitz continuous in the first variable, and N 2 is β 2 -Lipschitz continuous in the second variable, where f 1 : B 2 → B 1 is defined by f 1 (y) = A 1 ◦ f (y) = A 1 (f(y)) for all y ∈ B 2 and g 2 : B 1 → B 2 is defined by g 2 (x) = A 2 ◦ g(x) = A 2 (g(x)) for all x ∈ B 1 ; (H 4 ) for n = 0, 1, 2, . . ., M n i : B i × B i → 2 B i (i = 1, 2) are any nonlinear operators such that for all x ∈ B 1 , M n 1 (·, x) : B 1 → 2 B 1 is an (A 1 , η 1 )-accretive mapping with M n 1 (·, x) A 1 −G −→ M 1 (·, x), and M n 2 (y, ·) : B 2 → 2 B 2 is an (A 2 , η 2 )- accretive mapping with M n 2 (y, ·) A 2 −G −→ M 2 (y, ·) for all y ∈ B 2 , respectively; (H 5 ) there exist constants ν i (i = 1, 2), ρ ∈ (0, r 1 /m 1 ) and ∈ (0, r 2 /m 2 ) such that R ρλ 1 ,A 1 η 1 ,M 1 (·,x) (z) − R ρλ 1 ,A 1 η 1 ,M 1 (·,y ) (z) ≤ ν 2 x − y, ∀x, y, z ∈ B 1 , R λ 2 ,A 2 η 2 ,M 2 (x,·) (z) − R λ 2 ,A 2 η 2 ,M 2 (y,· ) (z) ≤ ν 1 x − y, ∀x, y, z ∈ B 2 , (3.3) and ν 2 + q 1 1 − q 1 δ 1 + c q 1 l q 1 p + τ q 1 −1 1 [(1−λ 1 )σ 1 l p +ρλ 1 β 1 ξ] r 1 −ρλ 1 m 1 + κλ 2 τ q 2 −1 2 q 2 √ σ q 2 2 l q 2 g −q 2 ι 2 q 2 1 +q 2 π 2 +c q 2 q 2 q 2 1 k(r 2 −λ 2 m 2 ) < 1, ν 1 + q 2 1 − q 2 δ 2 + c q 2 l q 2 h + τ q 2 −1 2 [(1−λ 2 )σ 2 l h +λ 2 β 2 ζ] r 2 −λ 2 m 2 + kλ 1 τ q 1 −1 1 q 1 σ q 1 1 l q 1 f −q 1 ρι 1 q 1 2 +q 1 ρπ 1 +c q 1 ρ q 1 q 1 2 κ(r 1 −ρλ 1 m 1 ) < 1 (3.4) where c q 1 , c q 2 are the constants as in Lemma 2.1 and k, κ ∈ (0, 1] are size constants. Then, there exist (x ∗ , y ∗ ) ∈ B 1 × B 2 u ∗ ∈ S(x ∗ ), v ∗ ∈ T (y ∗ ) such that (x ∗ , y ∗ , u ∗ , v ∗ ) is a solution of the Equation (1.1) and x n → x ∗ , y n → y ∗ , u n → u ∗ , v n → v ∗ , as n → ∞, where {x n }, {y n }, {u n } and {v n } are iterative sequences generated by Algo- rithm 3.1. Proof. Define · ∗ on B 1 × B 2 by (x, y) ∗ = x+ y, ∀(x, y) ∈ B 1 × B 2 . [...]... note that one can obtain the corresponding results of Theorems 3.1-3.2 when there are problems (1.1), (1.3)–(1.5) with (H, η)accretive mappings, (A, η)-monotone operators, (H, η)-monotone operators, H-accretive operators, generalized m-accretive operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, η-subdifferential operators or the classical m-accretive The results obtained... 1), 2 1 {dn } ⊂ H and {en } ⊂ H are two error sequences to take into account a possible inexact computation of the operator points, which satisfy the following conditions: ∞ lim dn = lim en = 0, n→∞ n→∞ ( dn − dn−1 + en − en−1 ) < ∞ n=1 Proof By the nonexpansivity of the resolvent operators associated with maximal monotone operators and the proof of Theorem 3.1, one can derive the result 2 Remark 3.3... A-monotone nonlinear relaxed cocoercive variational inclusions Central Eur J Math 5(2), 386–396 (2007) 18 Wang, Z, Wu, C: A system of nonlinear variational inclusions with (A, η)-monotone mappings J Inequal Appl 2008, Article ID 681734, 6 pp (2008) 19 Clarke, FH: Optimization and Nonsmooth Analysis Wiley, New York (1983) 20 Verma, RU: A generalization to variational convergence for operators Adv Nonlinear. .. grateful to the editor and referee for valuable comments and suggestions Author details 1 Department of Mathematics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, People’s Republic of China 2 Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju 660-701, Korea Authors contributions FL carried out the proof of convergence of. .. 1529–1538 (2006) 8 Lan, HY: New proximal algorithms for a class of (A, η)-accretive variational inclusion problems with non-accretive set-valued mappings J Appl Math Comput 25(1–2), 255– 267 (2007) 9 Lan, HY, Cai, LC: Variational convergence of a new proximal algorithm for nonlinear general A-monotone operator equation systems in Banach spaces Nonlinear Anal TMA 71(12), 6194–6201 (2009) 10 Li, HG, Xu, AJ,... for solving system of extended general variational inclusions J Inequal Appl 2011, Article ID 371241, 10 pp (2011) 13 Peng, JW, Zhu, DL, Zheng, XP: Existence of solutions and convergence of a multistep iterative algorithm for a system of variational inclusions with (H, η)-accretive operators Fixed Point Theory Appl 2007, Article ID 93678, 20 pp (2007) 14 Petrot, N: A resolvent operator technique for... application of Theorem 3.1 Theorem 3.2 Assume that H is a real Hilbert space and the following conditions hold: (H1 ) h : H → H is δ-strongly monotone and lh -Lipschitz continuous, f : H → H is lf -Lipschitz continuous and g : H → H is lg -Lipschitz continuous; (H2 ) N : H → H is (π1 , ι1 ) -relaxed cocoercive with respect to f and -Lipschitz continuous, and (π2 , ι2 ) -relaxed cocoercive with respect to g;... Jin, MM: Iterative algorithms for a new system of nonlinear vriational inclusions with (A, η)-accretive mappings in Banach spaces Comput Math Appl 54, 579–588 (2007) 6 Kazmi, KR, Bhat, MI: Iterative algorithm for a system of nonlinear variational-like inclusions Comput Math Appl 48(12), 1929–1935 (2004) 7 Lan, HY, Cho, YJ, Verma, RU: On nonlinear relaxed cocoercive variational inclusions involving (A,... for approximate solving of generalized system mixed variational inequality and fixed point problems Appl Math Lett 23(4), 440–445 (2010) 15 Tan, JF, Chang, SS: Iterative algorithms for finding common solutions to variational inclusion equilibrium and fixed point problems Fixed Point Theory Appl 2011, Article ID 915629, 17 pp (2011) 16 Verma, RU: Generalized system for relaxed cocoercive variational inequalities... results of [2, 3, 5, 12, 14, 17, 18] and many other recent works Acknowledgments This work was supported by the Sichuan Youth Science and Technology Foundation (08ZQ026-008), the Open Foundation of Artificial Intelligence of Key Laboratory of Sichuan Province (2009RZ001), the Scientific Research Fund of Sichuan Provincial Education Department (10ZA136), the Cultivation Project of Sichuan University of Science . manuscript No. (will be inserted by the editor) Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings Fang. resolvent operators asso- ciated with (H, η)-accretive mappings, (A, η)-monotone operators [8], (H , η)- monotone operators, H-accretive operators, generalized m-accretive operators, maximal η-monotone. η)-accretive mapping; Generalized resolvent operator technique; Gener- alized nonlinear relaxed cocoercive operator equation systems; New perturbed iterative algorithm with errors; Variational graphical