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Ishikawa-hybrid proximal point algorithm for NSVI system

Fixed Point Theory and Applications20122012:195

https://doi.org/10.1186/1687-1812-2012-195

  • Received: 11 April 2012
  • Accepted: 12 October 2012
  • Published:

Abstract

A nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm is developed and studied using the notion of an ( A , η ) -accretive mapping. Convergence analysis for the algorithm of solving the nonlinear set-valued inclusions system and existence analysis of solution for the system are explored along with some results on the resolvent operator corresponding to the ( A , η ) -accretive mapping in a Banach space. The result that the sequence generated by the algorithm converges linearly to a solution of the system with the convergence rate Ψ is proved.

MSC:49J40, 47H06.

Keywords

  • nonlinear set-valued inclusions system
  • ( A , η ) -accretive mapping
  • resolvent operator
  • Ishikawa-hybrid proximal point algorithm
  • convergence rate

1 Introduction

The nonlinear set-valued inclusions system, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality and variational inclusions system. In recent years, various variational inclusions systems and nonlinear set-valued inclusions systems have been intensively studied. For example, Kassay and Kolumbán [2], Chen, Deng and Tan [3], Yan, Fang and Huang [4], Fang, Huang and Thompson [5], Jin [6], Verma [7], Li, Xu and Jin [8], Kang, Cho and Liu [9]et al. introduced and studied various set-valued variational inclusions systems. For the past few years, many existence results and iterative algorithms for various variational inclusions systems have been studied. For details, please see [128] and the references therein.

Example 1.1 In 2001, Chen, Deng and Tan [3] have studied the problem associated with the following system of variational inequalities, which is finding ( x , y ) H × H (H, Hilbert space) such that
{ ρ T ( y ) + x y , w x ρ φ 1 ( x ) ρ φ 1 ( w ) , t S ( x ) + y x , w y t φ 2 ( y ) ρ φ 2 ( w ) ( w H ) ,
(1)

where φ i : H R is a proper, convex, lower semicontinuous functional and φ i ( ) denotes the subdifferential operator of φ i ( i = 1 , 2 ).

Example 1.2 Let X be a real q-uniformly smooth Banach space, and S , T , M 1 , M 2 : X X be four single-valued mappings. Find x , y X such that
{ 0 x y + ρ T ( y ) + ρ M 1 ( x ) , 0 y x + t S ( x ) + t M 2 ( y ) ,
(2)

which is studied by Jin in [6].

Inspired and motivated by Examples 1.1-1.2 and recent research work in this field (see [7, 8]), in this paper, we will introduce and discuss the problem associated with the following class of new nonlinear set-valued inclusions systems (NSVI Systems), which is finding ( x , y ) X × X for any f , g : X X such that z S ( x ) , w T ( y ) , and
{ f ( x ) F ( z , y ) + M ( y ) , g ( y ) G ( w , x ) + N ( x ) ,
(3)

where X is a real q-uniformly smooth Banach space, A , B : X X , η 1 , η 2 : X × X X , and F , G : X × X X are single-valued mappings; M : X 2 X is a set-valued ( A , η 1 ) -accretive mapping and N : X 2 X is a set-valued ( B , η 2 ) -accretive mapping, and S , T : X C B ( X ) are two set-valued mappings.

If f ( x ) = x , g ( y ) = y , F ( w , y ) = ρ T ( y ) y , G ( y , x ) = t S ( x ) x , N = t M 2 and M = ρ M 1 ( ) , then the problem (3) reduces to Example 1.1. If M 1 = M 2 = φ , φ i : H R is a proper, convex, lower semicontinuous functional and φ i ( ) denotes the subdifferential operator of φ i ( X = H , Hilbert space, and i = 1 , 2 ), then the problem (3) changes to Example 1.2.

If X is a real q-uniformly smooth Banach space, and G ( , ) = N ( y , g ( x ) ) , f ( u ) = u and S ( u ) = Q ( u ) ( u X ) , then the problem (3) reduces to the problem associated with the following variational inclusions:

For any u X , find x X and y = Q ( x ) such that
u N ( y , g ( x ) ) + M ( y ) ,
(4)

which is developed by Li in 2010 [8].

The main purpose of this paper is to introduce and study a generalized nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm using the notion of ( A , η ) -accretive due to Lan-Cho-Verma [10] in a Banach space, to analyse convergence for the algorithm of solving the system and existence of a solution for the system and to prove the result that sequence { ( x n , y n ) } n = 0 generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions system with the convergence rate Ψ .

2 Preliminaries

Let X be a real q-uniformly smooth Banach space with a dual space X , , be the dual pair between X and X , 2 X denote the family of all the nonempty subsets of X, and C B ( X ) denote the family of all nonempty closed bounded subsets of X. The generalized duality mapping J q : X 2 X is defined by
J q ( x ) = { f X : x , f = x q , f = x q 1 } , x X ,

where q > 1 is a constant. Let us recall the following results and concepts.

Definition 2.1 A single-valued mapping η : X × X X is said to be τ-Lipschitz continuous if there exists a constant τ > 0 such that
η ( x , y ) τ x y , x , y X .
Definition 2.2 A single-valued mapping A : X X is said to be
  1. (i)
    accretive if
    A ( x 1 ) A ( x 2 ) , J q ( x 1 x 2 ) 0 , x 1 , x 2 X ;
     
  2. (ii)

    strictly accretive if A is accretive and A ( x 1 ) A ( x 2 ) , J q ( x 1 x 2 ) = 0 if and only if x 1 = x 2 , x 1 , x 2 X ;

     
  3. (iii)
    r-strongly η-accretive if there exists a constant r > 0 such that
    A ( x 1 ) A ( x 2 ) , J q ( η ( x 1 , x 2 ) ) r x 1 x 2 q , x 1 , x 2 X ;
     
  4. (iv)
    γ-Lipschitz continuous if there exists a constant γ > 0 such that
    A ( x 1 ) A ( x 2 ) γ x 1 x 2 , x 1 , x 2 X ;
     
  5. (v)
    Let f : X X be a single-valued mapping. A is said to be ( σ , φ ) -relaxed cocoercive with respect to f if for any x 1 , x 2 X , there exist two constants σ , φ > 0 such that
    A ( x 1 ) A ( x 2 ) , J q ( f ( x 1 ) f ( x 2 ) ) σ A ( x 1 ) A ( x 2 ) q + φ x 1 x 2 q .
     
Definition 2.3 A set-valued mapping S : X C B ( X ) is said to be
  1. (i)
    D-Lipschitz continuous if there exists a constant α > 0 such that
    D ( S ( x ) , S ( y ) ) α x y , x , y X ,
     
where D ( , ) is the Hausdorff metric on C B ( X ) .
  1. (ii)
    β-strongly η-accretive if there exists a constant β > 0 such that
    u 1 u 2 , J q ( η ( x , y ) ) β x y q , x , y X , u 1 S ( x ) , u 2 S ( y ) .
     
Definition 2.4 Let A : X X and η : X × X X be single-valued mappings. A set-valued mapping M : X 2 X is said to be
  1. (i)
    accretive if
    u 1 u 2 , J q ( x , y ) 0 , x , y X , u 1 M ( x ) , u 2 M ( y ) ;
     
  2. (ii)
    η-accretive if
    u 1 u 2 , J q ( η ( x , y ) ) 0 , x , y X , u 1 M ( x ) , u 2 M ( y ) ;
     
  3. (iii)
    m-relaxed η-accretive, if there exists a constant m > 0 such that
    u 1 u 2 , J q ( η ( x , y ) ) m x y q , x , y X , u 1 M ( x ) , u 2 M ( y ) ;
     
  4. (iv)

    A-accretive if M is accretive and ( A + ρ M ) ( X ) = X for all ρ > 0 ;

     
  5. (v)

    ( A , η ) -accretive if M is m-relaxed η-accretive and ( A + ρ M ) ( X ) = X for every ρ > 0 .

     

Based on [10], we can define the resolvent operator R ρ , M A , η as follows.

Lemma 2.5 ([10])

Let η : X × X X be a τ-Lipschitz continuous mapping, A : X X be an r-strongly η-accretive mapping, and M : X 2 X be a set-valued ( A , η ) -accretive mapping. Then the generalized resolvent operator R ρ , M A , η : X X is τ q 1 / ( r m ρ ) -Lipschitz continuous; that is,
R ρ , M A , η ( x ) R ρ , M A , η ( y ) τ q 1 r m ρ x y for all x , y X ,

where ρ ( 0 , r / m ) , q > 1 .

Remark 2.6 The ( A , η ) -accretive mappings are more general than ( H , η ) -monotone mappings, A-monotone operators and η-subdifferential operators in a Banach space or a Hilbert space, and the resolvent operators associated with ( A , η ) -accretive mappings include as special cases the corresponding resolvent operators associated with them, respectively [36, 9, 25].

In the study of characteristic inequalities in q-uniformly smooth Banach spaces X, Xu [14] proved the following result.

Lemma 2.7 ([14])

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

Lemma 2.8 ([8])

Let a , b , c > 0 be real, for any real q 1 , if a q b q + c q , then
a b + c .

3 Existence theorem of solutions

Let us study the existence theorem of solutions for the inclusions system (3).

Theorem 3.1 Let X be a Banach space, f , g : X X be two single-valued mappings, F : X × X X be a ( μ 1 , ν 1 ) -Lipschitz continuous mapping and G : X × X X be a ( μ 2 , ν 2 ) -Lipschitz continuous mapping, η i : X × X X be a τ i -Lipschitz continuous mapping ( i = 1 , 2 ), A : X X be an r 1 -strongly η 1 -accretive mapping, B : X X be an r 2 -strongly η 2 -accretive mapping, M : X 2 X be a set-valued ( A , η 1 ) -accretive mapping and N : X 2 X be a set-valued ( B , η 2 ) -accretive mapping. Then the following statements are mutually equivalent:
  1. (i)

    An element ( x , y ) is a solution of the problem (3);

     
  2. (ii)
    For ( x , y ) X × X , z S ( x ) and w T ( y ) , the following relations hold:
    { x = R ρ 1 , M A , η 1 ( A ( x ) + ρ 1 f ( x ) ρ 1 F ( z , y ) ) , y = R ρ 2 , N B , η 2 ( B ( y ) + ρ 2 g ( y ) ρ 2 G ( w , x ) ) ,
    (5)
     
where ρ i > 0 is a constant ( i = 1 , 2 );
  1. (iii)
    For ( x , y ) X × X , z S ( x ) , w T ( y ) , and any 1 > λ > 0 , the following relations hold:
    { x = ( 1 λ ) x + λ R ρ 1 , M A , η 1 ( A ( x ) + ρ 1 f ( x ) ρ 1 F ( z , y ) ) , y = ( 1 λ ) y + λ R ρ 2 , N B , η 2 ( B ( y ) + ρ 2 g ( y ) ρ 2 G ( w , x ) ) ,
    (6)
     

where ρ i > 0 is a constant ( i = 1 , 2 );

Proof This directly follows from the definition of R ρ 1 , M A , η 1 , R ρ 2 , N B , η 2 , and the problem (3) for i = 1 , 2 . □

Theorem 3.2 Let X be a q-uniformly smooth Banach space. Let f , g : X X be two single-valued κ 1 or κ 2 -Lipschitz continuous mappings, respectively, η i : X × X X be a single-valued τ i -Lipschitz continuous mapping ( i = 1 , 2 ), F , G : X × X X be two single-valued ( μ 1 , ν 1 ) or ( μ 2 , ν 2 ) -Lipschitz continuous mappings, respectively. Let A : X X be single-valued r 1 -strongly η 1 -accretive, ω 1 -Lipschitz continuous, ( σ 1 , φ 1 ) -relaxed cocoercive with respect to f, and B : X X be single-valued r 2 -strongly η 2 -accretive, ω 2 -Lipschitz continuous, ( σ 2 , φ 2 ) -relaxed cocoercive with respect to g. Let S , T : X X be two set-valued γ 1 or γ 2 -Lipschitz continuous mappings, respectively. If M : X 2 X is a set-valued ( A , η 1 ) -accretive mapping and N : X 2 X is a set-valued ( B , η 2 ) -accretive mapping, and the following condition holds:
{ τ q ( ρ 1 μ 1 γ 1 + l 1 ) < τ ( r 1 m 1 ρ 1 ) , τ q ( ρ 2 μ 2 γ 2 + l 2 ) < τ ( r 2 m 2 ρ 2 ) , l 1 = ω 1 q + c q ρ 1 q κ 1 q + q σ 1 ω 1 q q φ 1 q , l 2 = ω 2 q + c q ρ 2 q κ 2 q + q σ 2 ω 2 q q φ 2 q ,
(7)

where c q > 0 is the same as in Lemma  2.7 and ρ i ( 0 , r i m i ) ( i = 1 , 2 ), then the problem (3) has a solution x , y X , z S ( x ) , w T ( y ) .

Proof Define two mappings Q 1 , Q 2 : X X as follows:
{ Q 1 ( x ) = ( 1 λ ) x + λ R ρ 1 , M A , η 1 ( A ( x ) + ρ 1 f ( x ) ρ 1 F ( z , y ) ) , Q 2 ( y ) = ( 1 λ ) y + λ R ρ 2 , N B , η 2 ( B ( y ) + ρ 2 g ( y ) ρ 2 G ( w , x ) ) ( x , y X , z S ( x ) , w T ( y ) ) .
(8)
For elements x 1 , x 2 , y 1 , y 2 X , if letting
Ω i = A ( x i ) ρ 1 f ( x i ) ρ 1 F ( z i , y i ) ( i = 1 , 2 ) ,
then by (8), Lemma 2.5 and Lemma 2.7, we have
Q 1 ( x 1 ) Q 1 ( x 2 ) = ( 1 λ ) x 1 + λ R ρ 1 , M A , η 1 ( Ω 1 ) ( 1 λ ) x 2 λ R ρ 1 , M A , η 1 ( Ω 2 ) ( 1 λ ) x 1 x 2 + λ R ρ 1 , M A , η 1 ( Ω 1 ) R ρ 1 , M A , η 1 ( Ω 2 ) ( 1 λ ) x 1 x 2 + λ τ q 1 r 1 m 1 ρ 1 [ ρ 1 ( F ( z 2 , y 2 ) F ( z 1 , y 1 ) ) + A ( x 1 ) A ( x 2 ) ρ 1 ( f ( x 1 ) f ( x 2 ) ) ] ,
(9)
and by ( μ 1 , ν 1 ) -Lipschitz continuity of F ( , ) and γ 1 -Lipschitz continuity of S, we obtain
F ( z 2 , y 2 ) F ( z 1 , y 1 ) μ 1 z 2 z 1 + ν 1 y 2 y 1 μ 1 γ 1 x 2 x 1 + ν 1 y 2 y 1 .
(10)
Since A is ω 1 -Lipschitz continuous and ( σ 1 , φ 1 ) -relaxed cocoercive with respect to f, and f is κ 1 -Lipschitz continuous so that for z 1 S ( x 1 ) , z 2 S ( x 2 ) , we have
(11)
Combining (9), (10) and (11), we can get
(12)
where
θ 1 = τ q 1 r 1 m 1 ρ 1 ( ρ 1 μ 1 γ 1 + ω 1 q + c q ρ 1 q κ 1 q + q σ 1 ω 1 q q φ 1 q ) .
For elements x 1 , x 2 , y 1 , y 2 X , z i S ( x i ) , y i T ( y i ) ( i = 1 , 2 ), if letting
Θ i = B ( y i ) + ρ 2 g ( y i ) ρ 2 G ( w i , x i ) ( i = 1 , 2 ) ,
then by using the same method as the one used above,
Q 2 ( y 1 ) Q 2 ( y 2 ) = ( 1 λ ) y 1 + λ R ρ 2 , N B , η 2 ( Θ 1 ) ( 1 λ ) y 2 λ R ρ 2 , N B , η 2 ( Θ 2 ) ( 1 λ ) y 1 y 2 + λ R ρ 2 , N B , η 2 ( Θ 1 ) R ρ 2 , N B , η 2 ( Θ 2 ) ( 1 λ ) y 1 y 2 + λ τ q 1 r 2 m 2 ρ 2 ( ρ 2 G ( w 2 , x 2 ) G ( w 1 , x 1 ) + B ( y 1 ) B ( y 2 ) ρ 2 ( g ( y 1 ) g ( y 2 ) ) ) λ τ q 1 r 2 m 2 ρ 2 ρ 2 ν 2 x 2 x 1 + [ ( 1 λ ) + λ θ 2 ] y 2 y 1
(13)
hold, where
θ 2 = τ q 1 r 2 m 2 ρ 2 ( ρ 2 μ 2 γ 2 + ω 2 q + c q ρ 2 q κ 2 q + q σ 2 ω 2 q q φ 2 q ) .
If setting
Γ 11 = θ 1 , Γ 12 = τ q 1 r 1 m 1 ρ 1 ρ 1 ν 1 , Γ 21 = τ q 1 r 2 m 2 ρ 2 ρ 2 ν 2 , Γ 22 = θ 2 ,
(14)
a = ( Q 1 ( x 1 ) Q 1 ( x 2 ) , Q 2 ( y 1 ) Q 2 ( y 2 ) ) T and b = ( x 1 x 2 , y 1 y 2 ) T , then from (12), (13) and (14), we have a ( 1 λ ) E + λ Ψ b , where
E = ( 1 0 0 1 ) , Ψ = ( Γ 11 Γ 12 Γ 21 Γ 22 ) , 0 < λ < 1 ,
(15)
where Ψ is called the matrix for nonlinear set-valued inclusions system. By using [16], we have
a ( 1 λ ) + λ Ψ b .
(16)
Letting
Ψ = max { Γ 11 , Γ 12 , Γ 21 , Γ 22 } .
It follows from (16), the assumption of the condition (7) and S ( x ) , T ( y ) C B ( X ) that 0 < Ψ < 1 , ( 1 λ ) + λ Ψ < 1 , and there exist x , y X and z S ( x ) , w T ( y ) such that
{ Q 1 ( x ) = x , Q 2 ( y ) = y .
Therefore, the following relations hold for Theorem 3.1(ii)-(iii):
{ x = R ρ 1 , M A , η 1 ( A ( x ) + ρ 1 f ( x ) ρ 1 F ( z , y ) ) , y = R ρ 2 , N B , η 2 ( B ( y ) + ρ 2 g ( y ) ρ 2 G ( w , x ) ) ,
(17)

where ρ i > 0 is a constant ( i = 1 , 2 ). Thus, by Theorem 3.1, we know that ( x , y , z , w ) is a solution of the problem (3). This completes the proof. □

4 Ishikawa-hybrid proximal algorithm

In 2008, Verma developed a hybrid version of the Eckstein-Bertsekas [11] proximal point algorithm, introduced the algorithm based on the ( A , η ) -maximal monotonicity framework [7] and studied convergence of the algorithm, and so did Li, Xu and Jin in [12]. Based on Theorem 3.1, we develop an Ishikawa-hybrid proximal point algorithm for finding an iterative sequence solving the problem (3) as follows.

Algorithm 4.1 Let X be a q-uniformly smooth Banach space. Let f , g : X X be two single-valued κ 1 or κ 2 -Lipschitz continuous mappings, respectively, η i : X × X X be a single-valued τ i -Lipschitz continuous mapping ( i = 1 , 2 ), F , G : X × X X be two single-valued ( μ 1 , ν 1 ) or ( μ 2 , ν 2 ) -Lipschitz continuous mappings, respectively. Let A : X X be single-valued r 1 -strongly η 1 -accretive, ω 1 -Lipschitz continuous, ( σ 1 , φ 1 ) -relaxed cocoercive with respect to f, and B : X X be single-valued r 2 -strongly η 2 -accretive, ω 2 -Lipschitz continuous, ( σ 2 , φ 2 ) -relaxed cocoercive with respect to g. Let S , T : X X be two set-valued γ 1 or γ 2 -Lipschitz continuous mappings, respectively, M : X 2 X be a set-valued ( A , η 1 ) -accretive mapping and N : X 2 X be a set-valued ( B , η 2 ) -accretive mapping. Suppose that { α i n } n = 0 , { β i n } n = 0 , { ξ i n } n = 0 , { ζ i n } n = 0 and { ρ i n } n = 0 ( i = 1 , 2 ) are ten nonnegative sequences such that

then we can get x 1 , y 1 X and z 1 S ( x 1 ) , w 1 T ( y 1 ) as follows.

Step 1: For arbitrarily chosen initial points x 0 X , y 0 X , we choose suitable z 0 S ( x 0 ) , w 0 T ( y 0 ) , setting
{ u 0 = ( 1 α 1 0 ) x 0 + α 1 0 e 1 0 , x 1 = ( 1 β 1 0 ) x 0 + β 1 0 d 1 0 ,
where e 1 0 , d 1 0 satisfy
{ e 1 0 R ρ 1 0 , M A , η 1 ( A ( x 0 ) + ρ 1 0 f ( x 0 ) + ρ 1 0 F ( z 0 , y 0 ) ) ξ 1 0 e 1 0 x 0 ( z 0 S ( x 0 ) ) , d 1 0 R ρ 1 0 , M A , η 1 ( A ( u 0 ) + ρ 1 0 f ( u 0 ) + ρ 1 0 F ( z 1 0 , y 0 ) ) ζ 1 0 d 1 0 u 0 ( z 1 0 S ( u 0 ) ) ,
and
{ v 0 = ( 1 α 2 0 ) y 0 + α 2 0 e 2 0 , y 1 = ( 1 β 2 0 ) y 0 + β 2 0 d 2 0 ,
where e 2 0 , d 2 0 satisfy
{ e 2 0 R ρ 2 0 , N B , η 2 ( B ( y 0 ) + ρ 2 0 g ( y 0 ) ρ 2 0 G ( w 0 , x 0 ) ) ξ 2 0 e 2 0 y 0 ( w 0 T ( y 0 ) , d 2 0 R ρ 2 0 , N B , η 2 ( B ( v 0 ) + ρ 2 0 g ( v 0 ) ρ 2 0 G ( w 2 0 , x 0 ) ) ζ 2 0 d 2 0 v 0 ( w 2 0 T ( v 0 ) ) .
By using Nadler [15], we can choose suitable z 1 S ( x 1 ) , w 1 T ( y 1 ) such that
{ z 0 z 1 ( 1 + 1 1 ) D ( S ( x 0 ) , S ( x 1 ) ) , w 0 w 1 ( 1 + 1 1 ) D ( T ( y 0 ) , T ( y 1 ) ) , z 1 0 z 1 1 ( 1 + 1 1 ) D ( S ( U 0 ) , S ( U 1 ) ) , w 2 0 w 2 1 ( 1 + 1 1 ) D ( T ( V 0 ) , T ( V 1 ) ) .

Therefore, we obtain x 1 , y 1 X and z 1 S ( x 1 ) , w 1 T ( y 1 ) and give the next step for generating sequences { x n } n = 2 , { y n } n = 2 , { z n } n = 2 and { w n } n = 2 .

Step 2: From x 1 , y 1 X and z 1 S ( x 1 ) , w 1 T ( y 1 ) , the sequences { x n } n = 2 , { y n } n = 2 , { z n } n = 2 and { w n } n = 2 are generated by the iterative procedure
{ u n = ( 1 α 1 n ) x n + α 1 n e 1 n , x n + 1 = ( 1 β 1 n ) x n + β 1 n d 1 n , e 1 n R ρ 1 n , M A , η 1 ( A ( x n ) + ρ 1 n f ( x n ) + ρ 1 n F ( z n , y n ) ) ξ 1 n e 1 n x n ( z n S ( x n ) ) , d 1 n R ρ 1 n , M A , η 1 ( A ( u n ) + ρ 1 n f ( u n ) ρ 1 n F ( z 1 n , y n ) ) ζ 1 n d 1 n u n ( z 1 n S ( u n ) ) ,
(18)
and
{ v n = ( 1 α 2 n ) y n + α 2 n e 2 n , y n + 1 = ( 1 β 2 n ) y n + β 2 n d 2 n , e 2 n R ρ 2 n , N B , η 2 ( B ( y n ) + ρ 2 n g ( y n ) ρ 2 n G ( w n , x n ) ) ξ 2 n e 2 n y n ( w n T ( y n ) ) , d 2 n R ρ 2 n , N B , η 2 ( B ( v n ) + ρ 2 n g ( v n ) ρ 2 n G ( w 2 n , x n ) ) ζ 2 n d 2 n v n ( w 2 n T ( v n ) ) .
(19)
By using Nadler [15], we can choose suitable z n + 1 S ( x n + 1 ) , w n + 1 T ( y n + 1 ) such that
{ z n z n + 1 ( 1 + 1 1 + n ) D ( S ( x n ) , S ( x n + 1 ) ) , w n w n + 1 ( 1 + 1 1 + n ) D ( T ( y n ) , T ( y n + 1 ) ) ,
(20)

for n = 0 , 1 , 2 ,  .

Remark 4.2 If we choose some suitable operators A, B, η 1 , η 1 , F, G, S, T, M, N, f, g and a space X, then Algorithm 4.1 can degenerate to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see [26, 810, 25]).

5 Convergence of Ishikawa-hybrid proximal Algorithm 4.1

In this section, we prove that { ( x n , y n , z n , w n ) } n = 0 generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution ( x , y , z , w ) of the problem (3) as the convergence rate Ψ .

Theorem 5.1 Let X be a q-uniformly smooth Banach space. Let f , g : X X be two single-valued κ 1 or κ 2 -Lipschitz continuous mappings, respectively, η i : X × X X be a single-valued τ i -Lipschitz continuous mapping ( i = 1 , 2 ), F , G : X × X X be two single-two valued ( μ 1 , ν 1 ) or ( μ 2 , ν 2 ) -Lipschitz continuous mappings, respectively. Let A : X X be a single-valued r 1 -strongly η 1 -accretive and ω 1 -Lipschitz continuous mapping, and let B : X X be a single-valued r 2 -strongly η 2 -accretive and ω 2 -Lipschitz continuous mapping. Let S , T : X X be two set-valued γ 1 or γ 2 -Lipschitz continuous mappings, respectively, A be ( σ 1 , φ 1 ) -relaxed cocoercive with respect to f and B be ( σ 2 , φ 2 ) -relaxed cocoercive with respect to g. Suppose that M : X 2 X is a set-valued ( A , η 1 ) -accretive mapping and N : X 2 X is a set-valued ( B , η 2 ) -accretive mapping, and the following conditions hold:
{ max { ρ 1 μ 1 γ 1 + l 1 , α 1 β 1 ν 1 θ 1 ( 1 + ρ 1 β 1 ρ 1 ) , β 2 ρ 2 ν 2 ( 1 + 2 α 2 θ 2 ) , ( ρ 2 μ 2 γ 2 + l 2 ) } < τ 1 q ( r 1 m 1 ρ 1 ) , 2 β 1 ( 1 α 1 ) α 1 + β 1 θ 1 + α 1 β 1 α 1 + β 1 θ 1 2 < 1 θ 1 , ( 1 β 2 + β 2 θ 2 ) + 3 ( 1 α 2 + α 2 θ 2 ) β 2 θ 2 + β 2 θ 2 < 1 , l 1 = ω 1 q + c q ρ 1 q κ 1 q + q σ 1 ω 1 q q φ 1 q , l 2 = ω 2 q + c q ρ 2 q κ 2 q + q σ 2 ω 2 q q φ 2 q , θ 1 = τ q 1 r 1 m 1 ρ 1 ( ρ 1 μ 1 γ 1 + l 1 ) , θ 2 = τ q 1 r 2 m 2 ρ 2 ( ρ 2 μ 2 γ 2 + l 2 ) ,
(21)
and eight nonnegative sequences { α i n } n = 0 , { β i n } n = 0 , { ξ i n } n = 0 , { ζ i n } n = 0 and { ρ i n } n = 0 ( i = 1 , 2 ) satisfy the following conditions:
(22)
(23)
Then the problem (3) has a solution ( x , y , z , w ) z S ( x ) , w T ( y ) , and the sequence { x n , y n } n = 0 generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution ( x , y ) of the problem (3) as the convergence rate
Ψ = max { 1 ( α 1 + β 1 ) + ( α 1 + β 1 ) θ 1 + ( 2 2 α 1 + α 1 θ 1 ) β 1 θ 1 , α 1 β 1 ν 1 θ 1 τ q 1 r 1 m 1 ρ 1 ( 1 + ρ 1 β 1 ρ 1 ) , 1 β 2 + 2 β 2 θ 2 ( 2 α 2 + α 2 θ 2 ) + β 2 θ 2 ( 1 α 2 + α 2 θ 2 ) , ( 1 β 2 + β 2 θ 2 ) + 3 ( 1 α 2 + α 2 θ 2 ) β 2 θ 2 + β 2 θ 2 } ,
(24)

where c q > 0 is the same as in Lemma  2.5, ρ i ( 0 , r i m i ) ( i = 1 , 2 ).

Proof Let ( x , y , z , w ) ( z S ( x ) , w T ( y ) ) be the solution of the problem (3), then for any λ > 0 ,
{ x = ( 1 λ ) x + λ R ρ 1 , M A , η 1 ( A ( x ) + ρ 1 f ( x ) ρ 1 F ( z , y ) ) , y = ( 1 λ ) y + λ R ρ 2 , N B , η 2 ( B ( y ) + ρ 2 g ( y ) ρ 2 G ( w , x ) ) .
(25)
For n 0 , we write
{ s 1 n = ( 1 α 1 n ) x n + α 1 n R ρ 1 n , M A , η 1 ( A ( x n ) + ρ 1 n f ( x n ) + ρ 1 n F ( z n , y n ) ) , t 1 n + 1 = ( 1 β 1 n ) x n + β 1 n R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) + ρ 1 n F ( z 2 n , y n ) ) ( z 2 n S ( s 1 n ) ) .
(26)
It follows from the hypotheses of the mappings A, f, F, S, M, η 1 and R ρ 1 n , M A , η 1 in Algorithm 4.1 that
that is,
(27)

where θ 1 ( n ) = τ q 1 r 1 m 1 ρ 1 n ( ρ 1 n μ 1 γ 1 + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q ) , z n S ( x n ) and x S ( x ) .

From (24)-(27) and (13), we have
t 1 n + 1 x ( 1 β 1 n ) x n x + β 1 n R ρ 1 n , M A , η 1 ( A ( x ) + ρ 1 n f ( x ) ρ 1 n F ( z , y ) ) R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) + ρ 1 n F ( z 2 n , y n ) ) ( ( 1 β 1 n ) + β 1 n τ q 1 r 1 m 1 ρ 1 n ( ρ 1 n μ 1 γ 1 + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q ) ) s 1 n x + β 1 n τ q 1 r 1 m 1 ρ 1 n ν 1 y n y ( ( 1 β 1 n ) + β 1 n θ 1 ( n ) ) s 1 n x + β 1 n τ q 1 r 1 m 1 ρ 1 n ν 1 y n y .
(28)
By Algorithm 4.1, x n + 1 x n = β 1 n ( d 1 n x n ) and u n x n = α 1 n ( e 1 n x n ) , we have
x n + 1 t 1 n + 1 ( 1 β 1 n ) x n + β 1 n d 1 n ( 1 β 1 n ) x n β 1 n R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) ρ 1 n F ( z 2 n , y n ) ) β 1 n d 1 n R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) ρ 1 n F ( z 2 n , y n ) ) β 1 n ( d 1 n R ρ 1 n , M A , η 1 ( A ( u n ) + ρ 1 n f ( u n ) ρ 1 n F ( z 1 n , y n ) ) + R ρ 1 n , M A , η 1 ( A ( u n ) + ρ 1 n f ( u n ) ρ 1 n F ( z 1 n , y n ) ) R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) ρ 1 n F ( z 2 n , y n ) ) ) β 1 n ζ 1 n d 1 n u n + β 1 n R ρ 1 n , M A , η 1 ( A ( u n ) + ρ 1 n f ( u n ) ρ 1 n F ( z 1 n , y n ) ) R ρ 1 n , M A , η 1 ( A ( s 1 n ) + ρ 1 n f ( s 1 n ) ρ 1 n F ( z 2 n , y n ) ) β 1 n ζ 1 n d 1 n u n + β 1 n τ q 1 r 1 m 1 ρ 1 n [ ρ 1 n μ 1 γ 1 s 1 n u n + ν 1 y n y n + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q s 1 n u n ] ζ 1 n x n + 1 x n + β 1 n ζ 1 n u n x n + β 1 n τ q 1 r 1 m 1 ρ 1 n [ ρ 1 n μ 1 γ 1 s 1 n u n + ν 1 y n y n + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q s 1 n u n ] + β 1 n τ q 1 r 1 m 1 ρ 1 n [ ρ 1 n μ 1 γ 1 + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q ] s 1 n u n ζ 1 n x n + 1 x + ζ 1 n x n x + β 1 n ζ 1 n u n x n + β 1 n τ q 1 r 1 m 1 ρ 1 n [ ρ 1 n μ 1 γ 1 + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q ] × ( s 1 n x + x x n + x n u n ) ζ 1 n x n + 1 x + ( ζ 1 n + β 1 n θ 1 ( n ) ) x n x + β 1 n ( ζ 1 n + θ 1 ( n ) ) ( u n x + x n x ) + β 1 n θ 1 s 1 n x ζ 1 n x n + 1 x + ( ζ 1 n + 2 β 1 n θ 1 ( n ) + β 1 n ζ 1 n ) x n x + β 1 n ( ζ 1 n + θ 1 ( n ) ) u n x + β 1 n θ 1 ( n ) s 1 n x .
(29)
It follows from (26)-(29) that
x n + 1 x x n + 1 t 1 n + 1 + t 1 n + 1 x ζ 1 n x n + 1 x + [ ζ 1 n + 2 β 1 n θ 1 ( n ) + β 1 n ζ 1 n + β 1 n ( ζ 1 n + θ 1 ( n ) ) 1 1 ξ 1 n ( 1 α 1 n + ξ 1 n + α 1 n θ 1 ( n ) ) + ( 1 β 1 n ) ( 1 α 1 n + α 1 n θ 1 ( n ) ) ] x n x + [ β 1 n ( ζ 1 n + θ 1 ( n ) ) 1 1 ξ 1 n α 1 n τ q 1 r 1 m 1 ρ 1 n ν 1 + ( 1 β 1 n ) β 1 n α 1 n θ 1 ( n ) ρ 1 n ν 1 τ q 1 r 1 m 1 ρ 1 n ] y n y ,
and
x n + 1 x 1 1 ζ 1 n [ ζ 1 n + 2 β 1 n θ 1 ( n ) + β 1 n ζ 1 n + β 1 n ( ζ 1 n + θ 1 ( n ) ) 1 1 ξ 1 n ( 1 α 1 n + ξ 1 n + α 1 n θ 1 ( n ) ) + ( 1 β 1 n ) ( 1 α 1 n + α 1 n θ 1 ( n ) ) ] x n x + 1 1 ζ 1 n [ β 1 n ( ζ 1 n + θ 1 ( n ) ) 1 1 ξ 1 n α 1 n τ q 1 r 1 m 1 ρ 1 n ν 1 + ( 1 β 1 n ) β 1 n α 1 n θ 1 ( n ) ρ 1 n ν 1 τ q 1 r 1 m 1 ρ 1 n ] y n y ,
(30)
where
θ 1 ( n ) = τ q 1 r 1 m 1 ρ 1 n ( ρ 1 n μ 1 γ 1 + ω 1 q + c q ( ρ 1 n ) q κ 1 q + q σ 1 ω 1 q q φ 1 q ) .
For n 0 , we write
{ s 2 n = ( 1 α 2 n ) y n + α 2 n R ρ 2 n , N B , η 2 ( B ( y n ) + ρ 2 n g ( y n ) ρ 2 n G ( w n , x n ) ) ( w n T ( y n ) ) , t 2 n + 1 = ( 1 β 2 n ) y n + β 2 n R ρ 2 n , N B , η 2 ( B ( s 2 n ) + ρ 2 n g ( s 2 n ) ρ 2 n G ( w 2 n , x n ) ) ( w 2 n T ( s 2 n ) ) .
(31)
By using the hypotheses of the mappings B, g, G, T, N, η 2 and R ρ 2 n , N B , η 2 in Theorem 5.1, and the same method as the one above, we can get
that is,
(32)

where α 2 n ( e 2 n y n ) = v n y n , and θ 2 ( n ) = τ q 1 r 2 m 2 ρ 2 n ( ρ 2 n μ 2 γ 2 + ω 2 q + c q ( ρ 2 n ) q κ 2 q + q σ 2 ω 2 q q φ 2 q ) .

Moreover, we have
t 2 n + 1 y ( 1 β 2 n ) y n + β 2 n d 2 n ( 1 β 2 n ) y β 2 n R ρ 2 n , N B , η 2 ( B ( y ) + ρ 2 n g ( y ) ρ 2 n G ( w , x ) ) ( 1 β 2 n ) y n y + β 2 n d 2 n R ρ 2 n , N B , η 2 ( B ( v n ) + ρ 2 n g ( v n ) ρ 2 n G ( w 2 n , x n ) ) + β 2 n R ρ 2 n , N B , η 2 ( B ( v n ) + ρ 2 n g ( v n ) ρ 2 n G ( w 2 n , x n ) ) R ρ 2 n , N B , η 2 ( B ( y ) + ρ 2 n g ( y ) ρ 2 n G ( w , x ) ) ( 1 β 2 n ) y n y + ζ 2 n β 2 n d 2 n y n + β 2 n ζ 2 n y n s 2 n + β 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 x n x + β 2 n θ 2 ( n ) v n y ( 1 β 2 n + ζ 2 n + β 2 n ζ 2 n ) y n y + β 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 x n x + ζ 2 n y n + 1 y + β 2 n ζ 2 n s 2 n y + β 2 n θ 2 ( n ) v n y ,

for (19) y n + 1 y n = β 2 n ( d 2 n y n ) .

It follows from (26) that
y n + 1 t 2 n + 1 ( 1 β 2 n ) y n + β 2 n d 2 n ( 1 β 2 n ) y n β 2 n R ρ 2 n , N B , η 2 ( B ( s 2 n ) + ρ 2 n g ( s 2 n ) ρ 2 n G ( w 2 n , x n ) ) β 2 n d 2 n R ρ 2 n , N B , η 2 ( B ( s 2 n ) + ρ 2 n g ( s 2 n ) ρ 2 n G ( w 2 n , x n ) ) β 2 n ζ 2 n d 2 n v n + β 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 x n x n + β 2 n θ 2 ( n ) s 2 n v n ζ 2 n ( y n + 1 y + y n y ) + ζ 2 n β 2 n ( y n y + v n y ) + β 2 n θ 2 ( n ) ( s 2 n y + y v n ) ζ 2 n y n + 1 y + ( ζ 2 n + ζ 2 n β 2 n ) y n y + ( ζ 2 n β 2 n + β 2 n θ 2 ( n ) ) v n y + β 2 n θ 2 ( n ) s 2 n y .
(33)
Combining (30), (31), (32), (33) and (19), we have
y n + 1 y 1 1 2 ζ 2 n [ ( 1 β 2 n + 2 β 2 n ζ 2 n + 2 ζ 2 n ) y n y + β 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 x n x + ( 2 β 2 n θ 2 ( n ) + ζ 2 n β 2 n ) v n y + β 2 n ( ζ 2 n + θ 2 ( n ) ) s 2 n y ] 1 1 2 ζ 2 n ( 1 β 2 n + 2 β 2 n ζ 2 n + 2 ζ 2 n + ( 2 β 2 n θ 2 ( n ) + ζ 2 n β 2 n ) 1 1 ξ 2 n ( 2 α 2 n + α 2 n θ 2 ( n ) ) + β 2 n ( ζ 2 n + θ 2 ( n ) ) [ ( 1 α 2 n ) + α 2 n θ 2 ( n ) ] ) y n y + 1 1 2 ζ 2 n ( β 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 + β 2 n ( ζ 2 n + θ 2 ( n ) ) α 2 n τ q 1 r 2 m 2 ρ 2 n ρ 2 n ν 2 + ( 2 β 2 n θ 2 ( n ) + ζ 2 n β 2 n ) 1 1 ξ 2 n α 2 n ρ 2 n ν 2 τ q 1 r 2 m 2 ρ 2 n ) x n x .
(34)
By using (22) and (23), let
(35)
where
Let A = ( x n + 1 x , y n + 1 y ) T and B = ( x n x , y n y ) T , then from (33), (34) and (35), we have a Ψ b , where
Ψ = ( a 11 a 12 a 21 a 22 ) ,
(36)
which is called the matrix for a nonlinear set-valued inclusions system involving ( A , η ) -accretive mappings. By using [16], we have
A Ψ B .
(37)
Let
Ψ = max { a 11 , a 12 , a 21 , a 22 } .
It follows from (21)-(23), Theorem 3.1 and [15] that 0 < Ψ < 1 and there exist x , y X and z S ( x ) , w T ( y ) [17] such that
{ Q 1 ( x ) = x , Q 2 ( y ) = y ;
and the sequence { x n , y n } n = 0 generated by Ishikawa-hybrid proximal Algorithm 4.1 converges linearly to a solution ( x , y ) of the problem (3) as the convergence rate
Ψ = max { 1 ( α 1 + β 1 ) + ( α 1 + β 1 ) θ 1 + ( 2 2 α 1 + α 1 θ 1 ) β 1 θ 1 , α 1 β 1 ν 1 θ 1 τ q 1 r 1 m 1 ρ 1 ( 1 + ρ 1 β 1 ρ 1 ) , 1 β 2 + 2 β 2 θ 2 ( 2 α 2 + α 2 θ 2 ) + β 2 θ 2 ( 1 α 2 + α 2 θ 2 ) , ( 1 β 2 + β 2 θ 2 ) + 3 ( 1 α 2 + α 2 θ 2 ) β 2 θ 2 + β 2 θ 2 } ,

where c q > 0 is the same as in Lemma 2.7, ρ i ( 0 , r i m i ) ( i = 1 , 2 ). This completes the proof. □

Remark 5.2 For a suitable choice of the mappings A, B, η i , F, G, M, N, S, T, f, g and X, we can obtain several known results in [7, 9, 11, 12] as special cases of Theorem 5.1.

Declarations

Authors’ Affiliations

(1)
College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

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© Li and Qiu; licensee Springer 2012

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