Open Access

Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings

Fixed Point Theory and Applications20122012:14

https://doi.org/10.1186/1687-1812-2012-14

Received: 24 April 2011

Accepted: 15 February 2012

Published: 15 February 2012

Abstract

In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of operator 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 cocoercive 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 generalize some well-known results in recent literatures.

2000 Mathematics Subject Classification: 47H05; 49J40

Keywords

(A, η)-accretive mappingGeneralized resolvent operator techniqueGeneralized nonlinear relaxed cocoercive operator equation systemsNew perturbed iterative algorithm with errorsVariational graphical convergence

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 ) - ω u ( t ) , u ( 0 ) = u 0

for almost all t [0, T], where T (0,1) is fixed, ω R (see [1]). For more general details on approximation solvability of general nonlinear inclusion problems, we refer the reader to [218] and the references therein.

On the other hand, it is well known that variational inequalities and variational inclusions provide mathematical models to some problems arising in economics, mechanics, and engineering science and have been studied extensively. There are many methods to find solutions of variational inequality and variational inclusion problems. Among these methods, the resolvent operator technique is very important. For some literature, we recommend to the following example, and the reader [215, 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] introduced a new concept of (A, η)-accretive mappings, which provides a unifying framework for maximal monotone operators, m-accretive operators, η-subdifferential operators, maximal η-monotone operators, 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 operator 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 nonlinear 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 problems for Lipschitz mappings in Hilbert spaces, and Agarwal and Verma [2] introduced and studied a new system of nonlinear (set-valued) variational inclusions involving (A, η )-maximal relaxed monotone and relative (A, η)-maximal monotone mappings in Hilbert spaces and proved its approximation solvability 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 1 × 2 :

Find ( x , y ) 1 × 2 and u S ( x ) , v T ( y ) such that
p ( x ) = R η 1 , M 1 ( , x ) ρ λ 1 , A 1 ( 1 - λ 1 ) A 1 ( p ( x ) ) + λ 1 ( A 1 ( f ( y ) ) - ρ N 1 ( u , y ) + a ) , h ( y ) = R η 2 , M 2 ( y , ) ϱ λ 2 , A 2 ( 1 - λ 2 ) A 2 ( h ( y ) ) + λ 2 ( A 2 ( g ( x ) ) - ϱ N 2 ( x , v ) + b ) ,
(1.1)

where for all ( x , y ) 1 × 2 , R η 1 , M 1 ( , x ) ρ λ 1 , A 1 = ( A 1 + ρ λ 1 M 1 ( , x ) ) - 1 and R η 2 , M 2 ( y , ) ϱ λ 2 , A 2 = ( A 2 + ϱ λ 2 M 2 ( y , ) ) - 1 are two resolvent operators and two constants ρ , ϱ > 0 , N 1 : 1 × 2 1 , N 2 : 1 × 2 2 , p : 1 1 , h : 2 2 , f : 2 1 , g : 1 2 are single-valued operators, λ1, λ2 > 0 are two constants, ( a , b ) 1 × 2 is an any given element, and S : 1 2 1 , T : 2 2 2 , A i : i i , η i : i × i i , M i : i × i 2 i ( i = 1 , 2 ) are any nonlinear operators such that for all x 1 , M 1 ( , x ) : 1 2 1 is an (A1,η1)-accretive mapping and M 2 ( y , ) : 2 2 1 is an (A2, η2)-accretive mapping for all y 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 ) ) + ρ N 1 ( u , y ) + ρ M 1 ( p ( x ) , x ) , b A 2 ( h ( y ) ) - A 2 ( g ( x ) ) + ϱ N 2 ( x , v ) + ϱ M 2 ( y , h ( y ) )
(1.2)

Remark 1.1. For appropriate and suitable choices of 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 i = ( i = 1 , 2 ) , p = f = h = g , N 1 ( x , ) = N 2 ( , y ) = N ( ) and M1(,x) = M1(), M2(y,), = M2() for all ( x , y ) 1 × 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 ) ) + ρ N ( y ) + ρ M 1 ( g ( x ) ) , 0 A 2 ( g ( y ) ) - A 2 ( g ( x ) ) + ϱ N ( x ) + ϱ 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 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 i = ( i = 1 , 2 ) is a Hilbert space, a = b = 0 , S : 1 1 and T : 2 2 are two single-valued mappings, p = f = h = g = S = T = I is the identity operator and M1(,x) = M2(y,) = M() for all ( x , y ) 1 × 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 ) + ρ N ( y , x ) + ρ M ( x ) , 0 A 2 ( y ) - A 2 ( x ) + ϱ N ( x , y ) + ϱ M ( y ) ,
(1.4)
The system (1.4) was introduced and studied by Wang and Wu [18] and contains 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 such that
0 N ( x , x ) + M ( x ) ,

which was considered by Verma [17].

Example 1.4. When i = , λ i = 1 ( i = 1 , 2 ) , p = h , A 1 = A 2 = I , N 1 ( x , ) = N 2 ( , y ) = N ( ) and M1(,x) = M1(),N2(y,) = M2() for all ( x , y ) 1 × 2 , the system (1.1) becomes to the following nonlinear operator equation systems: Finding ( x , y ) × such that
h ( x ) = J M 1 ρ f ( y ) - ρ N ( y ) , h ( y ) = J M 2 ϱ g ( x ) - ϱ N ( x ) ,
(1.5)
where J M 1 ρ = ( I + ρ M 1 ) - 1 and J M 2 ϱ = ( I + ϱ 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 ) + ρ N ( y ) + ρ M 1 ( h ( x ) ) , 0 h ( y ) - g ( x ) + ϱ N ( x ) + ϱ 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 errors 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 existence 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 be a real Banach space with dual space * , , be the dual pair between and * , C B ( ) denote the family of all nonempty closed bounded subsets of , and 2 denote the family of all the nonempty subsets of . The generalized duality mapping J q : 2 * is defined by
J q ( x ) = f * * : x , f * = x q , f * = x q - 1 , x ,

where q > 1 is a constant. In particular, J2 is the usual normalized duality mapping. It is known that, in general, J q (x) = ||x||q- 2J2(x) for all x ≠ 0, and J q is single-valued if * is strictly convex. In the sequel, we always suppose that is a real Banach space such that J q is single-valued and is a Hilbert space. If = , then J2 becomes the identity mapping on .

The modulus of smoothness of is the function X : [ 0 , ) [ 0 , ) defined by
X ( t ) = sup 1 2 x + y + x - y - 1 : x 1 , y t .

A Banach space is called uniformly smooth if lim t 0 X ( t ) t = 0 .

is called q-uniformly smooth if there exists a constant c > 0 such that X ( t ) c t q , q > 1 . Remark that J q is single-valued if is uniformly smooth. In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu [21] proved the following result:

Lemma 2.1. Let be a real uniformly smooth Banach space. Then, is q-uniformly smooth if and only if there exists a constant c q > 0 such that for all x , y ,
x + y q x q + q y , J q ( x ) + c q y q .

In the sequel, we give some concept and lemmas needed later.

Definition 2.1. Let be a q-uniformly smooth Banach space and T , A : be two single-valued mappings. T is said to be
  1. (i)
    accretive if
    T ( x ) - T ( y ) , J q ( x - y ) 0 , x , y ;
     
  2. (ii)
    strictly accretive if T is accretive and
    T ( x ) - T ( y ) , J q ( x - y ) = 0
     
if and only if x = y;
  1. (iii)
    r-strongly accretive if there exists a constant r > 0 such that
    T ( x ) - T ( y ) , J q ( x - y ) r x - y q , x , y ;
     
  2. (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 ) ) γ x - y q , x , y ;
     
  3. (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 ) ) - m T ( x ) - T ( y ) q , x , y ;
     
  4. (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 ) ) - π x - y q + ι T ( x ) - T ( y ) q , x , y ;
     
  5. (vii)
    s-Lipschitz continuous if there exists a constant s > 0 such that
    T ( x ) - T ( y ) s x - y , x , y .
     

In a similar way, we can define (relaxed) cocoercivity and Lipschitz continuity of the operator N ( , ) : × in the first and second arguments.

Remark 2.1. (1) The notion of the cocoercivity is applied in several directions, 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 cocoercivity. Several classes of relaxed cocoercive variational inequalities and variational inclusions have been studied in [2, 5, 710, 12, 1618].
  1. (2)

    When = , (i)-(iv) of Definition 2.1 reduce to the definitions of monotonicity, strict monotonicity, strong monotonicity, and strong monotonicity with respect to A, respectively (see [3, 18]).

     
Definition 2.2. A single-valued mapping η : × is said to be τ-Lipschitz continuous if there exists a constant τ > 0 such that
η ( x , y ) τ x - y , x , y .
Definition 2.3. Let be a q-uniformly smooth Banach space, η : × and A , H : be single-valued mappings. Then set-valued mapping M : 2 is said to be
  1. (i)
    η-accretive if
    u - v , J q ( η ( x , y ) ) 0 , x , y , u M ( x ) , v M ( y ) ;
     
  2. (ii)
    r-strongly η-accretive if there exists a constant r > 0 such that
    u - v , J q ( η ( x , y ) ) r x - y q , x , y , u M ( x ) , v M ( y ) ;
     
  3. (iii)
    m-relaxed η-accretive if there exists a constant m > 0 such that
    u - v , J q ( η ( x , y ) ) - m x - y q , x , y , u M ( x ) , v M ( y ) ;
     
  4. (iv)
    ξ - H ^ -Lipschitz continuous, if there exists a constant ξ > 0 such that
    H ^ ( M ( x ) , M ( y ) ) ξ x - y , x , y ,
     
where H ^ is the Hausdorff metric on C B ( ) ;
  1. (v)

    (A, η)-accretive if M is m-relaxed η-accretive and ( A + ρ M ) ( ) = 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 : be a strictly η-accretive mapping and M : 2 be an (A, η)-accretive mapping. The resolvent operator R η , M ρ , A : is defined by:
R η , M ρ , A ( u ) = ( A + ρ M ) - 1 ( u ) , u .

Remark 2.3. The resolvent operators associated with (A, η)-accretive mappings include as special cases the corresponding resolvent operators associated 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 operators, η-subdifferential operators, the classical m-accretive, and maximal monotone operators. See, for example, [1, 7, 8, 13] and the references therein.

Lemma 2.2. ([7]) Let be a q-uniformly smooth Banach space and η : × be τ-Lipschitz continuous, A : be a r-strongly η-accretive mapping and M : 2 be an (A, η)-accretive mapping. Then, the resolvent operator R η , M ρ , A : is τ q - 1 r - ρ m -Lipschitz continuous, i.e.,
R η , M ρ , A ( x ) - R η , M ρ , A ( y ) τ q - 1 r - ρ m x - y , x , y ,

where ρ 0 , r m is a constant.

Definition 2.5. Let M n , M : 2 be (A, η)-accretive mappings on for n = 0,1, 2,.... Let A : 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 : 2 be (A, η)-accretive mappings on for n = 0,1, 2,.... Then, the sequence M n A - G M if and only if
R η , M n ρ , A ( x ) R η , M ρ , A ( x ) , x ,

where R η , M ρ , A = ( A + ρ M n ) - 1 , R η , M ρ , A = ( A + ρ M ) - 1 , ρ > 0 is a constant, and A : 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 generated 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 η 1 , M 1 ( , x ) ρ λ 1 , A 1 ( s ) , s = ( 1 - λ 1 ) A 1 ( p ( x ) ) + λ 1 ( A 1 ( f ( y ) ) - ρ N 1 ( u , y ) + a ) , h ( y ) = R η 2 , M 2 ( y , ) ϱ λ 2 , A 2 ( t ) , t = ( 1 - λ 2 ) A 2 ( h ( y ) ) + λ 2 ( A 2 ( g ( x ) ) - ϱ 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 ) 1 × 2 , take u0 S(x0) and v0 T(y0).

Step 2. Choose sequences { d n } 1 and { e n } 2 are two error sequences to take into account a possible inexact computation of the operator 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 ) } 1 × 2 × 1 × 2 satisfy
s n = ( 1 - λ 1 ) A 1 ( p ( x n ) ) + λ 1 ( A 1 ( f ( y n ) ) - ρ N 1 ( u n , y n ) + a ) , t n = ( 1 - λ 2 ) A 2 ( h ( y n ) ) + λ 2 ( A 2 ( g ( x n ) ) - ϱ N 2 ( x n , v n ) + b ) , x n + 1 = ( 1 - k ) x n + k { x n - p ( x n ) + R η 1 , M 1 n ( , x n ) ρ λ 1 , A 1 ( s n ) } + d n , y n + 1 = ( 1 - κ ) y n + κ { y n - h ( y n ) + R η 2 , M 2 n ( y n , ) ρ λ 2 , A 2 ( t n ) } + e n ,
(3.1)

where R η 1 , M 1 n ( , x ) ρ λ 1 , A 1 = ( A 1 + ρ λ 1 M 1 n ( , x ) ) - 1 , R η 2 , M 2 n ( y , ) ϱ λ 2 , A 2 = ( A 2 + ϱ λ 2 M 2 n ( y , ) ) - 1 , λ 1 , λ 2 , ρ , ϱ are nonnegative constants and k, κ (0, 1] are size constants.

Step 4. Choose un+1 S(xn+1) and vn+1 T(yn+1) such that (see [22])
u n - u n + 1 1 + 1 n + 1 H ^ S ( x n ) , S ( x n + 1 ) , v n - v n + 1 1 + 1 n + 1 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 convergence of Algorithm 3.1.

Theorem 3.1. For i = 1, 2, let 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:

(H1) 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;

(H2) p is δ1-strongly accretive and l p -Lipschitz continuous, h is δ2-strongly accretive and l h -Lipschitz continuous, f is l f -Lipschitz continuous and g is l g -Lipschitz continuous, S : 1 C B ( 1 ) is ξ - H ^ -Lipschitz continuous and T : 2 C B ( 2 ) is ζ - H ^ -Lipschitz continuous;

(H3) N1 is (π1, ι1)-relaxed cocoercive with respect to f1 and ϖ2-Lipschitz continuous in the second argument, and N2 is (π2, ι2)-relaxed cocoercive with respect to g2 and ϖ1-Lipschitz continuous in the first argument, and N1 is β1-Lipschitz continuous in the first variable, and N2 is β2-Lipschitz continuous in the second variable, where f 1 : 2 1 is defined by f1(y) = A1 f(y) = A1(f(y)) for all y 2 and g 2 : 1 2 is defined by g2(x) = A2 g(x) = A2(g(x)) for all x 1 ;

(H4) for n = 0 , 1 , 2 , , M i n : i × i 2 i ( i = 1 , 2 ) are any nonlinear operators such that for all x 1 , M 1 n ( , x ) : 1 2 1 is an (A1, η1)-accretive mapping with M 1 n ( , x ) A 1 - G M 1 ( , x ) , and M 2 n ( y , ) : 2 2 2 is an (A2, η 2 )-accretive mapping with M 2 n ( y , ) A 2 - G M 2 ( y , ) for all y 2 , respectively;

(H5) there exist constants ν i (i = 1,2), ρ (0, r1/m1) and ϱ ( 0 , r 2 / m 2 ) such that
R η 1 , M 1 ( , x ) ρ λ 1 , A 1 ( z ) - R η 1 , M 1 ( , y ) ρ λ 1 , A 1 ( z ) R η 2 , M 2 ( x , ) ϱ λ 2 , A 2 ( z ) - R η 2 , M 2 ( y , ) ρ λ 2 , A 2 ( z ) ν 2 x - y , x , y , z 1 , ν 1 x - y , x , y , z 2 ,
(3.3)
and
ν 2 + 1 - q 1 δ 1 + c q 1 l p q 1 q 1 + τ 1 q 1 - 1 [ ( 1 - λ 1 ) σ 1 l p + ρ λ 1 β 1 ξ ] r 1 - ρ λ 1 m 1 + κ λ 2 τ 2 q 2 - 1 σ 2 q 2 l g q 2 - q 2 ϱ ι 2 ϖ 1 q 2 + q 2 ϱ π 2 + c q 2 ϱ q 2 ϖ 1 q 2 q 2 k ( r 2 - ϱ λ 2 m 2 ) < 1 , ν 1 + 1 - q 2 δ 2 + c q 2 l h q 2 q 2 + τ 2 q 2 - 1 [ ( 1 - λ 2 ) σ 2 l h + ϱ λ 2 β 2 ζ ] r 2 - ϱ λ 2 m 2 + k λ 1 τ 1 q 1 - 1 σ 1 q 1 l f q 1 - q 1 ρ ι 1 ϖ 2 q 1 + q 1 ρ π 1 + c q 1 ρ q 1 ϖ 2 q 1 q 1 κ ( r 1 - ρ λ 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 * ) 1 × 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 Algorithm 3.1.

Proof. Define ||||* on 1 × 2 by
( x , y ) * = x + y , ( x , y ) 1 × 2 .
It is easy to see that 1 × 2 , * is a Banach space. By the assumptions for relaxed cocoercivity and Lipschitz continuity of N with respect to both arguments, strongly accretivity of p and h, and Lipschitz continuity of S, T, p, f, g and h, Lemmas 2.1 and 2.2, and (3.1)-(3.3), now we know that
N 1 ( u n , y n - 1 ) - N 1 ( u n - 1 , y n - 1 ) β 1 u n - u n - 1 β 1 ( 1 + n - 1 ) H ^ ( S ( x n ) , S ( x n - 1 ) ) β 1 ξ ( 1 + n - 1 ) x n - x n - 1 , A 1 ( f ( y n ) ) - A 1 ( f ( y n - 1 ) ) - ρ N 1 ( u n , y n ) - N 1 ( u n , y n ) q 1 A 1 ( f ( y n ) ) - A 1 ( f ( y n - 1 ) ) q 1 + ρ q 1 c q 1 N 1 ( u n , y n ) - N 1 ( u n , y n - 1 ) q 1 - q 1 ρ N 1 ( u n , y n ) - N 1 ( u n , y n - 1 ) , J q 1 ( A 1 ( f ( y n ) ) - A 1 ( f ( y n - 1 ) ) ) σ 1 q 1 l f q 1 - q 1 ρ ι 1 ϖ 2 q 1 + q 1 ρ π 1 + c q 1 ρ q 1 ϖ 2 q 1 y n - y n - 1 q 1 , x n - x n - 1 - p ( x n ) - p ( x n - 1 ) 1 - q 1 δ 1 + c q 1 l p q 1 q 1 x n - x n - 1 ,
and
s n - s n - 1 = ( 1 - λ 1 ) A 1 ( p ( x n ) ) + λ 1 ( A 1 ( f ( y n ) ) - ρ N 1 ( u n , y n ) + a ) - ( 1 - λ 1 ) A 1 ( p ( x n - 1 ) ) - λ 1 ( A 1 ( f ( y n - 1 ) ) - ρ N 1 ( u n - 1 , y n - 1 ) + a ) ( 1 - λ 1 ) A 1 ( p ( x n ) ) - A 1 ( p ( x n - 1 ) ) + ρ λ 1 N 1 ( u n , y n - 1 ) - N 1 ( u n - 1 , y n - 1 ) + λ 1 A 1 ( f ( y n ) ) - A 1 ( f ( y n - 1 ) ) - ρ [ N 1 ( u n , y n ) - N 1 ( u n , y n - 1 ) ] ( 1 - λ 1 ) σ 1 l p + ρ λ 1 β 1 ξ ( 1 + n - 1 ) x n - x n - 1 + λ 1 σ 1 q 1 l f q 1 - q 1 ρ ι 1 ϖ 2 q 1 + q 1 ρ π 1 + c q 1 ρ q 1 ϖ 2 q 1 q 1 y n - y n - 1 ,
x n + 1 - x n ( 1 - k ) x n - x n - 1 + k x n - x n - 1 - p ( x n ) - p ( x n - 1 ) + k R η 1 , M 1 n ( , x n ) ρ λ 1 , A 1 ( s n ) - R η 1 , M 1 ( , x n ) ρ λ 1 , A 1 ( s n ) + d n - d n - 1 + k R η 1 , M 1 ( , x n ) ρ λ 1 , A 1 ( s n ) - R η 1 , M 1 ( , x n - 1 ) ρ λ 1 , A 1 ( S n ) + k R η 1 , M 1 ( , x n - 1 ) ρ λ 1 , A 1 ( s n ) - R η 1 , M 1 ( , x n - 1 ) ρ λ 1 , A 1 ( s n - 1 ) + k R η 1 , M 1 n - 1 ( , x n - 1 ) ρ λ 1 , A 1 ( s n - 1 ) - R η 1 , M 1 ( , x n - 1 ) ρ λ 1 , A 1 ( s n - 1 ) ( 1 - k ) x n - x n - 1 + k x n - x n - 1 - [ p ( x n ) - p ( x n - 1 ) ] + k ν 2 x n - x n - 1 + k τ 1 q 1 - 1 r 1 - ρ λ 1 m 1 s n - s n - 1 + k ( ε n + ε n - 1 ) + d n - d n - 1 [ 1 - k ( 1 - θ 1 n ) ] x n - x n - 1 + k ϑ 1 y n - y n - 1 + k ( ε n + ε n - 1 ) + d n - d n - 1 ,
(3.5)
where ε l = R η 1 , M 1 l ( , x l ) ρ λ 1 , A 1 ( s l ) - R η 1 , M 1 ( , x l ) ρ λ 1 , A 1 ( s l ) for l = n - 1, n and
θ 1 , n = ν 2 + 1 - q 1 δ 1 + c q 1 l p q 1 q 1 + τ 1 q 1 - 1 ( 1 - λ 1 ) σ 1 l p + ρ λ 1 β 1 ξ ( 1 + n - 1 ) r 1 - ρ λ 1 m 1 , ϑ 1 = λ 1 τ 1 q 1 - 1 σ 1 q 1 l f q 1 - q 1 ρ ι 1 ϖ 2 q 1 + q 1 ρ π 1 + c q 1 ρ q 1 ϖ 2 q 1 q 1 r 1 - ρ λ 1 m 1
Similarly, we get
y n + 1 - y n 1 - κ ( 1 - θ 2 n ) y n - y n - 1 + κ ϑ 2 x n - x n - 1 + κ ( ε n + ε n - 1 ) + e n - e n - 1 ,
(3.6)
where ε l = R η 2 , M 2 l ( y l , ) ϱ λ 2 , A 2 ( t l ) - R η 2 , M 2 ( y l , ) ϱ λ 2 , A 2 ( t l ) for l = n - 1, n and
θ 2 , n = ν + 1 - q 2 δ 2 + c q 2 l h q 2 q 2 + τ 2 q 2 - 1 [ ( 1 - λ 2 ) σ 2 l h + ϱ λ 2 β 2 ζ ( 1 + n - 1 ) ] r 2 - ϱ λ 2 m 2 , ϑ 2 = λ 2 τ 2 q 2 - 1 σ 2 q 2 l g q 2 - q 2 ϱ ι 2 ϖ 1 q 2 + q 2 ϱ π 2 + c q 2 ϱ q 2 ϖ 1 q 2 q 2 r 2 - ϱ λ 2 m 2 .
follows from (3.5) and (3.6) that
x n + 1 - x n + y n + 1 - y n θ n x n - x n - 1 + y n - y n - 1 + k ( ε n + ε n - 1 ) + κ ( ε n + ε n - 1 ) + d n - d n - 1 + e n - e n - 1 ,
(3.7)
where
θ n = max { 1 + κ ϑ 2 - k ( 1 - θ 1 , n ) , 1 + k ϑ 1 - κ ( 1 - θ 2 , n ) } .
Let
θ = max { 1 + κ ϑ 2 - k ( 1 - θ 1 ) , 1 + k ϑ 1 - κ ( 1 - θ 2 ) } ,
where
θ 1 = ν 2 + 1 - q 1 δ 1 + c q 1 l p q 1 q 1 + τ 1 q 1 - 1 [ ( 1 - λ 1 ) σ 1 l p + ρ λ 1 β 1 ξ ] r 1 - ρ λ 1 m 1 , θ 2 = ν 1 + 1 - q 2 δ 2 + c q 2 l h q 2 q 2 + τ 2 q 2 - 1 [ ( 1 - λ 2 ) σ 2 l h + ϱ λ 2 β 2 ζ ] r 2 - ϱ λ 2 m 2 .

Then, we know that θ n θ as n → ∞.

From the condition (3.4), we know that 0 < θ < 1 and so there exist n0 > 0 and θ0 (θ, 1) such that θ n θ0 for all nn0. Therefore, by (3.7), we have
( x n + 1 , y n + 1 ) - ( x n , y n ) * θ 0 ( x n , y n ) - ( x n - 1 , y n - 1 ) * + d n - d n - 1 + e n - e n - 1 + k ( ε n + ε n - 1 ) + κ ( ε n + ε n - 1 )
θ 0 n - n 0 ( x n 0 + 1 , y n 0 + 1 ) - ( x n 0 , y n 0 ) * + i = 1 n - n 0 θ 0 i - 1 ς n - ( i - 1 )
(3.8)
where ς n = ||d n - dn- 1|| + ||e n - en- 1|| + k n + ϵn- 1) + κ(ε n + εn-1) for all nn0. By (3.8), for any mn > n0, we have
( x m , y m ) - ( x n , y n ) * j = n m - 1 x j + 1 - x j + y j + 1 - y j j = n m - 1 θ 0 j - n 0 ( x n 0 + 1 , y n 0 + 1 ) - ( x n 0 , y n 0 ) * + j = n m - 1 i = 1 j - n 0 θ 0 i - 1 ς j - ( i - 1 ) .
(3.9)
It follows from the hypothesis of Algorithm 3.1, Lemma 2.3 and (3.9) that
lim n ( x m , y m ) - ( x n , y n ) * = 0 .

Hence, {(x n , y n )} is a Cauchy sequence, i.e., there exists ( x * , y * ) 1 × 2 such that (x n , y n ) → (x*, y*) as n → ∞.

Next, we prove that u n u* S(x*) and v n v* T(y*). In fact, because
u n - u n - 1 ( 1 + n - 1 ) H ^ S ( x n ) , S ( x n - 1 ) ξ ( 1 + n - 1 ) x n - x n - 1 ,
it follows that {u n } is also Cauchy sequence in 1 . Let u n u*. In the sequel, we will show that u* S(x*). Noting u n S(x n ), from the results in [22], we have
d ( u * , S ( x * ) ) = inf u n - y : y S ( x * ) u * - u n + d ( u n , S ( x * ) ) u * - u n + H ^ S ( x n ) , S ( x * ) u * - u n + ξ x n - x * 0 .

Hence d(u*,S(x*)) = 0 and therefore u* S(x*). Similarly, we have v n v* T(y*). By continuity and the hypothesis of Algorithm 3.1, we know that (x*, y*, u*, v*) satisfies the Equation (1.1). This completes the proof.

Remark 3.1. We note that Hilbert space and L p (or l p ) (2 ≤ p < ∞) spaces are 2-uniformly smooth Banach spaces and if i ( i = 1 , 2 ) is 2-uniformly smooth Banach space, we can choose constants ν i , λ i (i = 1,2), ρ and ϱ such that (3.4) hold. See, for example, [218] and the references therein.

Remark 3.2. Condition (3.4) of Theorem 3.1 holds for some suitable value of constants, for example, q1 = q2 = 2, c2 = 1, ν1 = ν2 = 0.021 = δ2 = 0.3, l p = l h = 0.61 = τ2 = 0.05, λ1 = λ2 = 0.01, σ1 = σ2 = 0.5, ρ = ϱ = 0 . 1 , β1 = β2 = 0.05, ξ = 0.7, ζ = 0.4, r1 = r2 = 0.3, m1 = m2 = 0.2, k = κ = 0.5, l f = 0.2, l g = 0.4, ι1 = ι2 = 0.05, ϖ1 = ϖ2 = 0.05 and π1 = π2 = 0.2.

From Theorem 3.1, we have the following results as an application of Theorem 3.1.

Theorem 3.2. Assume that is a real Hilbert space and the following conditions hold:

(H1) h : is δ-strongly monotone and l h -Lipschitz continuous, f : is l f -Lipschitz continuous and g : is l g -Lipschitz continuous;

(H2) N : is (π1, ι1)-relaxed cocoercive with respect to f and ϖ-Lipschitz continuous, and (π2, ι2)-relaxed cocoercive with respect to g;

(H3) for i = 1,2 and n = 0 , 1 , 2 , . . . , M i n , M i : 2 are maximal monotone operators with M i n A 1 - G M i ; ;

(H4) there exist positive constants ρ and ϱ such that
1 - 2 δ + l h 2 < min 1 - κ k l g 2 - 2 ϱ ι 2 ϖ 2 + 2 ϱ π 2 + ϱ 2 ϖ 2 , 1 - k κ l f 2 - 2 ρ ι 1 ϖ 2 + 2 ρ π 1 + ρ 2 ϖ 2 .

Then, the iterative sequences {(x n , y n )} generated as follows converges strongly to the common solution (x*, y*) of the system (1.5):

For any given ( x 0 , y 0 ) × , define an iterative sequence as follows:
{ x n + 1 = ( 1 k ) x n + k { x n h ( x n ) + J M 1 n ρ [ f ( y n ρ N ( y n ) ] } + d n , y n + 1 = ( 1 κ ) y n + κ { y n h ( y n ) + J M 2 n ϱ [ g ( x n ) ϱ N ( x n ) ] } + e n ,
(3.10)
where J M 1 n ρ = ( I + ρ M 1 n ) - 1 , J M 2 n ϱ = ( I + ϱ M 2 n ) - 1 , ρ , ϱ > 0 , k , κ ( 0 , 1 ) , { d n } and { e n } are two error sequences to take into account a possible inexact computation of the operator 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 < .

Proof. By the nonexpansivity of the resolvent operators associated withmaximal monotone operators and the proof of Theorem 3.1, one can derivethe result.

Remark 3.3. We 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 in this paper improve and generalize the corresponding results of [2, 3, 5, 12, 14, 17, 18] and many other recent works.

Declarations

Acknowledgements

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 and Engineering (2011PY01) and the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050). The authors are grateful to the editor and referee for valuable comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University of Science and Engineering
(2)
Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University

References

  1. Zeidler E: Nonlinear functional analysis and its applications. Volume I. Springer, New York; 1986.View ArticleGoogle Scholar
  2. Agarwal RP, Verma RU: General implicit variational inclusion problems based on A- maximal ( m )-relaxed monotonicity (AMRM) frameworks. Appl Math Comput 2009, 215: 367–379. 10.1016/j.amc.2009.04.078MathSciNetView ArticleGoogle Scholar
  3. Fang YP, Huang NJ: H -accretive operators and resolvent operator technique for solving variatonal inclusionsin Banach spaces. Appl Math Lett 2004, 17(6):647–653. 10.1016/S0893-9659(04)90099-7MathSciNetView ArticleGoogle Scholar
  4. He XF, Lou J, He Z: Iterative methods for solving variational inclusions in Banach spaces. J Comput Appl Math 2007, 203(1):80–86. 10.1016/j.cam.2006.03.011MathSciNetView ArticleGoogle Scholar
  5. Jin MM: Iterative algorithms for a new system of nonlinear vriational inclusions with ( A, η )-accretive mappings in Banach spaces. Comput Math Appl 2007, 54: 579–588. 10.1016/j.camwa.2006.12.030MathSciNetView ArticleGoogle Scholar
  6. Kazmi KR, Bhat MI: Iterative algorithm for a system of nonlinear variational-like inclusions. Comput Math Appl 2004, 48(12):1929–1935. 10.1016/j.camwa.2004.02.009MathSciNetView ArticleGoogle Scholar
  7. Lan HY, Cho YJ, Verma RU: On nonlinear relaxed cocoercive variational inclusions involving ( A, η )-accretive mappings in Banach spaces. Comput Math Appl 2006, 51(9–10):1529–1538. 10.1016/j.camwa.2005.11.036MathSciNetView ArticleGoogle Scholar
  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 2007, 25(1–2):255–267. 10.1007/BF02832351MathSciNetView ArticleGoogle Scholar
  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 2009, 71(12):6194–6201. 10.1016/j.na.2009.06.012MathSciNetView ArticleGoogle Scholar
  10. Li HG, Xu AJ, Jin MM: A hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving ( A , η )-accretive mappings. Fixed Point Theory Appl 2010, 2010: 24. Article ID 635382Google Scholar
  11. Liou YC: An iterative algorithm for mixed equilibrium problems and variational inclusions approach to variational inequalities. Fixed Point Theory Appl 2010, 2010: 15. Article ID 564361View ArticleGoogle Scholar
  12. Noor MA, Noor KI, Al-Said E: Resolvent iterative methods for solving system of extended general variational inclusions. J Inequal Appl 2011, 2011: 10. Article ID 371241 10.1186/1029-242X-2011-10MathSciNetView ArticleGoogle Scholar
  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, 2007: 20. Article ID 93678MathSciNetView ArticleGoogle Scholar
  14. Petrot N: A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems. Appl Math Lett 2010, 23(4):440–445. 10.1016/j.aml.2009.12.001MathSciNetView ArticleGoogle Scholar
  15. Tan JF, Chang SS: Iterative algorithms for finding common solutions to variational inclusion equilibrium and fixed point problems. Fixed Point Theory Appl 2011, 2011: 17. Article ID 915629 10.1186/1687-1812-2011-17MathSciNetView ArticleGoogle Scholar
  16. Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. J Optim Theory Appl 2004, 121: 203–210.MathSciNetView ArticleGoogle Scholar
  17. Verma RU: A -monotone nonlinear relaxed cocoercive variational inclusions. Central Eur J Math 2007, 5(2):386–396. 10.2478/s11533-007-0005-5View ArticleGoogle Scholar
  18. Wang Z, Wu C: A system of nonlinear variational inclusions with ( A , η )-monotone mappings. J Inequal Appl 2008, 2008: 6. Article ID 681734Google Scholar
  19. Clarke FH: Optimization and Nonsmooth Analysis. Wiley, New York; 1983.Google Scholar
  20. Verma RU: A generalization to variational convergence for operators. Adv Nonlinear var Inequal 2008, 11(2):97–101.MathSciNetGoogle Scholar
  21. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal 1991, 16(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar
  22. Nadler SB: Muliti-valued contraction mappings. Pac J Math 1969, 30: 475–488.MathSciNetView ArticleGoogle Scholar

Copyright

© Li et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.