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New general systems of set-valued variational inclusions involving relative (A,η)-maximal monotone operators in Hilbert spaces

Abstract

The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative (A,η)-maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative (A,η)-maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.

1 Introduction

Recently, some systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering science and other pure and applied sciences. Among these methods, the resolvent operator technique is very important. Huang and Fang [1] introduced a system of order complementarity problems and established some existence results for the system using fixed point theory. Verma [2] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of the systems of variational inequalities. Cho et al. [3] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. Further, the authors proved some existence and uniqueness theorems of solutions for the systems, and also constructed some iterative algorithms for approximating the solution of the systems of nonlinear variational inequalities, respectively.

Moreover, Fang et al. [4], Yan et al. [5], Fang and Huang [6] introduced and studied some new systems of variational inclusions involving H-monotone operators and (H,η)-monotone operators in Hilbert space, respectively. Using the corresponding resolvent operator technique associated with H-monotone operators, (H,η)-monotone operators, the authors proved the existence of solutions for the variational inclusion systems and constructed some algorithms for approximating the solutions of the systems and discussed convergence of the iteration sequences generated by the algorithms, respectively. Very recently, Lan et al. [7] introduced and studied a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces. By using the concept and properties of A-monotone operators, and the resolvent operator technique associated with A-monotone operators due to Verma [8], the authors constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions with A-monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. For some related work, see, for example, [132] and the references therein.

On the other hand, Cao [33] introduced and studied a new system of generalized quasi-variational-like-inclusions applying the η-proximal mapping technique. Further, Agarwal and Verma [34] introduced and studied relative (A,η)-maximal monotone operators and discussed the approximation solvability of a new system of nonlinear (set-valued) variational inclusions involving (A,η)-maximal relaxed monotone and relative (A,η)-maximal monotone operators in Hilbert spaces based on a generalized hybrid iterative algorithm and the general (A,η)-resolvent operator method.

Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative (A,η)-maximal monotone operators in Hilbert spaces: Find ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m and u i j U i j ( x j ) for any i,j=1,2,,m such that

0 F i ( u i 1 , u i 2 , , u i m ) + M i ( g i ( x i ) ) ,
(1.1)

where m is a given positive integer, F i : H 1 × H 2 ×× H m H i , A i : H i H i , g i : H i H i and η i : H i × H i H i are single-valued operators, U i j : H j 2 H j is a set-valued operator and M i : H i 2 H i is relative ( A i , η i )-maximal monotone.

We note that for appropriate and suitable choices of positive integer m, the operators F i , g i , A i , η i , M i , U i j , and H i for i,j=1,2,,m, one can know that the problem (1.1) includes a number of known general problems of variational character, including variational inequality (system) problems, variational inclusion (system) problems as special cases. For more details, see [131, 35] and the following examples.

Example 1.1 For i,j=1,2,,m, if U i j = T i j is single-valued operator, the problem (1.1) reduces to finding x j H j , such that

0 F i ( T i 1 x 1 , T i 2 x 2 , , T i m x m ) + M i ( g i ( x i ) ) .
(1.2)

Example 1.2 For i=1,2,,m, if H i =H and A i I, an identity operator, and M i = φ i , where φ i :HR{+} is proper and lower semi-continuous η i -subdifferentiable functional and φ i denotes η i -subdifferential operator, then the problem (1.1) reduces to finding x i H and u i j U i j ( x j ) for j=1,2,,m such that

F i ( u i 1 , u i 2 , , u i m ) , η i ( x , g i ( x i ) ) φ i ( g i ( x i ) ) φ i (x),xH.
(1.3)

The problem (1.3) is called a set-valued nonlinear generalized quasi-variational-like-inclusion system, which was considered and studied by Cao [33].

Example 1.3 When m=2 and g i I for i=1,2, then the problem (1.1) is equivalent to the following nonlinear set-valued variational inclusion system problem: Find ( x 1 , x 2 ) H 1 × H 2 and u 1 U 1 ( x 1 ), u 2 U 2 ( x 2 ) such that

0 F 1 ( x 1 , u 2 ) + M 1 ( x 1 ) , 0 F 2 ( u 1 , x 2 ) + M 2 ( x 2 ) ,
(1.4)

which was studied by Agarwal and Verma [34].

Example 1.4 If m=2 and M i ( x i )= φ i ( x i ), where φ i : H i R{+} is proper, convex, and lower semi-continuous functional and φ i denotes the subdifferential operator of φ i for all x i H i , i=1,2, then the problem (1.4) reduces to the following system of set-valued mixed variational inequalities: Find ( x 1 , x 2 ) H 1 × H 2 , u 1 U 1 ( x 1 ) and u 2 U 2 ( x 2 ) such that

F 1 ( x 1 , u 2 ) , x x 1 + φ 1 ( x ) φ 1 ( x 1 ) 0 , x H 1 , F 2 ( u 1 , x 2 ) , y x 2 + φ 2 ( y ) φ 2 ( x 2 ) 0 , y H 2 .
(1.5)

If U 1 = U 2 I, then the problem (1.5) reduces to finding ( x 1 , x 2 ) H 1 × H 2 such that

F 1 ( x 1 , x 2 ) , x x 1 + φ 1 ( x ) φ 1 ( x 1 ) 0 , x H 1 , F 2 ( x 1 , x 2 ) , y x 2 + φ 2 ( y ) φ 2 ( x 2 ) 0 , y H 2 ,
(1.6)

which is called the system of nonlinear variational inequalities considered by Cho et al. [3]. Some specializations of the problem (1.6) are dealt by Kim and Kim [35].

Example 1.5 If m=2 and U 1 = U 2 = g 1 = g 2 I, then the problem (1.1) reduces to the problem of finding ( x 1 , x 2 ) H 1 × H 2 such that

0 F 1 ( x 1 , x 2 ) + M 1 ( x 1 ) , 0 F 2 ( x 1 , x 2 ) + M 2 ( x 2 ) ,

which was introduced and studied by Fang et al. [4].

Moreover, by using the generalized resolvent operator technique associated with relative (A,η)-maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove convergence of the sequences generated by the algorithms.

2 Preliminaries

Throughout, let H and H i (i=1,2,,m) be real Hilbert spaces and endowed with the norm and inner product ,. Let 2 H and C(H) denote the family of all the nonempty subsets of H and the family of all closed subsets of H, respectively.

Definition 2.1 Let T:HH be a single-valued operator. Then the map T is said to be

  1. (i)

    r-strongly monotone, if there exists a constant r>0 such that

    T ( x ) T ( y ) , x y r x y 2 ,x,yH;
  2. (ii)

    β-Lipschitz continuous, if there exists a constant β>0 such that

    TxTyβxy,x,yH.

Definition 2.2 Let η:H×HH and A:HH be single-valued operators, M:H 2 H be set-valued operator. Then

  1. (i)

    η is said to be t-strongly monotone, if there exists a constant t>0 such that

    η ( x , y ) , x y t x y 2 ,x,yH;
  2. (ii)

    η is said to be τ-Lipschitz continuous, if there exists a constant τ>0 such that

    η ( x , y ) τxy,x,yH;
  3. (iii)

    A is said to be η-monotone, if

    A ( x ) A ( y ) , η ( x , y ) 0,x,yH;
  4. (iv)

    A is said to be strictly η-monotone, if A is η-monotone and

    A ( x ) A ( y ) , η ( x , y ) =0if and only ifx=y;
  5. (v)

    A is said to be (r,η)-strongly monotone, if there exists a constant r>0 such that

    A ( x ) A ( y ) , η ( x , y ) r x y 2 ,x,yH;
  6. (vi)

    M is said to be η-monotone with respect to A (or relative (A,η)-monotone) if

    u v , η ( A ( x ) , A ( y ) ) 0,x,yH,uM(x),vM(y);
  7. (vii)

    M is said to be relative (A,η)-maximal monotone, if M is η-monotone with respect to A (or relative (A,η)-monotone) and (A+λM)(H)=H, where λ>0 is an arbitrary constant.

Definition 2.3 For i,j=1,2,,m, let H i be a Hilbert space, A j : H j H j be single-valued operator, U i j : H j 2 H j be set-valued operator. Then nonlinear operator F i : H 1 × H 2 ×× H m H i is said to be

  1. (i)

    ( U i j , c j , μ j )-relaxed cocoercive with respect to A j (or relative ( U i j , c j , μ j )-relaxed cocoercive) in the j th argument, if there exist constants c j , μ j >0 such that for all x j 1 , x j 2 H j , and for any u j 1 U i j ( x j 1 ), u j 2 U i j ( x j 2 ),

    F i ( , u j 1 , ) F i ( , u j 2 , ) , A j ( x j 1 ) A j ( x j 2 ) ( c j ) F i ( , u j 1 , ) F i ( , u j 2 , ) 2 + μ j x j 1 x j 2 2 ;
  2. (ii)

    ζ i j -Lipschitz continuous in the j th argument, if there exists constant ζ i j >0 such that for all x j , y j H j ,

    F i ( x 1 , , x j 1 , x j , x j + 1 , , x m ) F i ( x 1 , , x j 1 , y j , x j + 1 , , x m ) x j y j .

Remark 2.1

  1. (i)

    When m=1 and U=I, then (i) and (ii) of Definition 2.3 reduce to corresponding concept of the relative relaxed cocoerciveness and Lipschitz continuity, respectively.

  2. (ii)

    If U i j = T i j is single-valued operator for i,j=1,2,,m, then F i is ( U i j , c j , μ j )-relaxed cocoercive with respect to A j in the j th argument reduce to ( T i j , c j , μ j )-relaxed cocoercive with respect to A j in the j th argument, that is, if there exist constants c j , μ j >0 such that for all x j 1 , x j 2 H j ,

    F i ( , T i j x j 1 , ) F i ( , T i j x j 2 , ) , A j ( x j 1 ) A j ( x j 2 ) ( c j ) F i ( , T i j x j 1 , ) F i ( , T i j x j 2 , ) 2 + μ j x j 1 x j 2 2 .

Lemma 2.1 ([34])

Let η:H×HH be a single-valued mapping, A:HH be a strictly η-monotone mapping and M:H 2 H be a relative (A,η)-maximal monotone mapping. Then the mapping (A+λM) is single-valued, where λ>0 is arbitrary constant.

Definition 2.4 Let η:H×HH be a single-valued mapping, A:HH be a strictly η-monotone mapping and M:H 2 H be a relative (A,η)-maximal monotone mapping. Then generalized resolvent operator R M , λ A , η :HH is defined by

R M , λ A , η (z)= ( A + λ M ) 1 (z),zH,

where λ>0 is a constant.

Lemma 2.2 ([34])

Let η:H×HH be a t-strongly monotone and τ-Lipschitz continuous mapping, A:HH be an r-strongly monotone mapping, and M:H 2 H be a relative (A,η)-maximal monotone mapping. Then generalized resolvent operator R M , λ A , η :HH is τ r t -Lipschitz continuous, that is,

R M , λ A , η ( x ) R M , λ A , η ( y ) τ r t xy,x,yH.

Definition 2.5 A set-valued operator U:H 2 H is said to be D-γ-Lipschitz continuous, if there exists a constant γ>0 such that

D ( U ( x ) , U ( y ) ) γxy,x,yH,

where D:C(H)×C(H)R{+} is called the Hausdorff pseudo-metric defined as follows:

D(U,V)=max { sup x U inf y V x y , sup y V inf x U x y } ,U,VC(H).

Furthermore, the Hausdorff pseudo-metric D reduces to the Hausdorff metric when C(H) is restricted to closed bounded subsets of the family CB(H).

Lemma 2.3 Let θ(0,1) be a constant. Then function f(λ)=1λ+λθ for λ[0,1] is nonnegative and strictly decrease and f(λ)[0,1]. Further, if λ0, then f(λ)(0,1).

Lemma 2.4 ([36])

Let { a n } and { b n } be two nonnegative real sequences satisfying

a n + 1 θ a n + b n

with 0<θ<1 and lim n b n =0. Then lim n a n =0.

3 Iterative algorithm and convergence analysis

In this section, we construct a class of new iterative algorithms for finding approximate solutions of the problems (1.1) and (1.2), respectively. Then the convergence criterion for the algorithms is also discussed.

Lemma 3.1 Let ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m and u i j U i j ( x j ) for i,j=1,2,,m, then ( x 1 , x 2 ,, x m , u 11 ,, u 1 m ,, u m 1 ,, u m m ) (denoted by ()) is a solution of the problem (1.1) if and only if () satisfy

g i ( x i ) = R M i , ρ i A i , η i [ A i ( g i ( x i ) ) ρ i F i ( u i 1 , , u i i 1 , u i i , u i i + 1 , , u i m ) ] ,
(3.1)

where R M i , ρ i A i , η i = ( A i + ρ i M i ) 1 and ρ i >0 is a constant for i=1,2,,m.

Proof It follows from the definition of generalized resolvent operator R M i , ρ i A i , η i that the proof can be obtained directly, and so it is omitted. □

Algorithm 3.1

Step 1. Setting ( x 1 0 , x 2 0 ,, x m 0 ) H 1 × H 2 ×× H m and choose u i j 0 U i j ( x j 0 ) for i,j=1,2,,m.

Step 2. Let

x i n + 1 = ( 1 λ ) x i n + λ { x i n g i ( x i n ) + R M i , ρ i A i , η i [ A i ( g i ( x i n ) ) ρ i F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) ] }
(3.2)

for all i=1,2,,m and n=0,1,2, , where λ(0,1] is a constant.

Step 3. By the results of Nadler [37], we can choose u i j n + 1 U i j ( x j n + 1 ) such that

u i j n + 1 u i j n ( 1 + 1 n + 1 ) D j ( U i j ( x j n + 1 ) , U i j ( x j n ) ) ,
(3.3)

where D j (,) is the Hausdorff pseudo-metric on C( H j ) and i,j=1,2,,m.

Step 4. If x i n + 1 and u i j n + 1 for i,j=1,2,,m satisfy (3.2) to sufficient accuracy, stop. Otherwise, set n:=n+1 and return to Step 2.

Remark 3.1 If R M i , ρ i A i , η i reduces to J ρ φ i = ( I + ρ φ i ) 1 , where φ i : H i R{+} is proper and lower semi-continuous η i -subdifferentiable functional, H i H for i=1,2,,m and λ=1, then Algorithm 3.1 reduces to Algorithm (I) of Cao [33].

When λ=1 and U i j = T i j is single-valued operator for i,j=1,2,,m, then Algorithm 3.1 reduces to the following algorithm for the problem (1.2).

Algorithm 3.2 For any given ( x 1 0 , x 2 0 ,, x m 0 ) H 1 × H 2 ×× H m , we compute x i n as follows:

x i n + 1 = x i n g i ( x i n ) + R M i , ρ i A i , η i [ A i ( g i ( x i n ) ) ρ i F i ( T i 1 x 1 n , , T i i 1 x i 1 n , T i i x i n , T i i + 1 x i + 1 n , , T i m x m n ) ] + w i n
(3.4)

for n=0,1,2, and i=1,2,,m, where w i n H i is error to take into account a possible inexact computation of the resolvent operator point satisfying conditions lim n w i n =0.

Remark 3.2

  1. (i)

    Let m=2, g i I, U i i I for i=1,2, then Algorithm 3.1 reduces to Algorithm 4.3 of Agarwal and Verma [34].

  2. (ii)

    If for appropriate and suitable choices of positive integer m and mappings F i , g i , A i , η i , M, U i j , and H i for i,j=1,2,,m, one can know that Algorithms 3.1-3.2 are extending a number of known algorithms.

In the sequel, we provide main result concerning the problem (1.1) with respect to Algorithm 3.1.

Theorem 3.1 For i=1,2,,m, let η i : H i × H i H i be τ i -Lipschitz continuous and t i -strongly monotone operator, A i : H i H i be β i -Lipschitz continuous and r i -strongly monotone operator, g i : H i H i be ξ i -Lipschitz continuous and δ i -strongly monotone operator and M i : H i 2 H i be relative ( A i , η i )-maximal monotone. Suppose that U i j : H j C H j is D j - γ i j -Lipschitz continuous, F i : H 1 × H 2 ×× H m H i is ( U i i , c i , μ i )-relaxed cocoercive with respect to A i in the ith argument and ζ i j -Lipschitz continuous in the jth for i,j=1,2,,m. If there exists constant ρ i >0 for such that

θ j = τ j r j t j β j 2 ξ j 2 2 ρ j μ j δ j 2 + 2 ρ j c j ζ j j 2 γ j j 2 + ρ j 2 ζ j j 2 γ j j 2 + 1 2 δ j + ξ j 2 + i = 1 , i j m ρ i τ i ζ i j γ i j r i t i < 1
(3.5)

for all j=1,2,,m, then the problem (1.1) admits a solution (), i.e. ( x 1 , x 2 ,, x m , u 11 ,, u 1 m ,, u m 1 ,, u m m ), where ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m and u i j U i j ( x j ) for i,j=1,2,,m. Moreover, iterative sequences { x j n } and { u i j n } generated by Algorithm  3.1 strongly converge to x j and u i j for i,j=1,2,,m, respectively.

Proof For i=1,2,,m, applying Algorithm 3.1 and Lemma 2.2, we have

x i n + 1 x i n ( 1 λ ) x i n x i n 1 + λ x i n x i n 1 ( g i ( x i n ) g i ( x i n 1 ) ) + λ R M i , ρ i A i , η i [ A i ( g i ( x i n ) ) ρ i F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) ] R M i , ρ i A i , η i [ A i ( g i ( x i n 1 ) ) ρ i F i ( u i 1 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n 1 ) ] ( 1 λ ) x i n x i n 1 + λ x i n x i n 1 ( g i ( x i n ) g i ( x i n 1 ) ) + λ τ i r i t i A i ( g i ( x i n ) ) A i ( g i ( x i n 1 ) ) ρ i [ F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) ] + λ τ i ρ i r i t i F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) F i ( u i 1 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n 1 ) .
(3.6)

By ξ i -Lipschitz continuity and δ i -strongly monotonicity of g i , we get

x i n x i n 1 ( g i ( x i n ) g i ( x i n 1 ) ) 2 = x i n x i n 1 2 2 g i ( x i n ) g i ( x i n 1 ) , x i n x i n 1 + g i ( x i n ) g i ( x i n 1 ) 2 ( 1 2 δ i + ξ i 2 ) x i n x i n 1 2 .
(3.7)

Since A i is β i -Lipschitz continuous, F i is ( U i i , c i , μ i )-relaxed cocoercive with respect to A i in the i th argument and F i is ζ i j -Lipschitz continuous in the j th argument, then we have

A i ( g i ( x i n ) ) A i ( g i ( x i n 1 ) ) ρ i [ F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) ] 2 = A i ( g i ( x i n ) ) A i ( g i ( x i n 1 ) ) 2 2 ρ i F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) , A i ( g i ( x i n ) ) A i ( g i ( x i n 1 ) ) + ρ i 2 F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) 2 β i 2 g i ( x i n ) g i ( x i n 1 ) 2 2 ρ i [ ( c i ) F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) 2 + μ i g i ( x i n ) g i ( x i n 1 ) 2 ] + ρ i 2 ζ i i 2 u i i n u i i n 1 2 ( β i 2 ξ i 2 2 ρ i μ i δ i 2 ) x i n x i n 1 2 + ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) u i i n u i i n 1 2 .
(3.8)

By D j - γ i j -Lipschitz continuity of the U i j and (3.3), we get

F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) F i ( u i 1 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n 1 ) F i ( u i 1 n , u i 2 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) F i ( u i 1 n 1 , u i 2 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) + + F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n , , u i m n ) + F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n , , u i m n ) F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n ) + + F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n ) F i ( u i 1 n 1 , u i 2 n 1 , , u i i 1 n 1 , u i i n 1 , u i i + 1 n 1 , , u i m n 1 ) ζ i 1 u i 1 n u i 1 n 1 + + ζ i i 1 u i i 1 n u i i 1 n 1 + ζ i i + 1 u i i + 1 n u i i + 1 n 1 + + ζ i m u i m n u i m n 1 = j = 1 , j i m ζ i j u i j n u i j n 1 j = 1 , j i m ζ i j ( 1 + 1 n ) D j ( U i j ( x j n ) , U i j ( x j n 1 ) ) ( 1 + 1 n ) j = 1 , j i m ζ i j γ i j x j n x j n 1
(3.9)

and

u i i n u i i n 1 ( 1 + 1 n ) D i ( U i i ( x i n ) , U i i ( x i n 1 ) ) ( 1 + 1 n ) γ i i x i n x i n 1 .
(3.10)

Combining (3.8) and (3.10), we have

A i ( g i ( x i n ) ) A i ( g i ( x i n 1 ) ) ρ i [ F i ( u i 1 n , , u i i 1 n , u i i n , u i i + 1 n , , u i m n ) F i ( u i 1 n , , u i i 1 n , u i i n 1 , u i i + 1 n , , u i m n ) ] 2 [ β i 2 ξ i 2 2 ρ i μ i δ i 2 + ( 1 + 1 n ) 2 γ i i 2 ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) ] x i n x i n 1 2 .
(3.11)

It follows from (3.6)-(3.9), and (3.11), that

x i n + 1 x i n ( 1 λ + λ 1 2 δ i + ξ i 2 ) x i n x i n 1 + λ τ i r i t i [ β i 2 ξ i 2 2 ρ i μ i δ i 2 + ( 1 + n 1 ) 2 γ i i 2 ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) x i n x i n 1 + ( 1 + 1 n ) ρ i j = 1 , j i m ζ i j γ i j x j n x j n 1 ] ,

which implies that

j = 1 m x j n + 1 x j n = i = 1 m x i n + 1 x i n i = 1 m [ ( 1 λ + λ 1 2 δ i + ξ i 2 ) x i n x i n 1 + λ τ i r i t i ( β i 2 ξ i 2 2 ρ i μ i δ i 2 + ( 1 + 1 n ) 2 γ i i 2 ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) x i n x i n 1 + ( 1 + 1 n ) ρ i j = 1 , j i m ζ i j γ i j x j n x j n 1 ) ] = i = 1 m [ ( 1 λ + λ 1 2 δ i + ξ i 2 ) + λ τ i r i t i β i 2 ξ i 2 2 ρ i μ i δ i 2 + ( 1 + 1 n ) 2 γ i i 2 ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) ] x i n x i n 1 + ( 1 + 1 n ) λ i = 1 m j = 1 , j i m ρ i τ i ζ i j γ i j r i t i x j n x j n 1 = j = 1 m [ ( 1 λ + λ 1 2 δ j + ξ j 2 ) + λ τ j r j t j β j 2 ξ j 2 2 ρ j μ j δ j 2 + ( 1 + 1 n ) 2 γ j j 2 ( 2 ρ j c j ζ j j 2 + ρ j 2 ζ j j 2 ) ] x j n x j n 1 + ( 1 + 1 n ) λ j = 1 m i = 1 , i j m ρ i τ i ζ i j γ i j r i t i x j n x j n 1 = j = 1 m [ ( 1 λ ) + λ ( 1 2 δ j + ξ j 2 + τ j r j t j β j 2 ξ j 2 2 ρ j μ j δ j 2 + ( 1 + 1 n ) 2 γ j j 2 ( 2 ρ j c j ζ j j 2 + ρ j 2 ζ j j 2 ) ) + ( 1 + 1 n ) i = 1 , i j m ρ i τ i ζ i j γ i j r i t i ] x j n x j n 1 = j = 1 m [ 1 λ + λ θ j n ] x j n x j n 1 f n ( λ ) j = 1 m x j n x j n 1 ,
(3.12)

where

θ j n = τ j r j t j β j 2 ξ j 2 2 ρ j μ j δ j 2 + ( 1 + 1 n ) 2 γ j j 2 ( 2 ρ j c j ζ j j 2 + ρ j 2 ζ j j 2 ) + 1 2 δ j + ξ j 2 + ( 1 + 1 n ) i = 1 , i j m ρ i τ i ζ i j γ i j r i t i

and

f n (λ)= max 1 j m { 1 λ + λ θ j n } .

By condition (3.5), we know that sequence { θ j n } is monotone decreasing and θ j n θ j as n. Thus,

f(λ)= lim n f n (λ)= max 1 j m {1λ+λ θ j }.

Since 0< θ j <1 for j=1,2,,m, we get θ= max 1 j m { θ j }(0,1), by Lemma 2.3, we have f(λ)=1λ+λθ(0,1). From (3.12), it follows that { x j n } is a Cauchy sequence and there exists x j H j such that x j n x j as n for j=1,2,,m.

Next, we show that u i j n u i j U i j ( x j ) as n for i,j=1,2,,m.

It follows from (3.9) and (3.10) that { u i j n } are also Cauchy sequences. Hence, there exists u i j H j such that u i j n u i j as n for i,j=1,2,,m. Furthermore,

d ( u i j , U i j ( x j ) ) = inf { u i j t : t U i j ( x j ) } u i j u i j n + d ( u i j n , U i j ( x j ) ) u i j u i j n + D j ( U i j ( x j n ) , U i j ( x j ) ) u i j u i j n + γ i j x j n x j 0 ( n ) .

Since U i j ( x j ) is closed for i,j=1,2,,m, we have u i j U i j ( x j ) for i,j=1,2,,m. Using continuity, ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m and u i j U i j ( x j ) for i,j=1,2,,m satisfy (3.1) and so in light of Lemma 3.1, () is a solution to the problem (1.1). This completes the proof. □

Remark 3.3 If the generalized resolvent operator R M i , ρ i A i , η i reduces to J ρ φ i = ( I + ρ φ i ) 1 , where φ i : H i R{+} is proper and lower semi-continuous η i -subdifferentiable functional, H i =H for i=1,2,,m, λ=1 and ( U i i , c i , μ i )-relaxed cocoerciveness with respect to A i in the i th argument of F i reduces to μ i -( U i i , A i )-strongly monotonicity (right now, c i =0, A i g i ), then Theorem 3.1 reduces to Theorem 3.1 of Cao [33].

Theorem 3.2 Assume that η i , A i , g i , M i are the same as in the Theorem  3.1 for i=1,2,,m. Suppose that T i j : H j H j is γ i j -Lipschitz continuous, F i : H 1 × H 2 ×× H m H i is ( T i i , c i , μ i )-relaxed cocoercive with respect to A i in the ith argument and ζ i j -Lipschitz continuous in the jth for i,j=1,2,,m. If there exists constant ρ i >0 for such that

θ j = τ j r j t j β j 2 ξ j 2 2 ρ j μ j δ j 2 + 2 ρ j c j ζ j j 2 γ j j 2 + ρ j 2 ζ j j 2 γ j j 2 + 1 2 δ j + ξ j 2 + i = 1 , i j m ρ i τ i ζ i j γ i j r i t i < 1

for j=1,2,,m, then the problem (1.2) has a unique solution ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m . Moreover, the iterative sequences { x j n } generated by Algorithm  3.2 strongly converge to x j for j=1,2,,m.

Proof Define the norm on product space H 1 × H 2 ×× H m by

( x 1 , x 2 , , x m ) = j = 1 m x j ,( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m .

It is easy to see that ( H 1 × H 2 ×× H m , ) is a Banach space. Set

y i = x i g i ( x i ) + R M i , ρ i A i , η i [ A i ( g i ( x i ) ) ρ i F i ( T i 1 x 1 , , T i i 1 x i 1 , T i i x i , T i i + 1 x i + 1 , , T i m x m ) ] .

Let G: H 1 × H 2 ×× H m H 1 × H 2 ×× H m be defined by

G( x 1 , x 2 ,, x m )=( y 1 , y 2 ,, y m ),( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m .

For any ( x 1 1 , x 2 1 ,, x m 1 ),( x 1 2 , x 2 2 ,, x m 2 ) H 1 × H 2 ×× H m , it follows from Lemma 2.2 that

G ( x 1 1 , x 2 1 , , x m 1 ) G ( x 1 2 , x 2 2 , , x m 2 ) = i = 1 m y i 1 y i 2 i = 1 m { x i 1 x i 2 ( g i ( x i 1 ) g i ( x i 2 ) ) + R M i , ρ i A i , η i [ A i ( g i ( x i 1 ) ) ρ i F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 1 , T i i + 1 x i + 1 1 , , T i m x m 1 ) ] R M i , ρ i A i , η i [ A i ( g i ( x i 2 ) ) ρ i F i ( T i 1 x 1 2 , , T i i 1 x i 1 2 , T i i x i 2 , T i i + 1 x i + 1 2 , , T i m x m 2 ) ] } i = 1 m { x i 1 x i 2 ( g i ( x i 1 ) g i ( x i 2 ) ) + τ i r i t i A i ( g i ( x i 1 ) ) A i ( g i ( x i 2 ) ) ρ i [ F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 1 , T i i + 1 x i + 1 1 , , T i m x m 1 ) F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 2 , T i i + 1 x i + 1 1 , , T i m x m 1 ) ] + τ i ρ i r i t i F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 2 , T i i + 1 x i + 1 1 , , T i m x m 1 ) F i ( T i 1 x 1 2 , , T i i 1 x i 1 2 , T i i x i 2 , T i i + 1 x i + 1 2 , , T i m x m 2 ) } .
(3.13)

By ξ i -Lipschitz continuity and δ i -strongly monotonicity of g i , we get

x i 1 x i 2 ( g i ( x i 1 ) g i ( x i 2 ) ) 1 2 δ i + ξ i 2 x i 1 x i 2 .
(3.14)

Since A i is β i -Lipschitz continuous, F i is ( T i i , c i , μ i )-relaxed cocoercive with respect to A i in the i th argument and F i is ζ i j -Lipschitz continuous in the j th argument and T i j : H j H j is γ i j -Lipschitz continuous, then we have

A i ( g i ( x i 1 ) ) A i ( g i ( x i 2 ) ) ρ i [ F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 1 , T i i + 1 x i + 1 1 , , T i m x m 1 ) F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 2 , T i i + 1 x i + 1 1 , , T i m x m 1 ) ] 2 β i 2 g i ( x i 1 ) g i ( x i 2 ) 2 2 ρ i [ ( c i ) F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 1 , T i i + 1 x i + 1 1 , , T i m x m 1 ) F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 2 , T i i + 1 x i + 1 1 , , T i m x m 1 ) 2 + μ i g i ( x i 1 ) g i ( x i 2 ) 2 ] + ρ i 2 ζ i i 2 T i i x i 1 T i i x i 2 2 ( β i 2 ξ i 2 2 ρ i μ i δ i 2 ) x i 1 x i 2 2 + ( 2 ρ i c i ζ i i 2 + ρ i 2 ζ i i 2 ) T i i x i 1 T i i x i 2 2 ( β i 2 ξ i 2 2 ρ i μ i δ i 2 + 2 ρ i c i ζ i i 2 γ i i + ρ i 2 ζ i i 2 γ i i ) x i 1 x i 2 2
(3.15)

and

F i ( T i 1 x 1 1 , , T i i 1 x i 1 1 , T i i x i 2 , T i i + 1 x i + 1 1 , , T i m x m 1 ) F i ( T i 1 x 1 2 , , T i i 1 x i 1 2 , T i i x i 2 , T i i + 1 x i + 1 2 , , T i m x m 2 ) ζ i 1 T i 1 x 1 1 T i 1 x 1 2 + + ζ i i 1 T i i 1 x i 1 1 T i i 1 x i 1 2 + ζ i i + 1 T i i + 1 x i + 1 1 T i i + 1 x i i + 1 2 + + ζ i m T i m x m 1 T i m x m 2 = j = 1 , j i m ζ i j T i j x j 1 T i j x j 2 j = 1 , j i m ζ i j γ i j x j 1 x j 2 .
(3.16)

From (3.13)-(3.16), we have

G ( x 1 1 , x 2 1 , , x m 1 ) G ( x 1 2 , x 2 2 , , x m 2 ) i = 1 m ( 1 2 δ i + ξ i 2 + τ i r i t i β i 2 ξ i 2 2 ρ i μ i δ i 2 + 2 ρ i c i ζ i i 2 γ i i 2 + ρ i 2 ζ i i 2 γ i i 2 ) x i 1 x i 2 + j = 1 , j i m ρ j τ j ζ i j γ i j r j t j x j 1 x j 2 = j = 1 m θ j x j 1 x j 2 θ j = 1 m x j 1 x j 1 = θ ( x 1 1 , x 2 1 , , x m 1 ) ( x 1 2 , x 2 2 , , x m 2 ) ,

where θ= max 1 j m θ j . It follows from assumption (3.5) that 0<θ<1. This shows that G: H 1 × H 2 ×× H m H 1 × H 2 ×× H m is a contractive operator, and so there exists a unique ( x 1 , x 2 ,, x m ) H 1 × H 2 ×× H m such that G( x 1 , x 2 ,, x m )=( x 1 , x 2 ,, x m ). Thus, ( x 1 , x 2 ,, x m ) is the unique solution of the problem (1.2).

Now we prove that x i n x i as n for i=1,2,,m. In fact, it follows from (3.4) and Lemma 2.2 that

x i n + 1 x x i n x i ( g i ( x i n ) g i ( x i ) ) + R M i , ρ i A i , η i [ A i ( g i ( x i n ) ) ρ i F i ( T i 1 x 1 n , , T i i 1 x i 1 n , T i i x i n , T i i + 1 x i + 1 n , , T i m x m n ) ] R M i , ρ i A i , η i [ A i ( g i ( x i ) ) ρ i F i ( T i 1 x 1 , , T i i 1 x i 1 , T i i x i , T i i + 1 x i + 1 , , T i m x m ) ] + w i n x i n x i ( g i ( x i n ) g i ( x i ) ) + w i n + τ i r i t i A i ( g i ( x i n ) ) A i ( g i ( x i ) ) ρ i [ F i ( T i 1 x 1 n , , T i i 1 x i 1 n , T i i x i n , T i i + 1 x i + 1 n , , T i m x m n ) F i ( T i 1 x 1 n , , T i i 1 x i 1 n , T i i x i , T i i + 1 x i + 1 n , , T i m x m n ) ] + τ i ρ i r i t i F i ( T i 1 x 1 n , , T i i 1 x i 1 n , T i i x i , T i i + 1 x i + 1 n , , T i m x m n ) F i ( T i 1 x 1 , , T i i 1 x i 1 , T i i x i , T i i + 1 x i + 1 , , T i m x m ) .
(3.17)

Following very similar arguments from (3.14)-(3.16), we have

x i n + 1 x i 1 2 δ i + ξ i 2 x i n x i + τ i r i t i [ β i 2 ξ i 2 2 ρ i μ i δ i 2 + 2 ρ i c i ζ i i 2 γ i i 2 + ρ i 2 ζ i i 2 γ i i 2 x i n x i + ρ i j = 1 , j i m ζ i j γ i j x j n x j ] + w i n ,
(3.18)

which implies that

j = 1 m x j n + 1 x j = j = 1 m θ j x j n x j + j = 1 m w j n θ j = 1 m x j n x j + j = 1 m w j n ,

where a n = j = 1 m x j n x j , b n = j = 1 m w j n . The condition of Algorithm 3.2 yields lim n b n =0. Now Lemma 2.4 implies that lim n a n =0, and so x j n x j as n for j=1,2,,m. This completes the proof. □

Remark 3.4 If m=2, g 1 = g 2 = U 11 = U 22 I (right now, δ i = ξ i = ζ i i =1 for i=1,2), then Theorem 3.1 reduces to Theorem 4.5 based on Algorithm 4.3 of Agarwal and Verma [34]. Our presented results improve and extend some known results in the literature.

References

  1. 1.

    Huang NJ, Fang YP: Fixed point theorems and new system of multi-valued generalized order complementarity problems. Positivity 2003, 7: 257-285. 10.1023/A:1026222030596

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities. Comput. Math. Appl. 2001, 41: 1025-1031. 10.1016/S0898-1221(00)00336-9

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Cho YJ, Fang YP, Huang NJ, Hwang HJ: Algorithms for systems of nonlinear variational inequalities. J. Korean Math. Soc. 2004,41(3):489-499.

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Fang YP, Huang NJ, Thompson HB:A new system of variational inclusions with (H,η)-monotone operators in Hilbert spaces. Comput. Math. Appl. 2005, 49: 365-374. 10.1016/j.camwa.2004.04.037

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Yan WY, Fang YP, Huang NJ: A new system of set-valued variational inclusions with H -monotone operators. Math. Inequal. Appl. 2005,8(3):537-546.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Fang YP, Huang NJ: H -Monotone operator and resolvent operator technique for variational inclusion. Appl. Math. Comput. 2003,145(2-3):795-803. 10.1016/S0096-3003(03)00275-3

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Lan HY, Cho YJ, Kim JH: On a new system of nonlinear A -monotone multi-valued variational inclusions. J. Math. Anal. Appl. 2007,327(1):481-494. 10.1016/j.jmaa.2005.11.067

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Verma RU: A -Monotonicity and applications to nonlinear variational inclusion problems. J. Appl. Math. Stoch. Anal. 2004,17(2):193-195.

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    Verma RU:Sensitivity analysis for generalized strongly monotone variational inclusions based on (A,η)-resolvent operator technique. Appl. Math. Lett. 2006, 19: 1409-1413. 10.1016/j.aml.2006.02.014

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Agarwal RP, Huang NJ, Cho YJ: Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings. J. Inequal. Appl. 2002,7(6):807-828.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Ding XP, Luo CL: Perturbed proximal point algorithms for generalized quasi-variational-like inclusions. J. Comput. Appl. Math. 2000, 210: 153-165.

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Fang YP, Huang NJ: H -Monotone operator and system of variational inclusions. Commun. Appl. Nonlinear Anal. 2004,11(1):93-101.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Huang NJ, Fang YP: A new class of general variational inclusions involving maximal η -monotone mappings. Publ. Math. (Debr.) 2003, 62: 83-98.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Jin MM: Generalized nonlinear implicit quasi-variational inclusions with relaxed monotone mappings. Adv. Nonlinear Var. Inequal. 2004,7(2):173-181.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Verma RU:Approximation solvability of a class of nonlinear set-valued mappings inclusions involving (A,η)-monotone mappings. J. Math. Anal. Appl. 2008, 337: 969-975. 10.1016/j.jmaa.2007.01.114

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. J. Optim. Theory Appl. 2004, 121: 203-210.

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Kasay G, Kolumban J: System of multi-valued variational inequalities. Publ. Math. (Debr.) 2000, 56: 185-195.

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Ding XP: Perturbed proximal point algorithms for generalized quasi-variational inclusions. J. Math. Anal. Appl. 1997, 210: 88-101. 10.1006/jmaa.1997.5370

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Qin XL, Kang SM, Su YF, Shang MJ: Strong convergence of an iterative method for variational inequality problems and fixed point problems. Arch. Math. 2009,45(2):147-158.

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Alimohammady M, Roohi M:A system of generalized variational inclusion problems involving (A,η)-monotone mappings. Filomat 2009,23(1):13-20. 10.2298/FIL0901013A

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Katchang P, Kumam P: A general iterative method of fixed points for mixed equilibrium problems and variational inclusions problems. J. Inequal. Appl. 2010. Article ID 370197, 2010:

    Google Scholar 

  22. 22.

    Jin MM: Perturbed iterative algorithms for generalized nonlinear set-valued quasivariational inclusions involving generalized m -accretive mappings. J. Inequal. Appl. 2007. Article ID 29863, 2007:

    Google Scholar 

  23. 23.

    Ding K, Yan WY, Huang NJ: A new system of generalized nonlinear relaxed cocoercive variational inequalities. J. Inequal. Appl. 2006. Article ID 40591, 2006:

    Google Scholar 

  24. 24.

    Peng JW, Zhao LJ: General system of A -monotone nonlinear variational inclusions problems with applications. J. Inequal. Appl. 2009. Article ID 364615, 2009:

    Google Scholar 

  25. 25.

    Kazmi KR, Bhat MI: Iterative algorithms for a system of nonlinear variational-like inclusions. Comput. Math. Appl. 2004, 48: 1929-1935. 10.1016/j.camwa.2004.02.009

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Eam CK, Suantai S: A new approximation method for solving variational inequalities and fixed points of nonexpansive mappings. J. Inequal. Appl. 2009. Article ID 520301, 2009:

    Google Scholar 

  27. 27.

    Cho YJ, Petrot N: Regularization and iterative method for general variational inequalities problem in Hilbert spaces. J. Inequal. Appl. 2011. Article ID 21, 2011:

    Google Scholar 

  28. 28.

    Lan HY: Projection iterative approximations a new class of general random implicit quasi-variational inequalities. J. Inequal. Appl. 2006. Article ID 81261, 2006:

    Google Scholar 

  29. 29.

    Noor MA, Noor KI, Kamal R: General variational inclusions involving difference of operators. J. Inequal. Appl. 2014. Article ID 98, 2014:

    Google Scholar 

  30. 30.

    Witthayarat U, Cho YJ, Kumam P: Approximation algorithm for fixed points of nonlinear operators and solutions of mixed equilibrium problems and variational inclusion problems with applications. J. Nonlinear Sci. Appl. 2012 special issue,5(6):475-494. special issue

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Ahmad R, Dilshad M: H(,) - η -Cocoercive operators and variational-like inclusions in Banach spaces. J. Nonlinear Sci. Appl. 2012 special issue,5(5):334-344.

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Kavitha V, Arjunan MM, Ravichandran C: Existence results for a second order impulsive neutral functional integrodifferential inclusions in Banach spaces with infinite delay. J. Nonlinear Sci. Appl. 2012 special issue,5(5):321-333.

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Cao HW: A new system of generalized quasi-variational-like inclusions with noncompact valued mappings. J. Inequal. Appl. 2012. Article ID 41, 2012:

    Google Scholar 

  34. 34.

    Agarwal RP, Verma RU:General system of (A,η)-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms. Commun. Nonlinear Sci. Numer. Simul. 2010,15(2):238-251. 10.1016/j.cnsns.2009.03.037

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Kim JK, Kim DS: A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. J. Convex Anal. 2004,11(1):235-243.

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Appl. 1995,194(1):114-125. 10.1006/jmaa.1995.1289

    Article  MathSciNet  MATH  Google Scholar 

  37. 37.

    Nadler SP: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475-488. 10.2140/pjm.1969.30.475

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Cultivation Project of Sichuan University of Science and Engineering (2011PY01) and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01).

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Correspondence to Heng-you Lan.

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TX carried out the proof of the corollaries and gave some examples to show the main results. HL conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.

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Xiong, T., Lan, H. New general systems of set-valued variational inclusions involving relative (A,η)-maximal monotone operators in Hilbert spaces. J Inequal Appl 2014, 407 (2014). https://doi.org/10.1186/1029-242X-2014-407

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Keywords

  • general system of set-valued variational inclusions
  • relative (A,η)-maximal monotone operator
  • generalized resolvent operator technique
  • relative relaxed cocoercive
  • iterative algorithm
  • convergence criteria