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Affine algorithms for the split variational inequality and equilibrium problems

Abstract

An affine algorithm for the split variational inequality and equilibrium problems is presented. Strong convergence result is given.

1 Introduction

In the present manuscript, we focus on the following split variational inequality and equilibrium problem: Finding a point x such that

x GVI(B,ψ,C)andψ ( x ) EP(F,A),
(1.1)

where GVI(B,ψ,C) is the solution set of the generalized variational inequality of finding uC, ψ(u)C such that

B u , ψ ( v ) ψ ( u ) 0,ψ(v)C,
(1.2)

and EP(F,A) is the solution set of the equilibrium problem, which is to find x C such that

F ( x , y ) + A x , y x 0,yC.
(1.3)

Our main motivations are inspired by the following reasons.

Reason 1 Recently, the split problems have been considered by some authors. Especially, the split feasibility problem which can mathematically be formulated as the problem of finding a point x ˜ with the property

x ˜ Candg( x ˜ )Q

has received much attention due to its applications in signal processing and image reconstruction with particular progress in intensity modulated radiation therapy [113]. Note that the involved operator g is a bounded linear operator. However, in the present paper, the involved mapping ψ in (1.1) is a nonlinear mapping.

Reason 2 The variational inequality problem [1424] and equilibrium problem [2327], which include the fixed point problems and optimization problems [2830], have been studied by many authors. It is an interesting topic associated with the analytical and algorithmic approach to the variational inequality and equilibrium problems.

Motivated and inspired by the results in the literature, we present an affine algorithm for solving the split problem (1.1). Strong convergence theorem is given under some mild assumptions.

2 Preliminaries

Let H be a real Hilbert space with the inner product , and the norm , respectively. Let C be a nonempty closed convex subset of H.

2.1 Monotonicity and convexity

An operator A:CH is said to be monotone if xy,AxAy0 for all x,yC. A:CH is said to be strongly monotone if there exists a constant γ>0 such that xy,AxAyγ x y 2 for all x,yC. A:CH is called an inverse-strongly-monotone operator if there exists α>0 such that xy,AxAyα A x A y 2 for all x,yC. Let g:CC be a nonlinear operator. A:CH is said to be α-inverse strongly g-monotone iff g(x)g(y),AxAyα A x A y 2 for all x,yC and for some α>0. Let B be a mapping of H into 2 H . The effective domain of B is denoted by dom(B), that is, dom(B)={xH:Bx}. A multi-valued mapping B is said to be a monotone operator on H iff xy,uv0 for all x,ydom(B), uBx, and vBy. A monotone operator B on H is said to be maximal iff its graph is not strictly contained in the graph of any other monotone operator on H.

A function F:HR is said to be convex if for any x,yH and for any λ[0,1], F(λx+(1λ)y)λF(x)+(1λ)F(y).

2.2 Nonexpansivity and continuity

A mapping T:CC is said to be nonexpansive [3138] if TxTyxy for all x,yC. We use Fix(T) to denote the set of fixed points of T. T:CC is called a firmly nonexpansive mapping if, for all x,yC, T x T y 2 xy,TxTy. It is known that T is firmly nonexpansive if and only if a mapping 2TI is nonexpansive, where I is the identity mapping on H. T:CH is said to be L-Lipschitz continuous if there exists a constant L>0 such that TxTyLxy for all x,yC. In such a case, T is said to be L-Lipschitz continuous. Given a nonempty, closed convex subset C of H, the mapping that assigns every point xH to its unique nearest point in C is called a metric projection onto C and denoted by P C , that is, P C xC and x P C x=inf{xy:yC}. The metric projection P C is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is x P C x,y P C x0 for all xH, yC.

2.3 Equilibrium problem

In this paper, we consider the split problem (1.1). In the sequel, we assume that the solution set S of (1.1) is nonempty.

Problem 2.1 Assume that

  1. (A1)

    B:CH is an α-inverse strongly ψ-monotone mapping;

  2. (A2)

    ψ:CC is a weakly continuous and γ-strongly monotone mapping such that R(ψ)=C;

  3. (A3)

    F:C×CR is a bifunction;

  4. (A4)

    A:CH is a β-inverse-strongly monotone mapping.

Our objective is to

find  x GVI(B,ψ,C) such that ψ ( x ) EP(F,A),

where F satisfies the following conditions:

  1. (F1)

    F(x,x)=0 for all xC;

  2. (F2)

    F is monotone, i.e., F(x,y)+F(y,x)0 for all x,yC;

  3. (F3)

    for each x,y,zC, lim t 0 F(tz+(1t)x,y)F(x,y);

  4. (F4)

    for each xC, yF(x,y) is convex and lower semicontinuous.

In order to solve Problem 2.1, we need the following useful lemmas.

2.4 Useful lemmas

The following three lemmas are important tools for our main results in the next section. Note that these lemmas are used extensively in the literature.

Lemma 2.2 (Combettes and Hirstoaga’s lemma [26])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:C×CR be a bifunction which satisfies conditions (F1)-(F4). Let λ>0 and xC. Then there exists zC such that

F(z,y)+ 1 λ yz,zx0,yC.

Further, if T λ (x)={zC:F(z,y)+ 1 λ yz,zx0 for all yC}, then the following hold:

  1. (a)

    T λ is single-valued and T λ is firmly nonexpansive;

  2. (b)

    EP(F) is closed and convex and EP(F)=Fix( T λ ).

Lemma 2.3 (Suzuki’s lemma [39])

Let { x n } and { y n } be bounded sequences in a Banach space X and let { β n } be a sequence in [0,1] with 0< lim inf n β n lim sup n β n <1. Suppose x n + 1 =(1 β n ) y n + β n x n for all n0 and lim sup n ( y n + 1 y n x n + 1 x n )0. Then lim n y n x n =0.

Lemma 2.4 (Xu’s lemma [40])

Assume that { a n } is a sequence of nonnegative real numbers such that a n + 1 (1 γ n ) a n + δ n γ n , where { γ n } is a sequence in (0,1) and { δ n } is a sequence such that n = 1 γ n = and lim sup n δ n 0 (or n = 1 | δ n γ n |<). Then lim n a n =0.

3 Algorithms and convergence analysis

In this section, we first present our algorithm for solving Problem 2.1. Assume that the conditions in Problem 2.1 are all satisfied.

Algorithm 3.1 Let C be a nonempty closed and convex subset of a real Hilbert space H.

Step 0. (Initialization)

x 0 C.

Step 1. (Projection step) For { x n }, let the sequence { u n } be generated iteratively by

u n = P C [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] ,n0,

where P C is the metric projection, { α n }[0.1] is a real number sequence, φ:CH is an L-Lipschitz continuous mapping and δ>0 is a constant.

Step 2. (Proximal step) Find { z n } such that

F( z n ,y)+A u n ,y z n + 1 λ n y z n , z n u n 0,yC,

where { λ n }(0,) is a real number sequence.

Step 3. (Affine step) For the above sequences { x n } and { z n }, let the (n+1)th sequence { x n + 1 } be generated by

ψ( x n + 1 )= β n ψ( x n )+(1 β n ) z n ,n0,

where { β n }[0,1] is a real number sequence.

Theorem 3.2 Suppose S. Assume that the following restrictions are satisfied:

  1. (C1)

    λ n (a,b)(0,2β), μ n (c,d)(0,2α) and γ(Lδ,2α);

  2. (C2)

    lim n ( μ n + 1 μ n )=0 and lim n ( λ n + 1 λ n )=0;

  3. (C3)

    lim n α n =0 and n α n =;

  4. (C4)

    β n [ ξ 1 , ξ 2 ](0,1).

Then the sequence { x n } generated by Algorithm 3.1 converges strongly to x S, which solves the following variational inequality:

δ φ ( x ) ψ ( x ) , ψ ( x ) ψ ( x ) 0,xΩ.
(3.1)

Remark 3.3 The solution of variational inequality (3.1) is unique. As a matter of fact, if x ˜ S also solves (3.1), we have

δ φ ( x ) ψ ( x ) , ψ ( x ˜ ) ψ ( x ) 0and δ φ ( x ˜ ) ψ ( x ˜ ) , ψ ( x ) ψ ( x ˜ ) 0.

Adding up the above two inequalities, we deduce

δ φ ( x ˜ ) ψ ( x ˜ ) δ φ ( x ) + ψ ( x ) , ψ ( x ) ψ ( x ˜ ) 0.

It follows that

ψ ( x ) ψ ( x ˜ ) 2 δ φ ( x ) φ ( x ˜ ) , ψ ( x ) ψ ( x ˜ ) δ φ ( x ) φ ( x ˜ ) ψ ( x ) ψ ( x ˜ ) ,

which implies that

ψ ( x ) ψ ( x ˜ ) δ φ ( x ) φ ( x ˜ ) .

Since ψ is γ-strongly monotone, we have

γ x x ˜ 2 ψ ( x ) ψ ( x ˜ ) , x x ˜ ψ ( x ) ψ ( x ˜ ) x x ˜ .

Hence,

γ x x ˜ ψ ( x ) ψ ( x ˜ ) δ φ ( x ) φ ( x ˜ ) δL x x ˜ .

This deduces the contraction because of δL<γ by the assumption. Therefore, x = x ˜ . So, the solution of variational inequality (3.1) is unique.

Remark 3.4 Using the characteristic inequality of the projection, we have

x ˘ GVI(B,ψ,C)ψ( x ˘ )= P C ( ψ ( x ˘ ) ν B x ˘ ) ,ν>0.

Remark 3.5

( ψ ( x ) μ B x ) ( ψ ( y ) μ B y ) 2 ψ ( x ) ψ ( y ) 2 +μ(μ2α) B x B y 2 .

In fact,

( ψ ( x ) μ B x ) ( ψ ( y ) μ B y ) 2 = ψ ( x ) ψ ( y ) 2 2 μ B x B y , ψ ( x ) ψ ( y ) + μ 2 B x B y 2 ψ ( x ) ψ ( y ) 2 2 μ α B x B y 2 + μ 2 B x B y 2 ψ ( x ) ψ ( y ) 2 + μ ( μ 2 α ) B x B y 2 .

Next, we prove Theorem 3.2.

Proof Let x Ω. Hence x GVI(B,ψ,C) and ψ( x )EP(F,A). Since μ n >0, from Remark 3.4 we have ψ( x )= P C [ψ( x ) μ n B x ] for all n0. Thus,

u n ψ ( x ) = P C [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] P C [ ψ ( x ) μ n B x ] α n ( δ φ ( x n ) ψ ( x ) + μ n B x ) + ( 1 α n ) ( ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) ) α n δ φ ( x n ) δ φ ( x ) + α n δ φ ( x ) ψ ( x ) + μ n B x + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) α n δ L x n x + α n δ φ ( x ) ψ ( x ) + μ n B x + ( 1 α n ) ψ ( x n ) ψ ( x ) α n δ L / γ ψ ( x n ) ψ ( x ) + α n δ φ ( x ) ψ ( x ) + μ n B x + ( 1 α n ) ψ ( x n ) ψ ( x ) = [ 1 ( 1 δ L / γ ) α n ] ψ ( x n ) ψ ( x ) + α n δ φ ( x ) ψ ( x ) + μ n B x [ 1 ( 1 δ L / γ ) α n ] ψ ( x n ) ψ ( x ) + α n ( δ φ ( x ) ψ ( x ) + 2 α B x ) .
(3.2)

By Algorithm 3.1, we have z n = T λ n (I λ n A) u n for all n0. Noting that ψ( x )EP(F,A), we deduce ψ( x )= T λ n (I λ n A)ψ( x ) for all n0. It follows that

ψ ( x n + 1 ) ψ ( x ) β n ψ ( x n ) ψ ( x ) + ( 1 β n ) T λ n ( I λ n A ) u n T λ n ( I λ n A ) ψ ( x ) β n ψ ( x n ) ψ ( x ) + ( 1 β n ) u n ψ ( x ) β n ψ ( x n ) ψ ( x ) + ( 1 β n ) [ 1 ( 1 δ L / γ ) α n ] ψ ( x n ) ψ ( x ) + ( 1 β n ) α n ( δ φ ( x ) ψ ( x ) + 2 α B x ) = [ 1 ( 1 δ L / γ ) ( 1 β n ) α n ] ψ ( x n ) ψ ( x ) + ( 1 δ L / γ ) ( 1 β n ) α n δ φ ( x ) ψ ( x ) + 2 α B x 1 δ L / γ .

By induction

ψ ( x n ) ψ ( x ) max { ψ ( x 0 ) ψ ( x ) , δ φ ( x ) ψ ( x ) + 2 α B x 1 δ L / γ } .

Hence, {ψ( x n )} is bounded. Since ψ is γ-strongly monotone, we can get (by a similar technique as that in Remark 3.3) γ x n x ψ( x n )ψ( x ). So, x n x 1 γ ψ( x n )ψ( x ) 1 γ max{ψ( x 0 )ψ( x ), δ φ ( x ) ψ ( x ) + 2 α B x 1 δ L / γ }. This implies that { x n } is bounded. Next, we show x n + 1 x n 0. From Step 2 in Algorithm 3.1, we have

F( z n ,y)+ 1 λ n y z n , z n ( u n λ n A u n ) 0,yC.

Taking y= z n + 1 , we get

F( z n , z n + 1 )+ 1 λ n z n + 1 z n , z n ( u n λ n A u n ) 0.

Similarly, we also have

F( z n + 1 , z n )+ 1 λ n + 1 z n z n + 1 , z n + 1 ( u n + 1 λ n + 1 A u n + 1 ) 0.

Adding up the above two inequalities, we get

F( z n , z n + 1 )+F( z n + 1 , z n )+A u n A u n + 1 , z n + 1 z n + z n + 1 z n , z n u n λ n z n + 1 u n + 1 λ n + 1 0.

By the monotonicity of F, we have

F( z n , z n + 1 )+F( z n + 1 , z n )0.

So,

A u n A u n + 1 , z n + 1 z n + z n + 1 z n , z n u n λ n z n + 1 u n + 1 λ n + 1 0.

Thus,

λ n A u n A u n + 1 , z n + 1 z n + z n + 1 z n , z n z n + 1 + z n + 1 u n λ n λ n + 1 ( z n + 1 u n + 1 ) 0.

It follows that

z n + 1 z n 2 λ n A u n A u n + 1 , z n + 1 z n + z n + 1 z n , u n + 1 u n + ( 1 λ n λ n + 1 ) ( z n + 1 u n + 1 ) = ( I λ n A ) u n + 1 ( I λ n A ) u n , z n + 1 z n + z n + 1 z n , ( 1 λ n λ n + 1 ) ( z n + 1 u n + 1 ) ( I λ n A ) u n + 1 ( I λ n A ) u n z n + 1 z n + | 1 λ n λ n + 1 | z n + 1 z n z n + 1 u n + 1 z n + 1 z n ( u n + 1 u n + | 1 λ n λ n + 1 | z n + 1 u n + 1 )

and hence

z n + 1 z n u n + 1 u n + | 1 λ n λ n + 1 | z n + 1 u n + 1 u n + 1 u n + 1 a | λ n + 1 λ n | z n + 1 u n + 1 .

By Algorithm 3.1, we have

u n + 1 u n = P C [ α n + 1 δ φ ( x n + 1 ) + ( 1 α n + 1 ) ( ψ ( x n + 1 ) μ n + 1 B x n + 1 ) ] P C [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] [ α n + 1 δ φ ( x n + 1 ) + ( 1 α n + 1 ) ( ψ ( x n + 1 ) μ n + 1 B x n + 1 ) ] [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] α n + 1 δ φ ( x n + 1 ) φ ( x n ) + δ | α n + 1 α n | φ ( x n ) + ( 1 α n + 1 ) ψ ( x n + 1 ) μ n + 1 B x n + 1 ( ψ ( x n ) μ n + 1 B x n ) + | α n + 1 α n | ψ ( x n ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) α n + 1 δ L x n + 1 x n + ( 1 α n + 1 ) ψ ( x n + 1 ) ψ ( x n ) + | α n + 1 α n | ( δ φ ( x n ) + ψ ( x n ) ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) α n + 1 ( δ L / γ ) ψ ( x n + 1 ) ψ ( x n ) + ( 1 α n + 1 ) ψ ( x n + 1 ) ψ ( x n ) + | α n + 1 α n | ( δ φ ( x n ) + ψ ( x n ) ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) = [ 1 ( 1 δ L / γ ) α n + 1 ] ψ ( x n + 1 ) ψ ( x n ) + | α n + 1 α n | ( δ φ ( x n ) + ψ ( x n ) ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) .

Therefore,

z n + 1 z n [ 1 ( 1 δ L / γ ) α n + 1 ] ψ ( x n + 1 ) ψ ( x n ) + | α n + 1 α n | ( δ φ ( x n ) + ψ ( x n ) ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) + 1 a | λ n + 1 λ n | z n + 1 u n + 1 .

It follows that

z n + 1 z n ψ ( x n + 1 ) ψ ( x n ) | α n + 1 α n | ( δ φ ( x n ) + ψ ( x n ) ) + | μ n + 1 μ n | B ( x n ) + | α n + 1 μ n + 1 α n μ n | B ( x n ) + 1 a | λ n + 1 λ n | z n + 1 u n + 1 .

Since lim n α n =0, lim n ( μ n + 1 μ n )=0, lim n ( λ n + 1 λ n )=0 and the sequences {φ( x n )}, {ψ( x n )}, { z n }, { u n } and {B x n } are bounded, we have

lim sup n ( z n + 1 z n ψ ( x n + 1 ) ψ ( x n ) ) 0.

By Lemma 2.3, we obtain

lim n z n ψ ( x n ) =0.

Hence,

lim n ψ ( x n + 1 ) ψ ( x n ) = lim n (1 β n ) z n ψ ( x n ) =0.

This together with the γ-strong monotonicity of ψ implies that

lim n x n + 1 x n =0.

By the convexity of the norm, we have

ψ ( x n + 1 ) ψ ( x ) 2 = β n ( ψ ( x n ) ψ ( x ) ) + ( 1 β n ) ( z n ψ ( x ) ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) z n ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) [ α n ( δ φ ( x n ) ψ ( x ) + μ n B x ) + ( 1 α n ) ( ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) ) ] 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) [ α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) 2 + 2 α n ( 1 α n ) δ φ ( x n ) ψ ( x ) + μ n B x ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) ] β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) ( 1 α n ) ( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) 2 + α n M ,
(3.3)

where M>0 is some constant. From Remark 3.5, we derive

( ψ ( x n ) μ n B x n ) ( ψ ( x ) μ n B x ) 2 ψ ( x n ) ψ ( x ) 2 + μ n ( μ n 2α) B x n B x 2 .

Thus,

ψ ( x n + 1 ) ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) ( 1 α n ) ( ψ ( x n ) ψ ( x ) 2 + μ n ( μ n 2 α ) B x n B x 2 ) + α n M ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) ( 1 α n ) μ n ( μ n 2 α ) B x n B x 2 + α n M .

So,

( 1 β n ) ( 1 α n ) μ n ( 2 α μ n ) B x n B x 2 ψ ( x n ) ψ ( x ) 2 ψ ( x n + 1 ) ψ ( x ) 2 + α n M ( ψ ( x n ) ψ ( x ) + ψ ( x n + 1 ) ψ ( x ) ) ψ ( x n + 1 ) ψ ( x n ) + α n M .

Since α n 0, ψ( x n + 1 )ψ( x n )0 and lim inf n (1 β n )(1 α n ) μ n (2α μ n )>0, we obtain

lim n B x n B x =0.

Set y n =ψ( x n ) μ n B x n (ψ( x ) μ n B x ) for all n. By using the property of projection, we get

u n ψ ( x ) 2 = P C [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] P C [ ψ ( x ) μ n B x ] 2 α n ( δ φ ( x n ) ψ ( x ) + μ n B x ) + ( 1 α n ) y n , u n ψ ( x ) = 1 2 { α n ( δ φ ( x n ) ψ ( x ) + μ n B x ) + ( 1 α n ) y n 2 + u n ψ ( x ) 2 α n ( δ φ ( x n ) ψ ( x ) + μ n B x ) + ( 1 α n ) y n u n + ψ ( x ) 2 } 1 2 { α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + ( 1 α n ) ψ ( x n ) ψ ( x ) 2 + u n ψ ( x ) 2 α n ( δ φ ( x n ) ψ ( x ) + μ n B x y n ) + ψ ( x n ) u n μ n ( B x n B x ) 2 } = 1 2 { α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + ( 1 α n ) ψ ( x n ) ψ ( x ) 2 + u n ψ ( x ) 2 ψ ( x n ) u n 2 μ n 2 B x n B x α n 2 δ φ ( x n ) ψ ( x ) + μ n B x y n 2 + 2 μ n α n B x n B x , δ φ ( x n ) ψ ( x ) + μ n B x y n + 2 μ n ψ ( x n ) u n , B x n B x 2 α n ψ ( x n ) u n , δ φ ( x n ) ψ ( x ) + μ n B x y n } .
(3.4)

It follows that

u n ψ ( x ) 2 α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + ( 1 α n ) ψ ( x n ) ψ ( x ) 2 ψ ( x n ) u n 2 + 2 μ n α n B x n B x δ φ ( x n ) ψ ( x ) + μ n B x y n + 2 μ n ψ ( x n ) u n B x n B x + 2 α n ψ ( x n ) u n δ φ ( x n ) ψ ( x ) + μ n B x y n .
(3.5)

From (3.3) and (3.5), we have

ψ ( x n + 1 ) ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) u n ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + ( 1 α n ) ( 1 β n ) ψ ( x n ) ψ ( x ) 2 ( 1 β n ) ψ ( x n ) u n 2 + 2 μ n ( 1 β n ) α n B x n B x δ φ ( x n ) ψ ( x ) + μ n B x y n + 2 μ n ( 1 β n ) ψ ( x n ) u n B x n B x + 2 ( 1 β n ) α n ψ ( x n ) u n δ φ ( x n ) ψ ( x ) + μ n B x y n ψ ( x n ) ψ ( x ) 2 + α n δ φ ( x n ) ψ ( x ) + μ n B x 2 ( 1 β n ) ψ ( x n ) u n 2 + 2 μ n α n B x n B x δ φ ( x n ) ψ ( x ) + μ n B x y n + 2 μ n ψ ( x n ) u n B x n B x + 2 α n ψ ( x n ) u n δ φ ( x n ) ψ ( x ) + μ n B x y n .

Then we obtain

( 1 β n ) ψ ( x n ) u n 2 ( ψ ( x n ) ψ ( x ) + ψ ( x n + 1 ) ψ ( x ) ) ψ ( x n + 1 ) ψ ( x n ) + α n δ φ ( x n ) ψ ( x ) + μ n B x 2 + 2 μ n α n B x n B x δ φ ( x n ) ψ ( x ) + μ n B x y n + 2 μ n ψ ( x n ) u n B x n B x + 2 α n ψ ( x n ) u n δ φ ( x n ) ψ ( x ) + μ n B x y n .

Since lim n α n =0, lim n ψ( x n + 1 )ψ( x n )=0 and lim n B x n B x =0, we have

lim n ψ ( x n ) u n =0.
(3.6)

Next, we prove lim sup n δφ( x )ψ( x ), u n ψ( x )0, where x is the unique solution of (3.1). We take a subsequence { u n i } of { u n } such that

lim sup n δ φ ( x ) ψ ( x ) , u n ψ ( x ) = lim i δ φ ( x ) ψ ( x ) , u n i ψ ( x ) = lim i δ φ ( x ) ψ ( x ) , ψ ( x n i ) ψ ( x ) .
(3.7)

Since { x n i } is bounded, there exists a subsequence { x n i j } of { x n i } which converges weakly to some point zC. Without loss of generality, we may assume that x n i z. This implies that ψ( x n i )ψ(z) due to the weak continuity of ψ. Now, we show zS. We firstly show zEP(F,A). Since z n = T λ n ( u n λ n A u n ), for any yC, we have

F( z n ,y)+ 1 λ n y z n , z n ( u n λ n A u n ) 0.

From the monotonicity of F, we have

1 λ n y z n , z n ( u n λ n A u n ) F(y, z n ),yC.

Hence,

y z n i , z n i u n i λ n i + A u n i F(y, z n i ),yC.
(3.8)

Put v t =ty+(1t)z for all t(0,1] and yC. Then we have v t C. So, from (3.8) we have

v t z n i , A v t v t z n i , A v t v t z n i , z n i u n i λ n i + A u n i + F ( v t , z n i ) = v t z n i , A v t A z n i + v t z n i , A z n i A u n i v t z n i , z n i u n i λ n i + F ( v t , z n i ) .
(3.9)

Note that A z n i A u n i 1 β z n i u n i 0. Further, from the monotonicity of A, we have v t z n i ,A v t A z n i 0. Letting i in (3.9), we have v t z,A v t F( v t ,z). This together with (F1), (F4) implies that

0 = F ( v t , v t ) t F ( v t , y ) + ( 1 t ) F ( v t , z ) t F ( v t , y ) + ( 1 t ) v t z , A v t = t F ( v t , y ) + ( 1 t ) t y z , A v t ,

and hence 0F( v t ,y)+(1t)A v t ,yz. Letting t0, we have 0F(z,y)+yz,Az. This implies that zEP(F,A). Next, we only need to prove zGVI(B,ψ,C). Set

Rv={ B v + N C ( v ) , v C , , v C .

By [41], we know that R is maximal ψ-monotone. Let (v,w)G(R). Since wBv N C (v) and x n C, we have ψ(v)ψ( x n ),wBv0. Noting that u n = P C [ α n δφ( x n )+(1 α n )(ψ( x n ) μ n B x n )], we get

ψ ( v ) u n , u n [ α n δ φ ( x n ) + ( 1 α n ) ( ψ ( x n ) μ n B x n ) ] 0.

It follows that

ψ ( v ) u n , u n ψ ( x n ) μ n + B x n α n μ n ( δ φ ( x n ) ψ ( x n ) + μ n B x n ) 0.

Then

ψ ( v ) ψ ( x n i ) , w ψ ( v ) ψ ( x n i ) , B v ψ ( v ) ψ ( x n i ) , B v ψ ( v ) u n i , u n i ψ ( x n i ) μ n i ψ ( v ) u n i , B x n i + α n i μ n i ψ ( v ) u n i , δ φ ( x n i ) ψ ( x n i ) + μ n i B x n i = ψ ( v ) ψ ( x n i ) , B v B x n i + ψ ( v ) ψ ( x n i ) , B x n i ψ ( v ) u n i , u n i ψ ( x n i ) μ n i ψ ( v ) u n i , B x n i + α n i μ n i ψ ( v ) u n i , δ φ ( x n i ) ψ ( x n i ) + μ n i B x n i ψ ( v ) u n i , u n i ψ ( x n i ) μ n i ψ ( x n i ) u n i , B x n i + α n i μ n i ψ ( v ) u n i , δ φ ( x n i ) ψ ( x n i ) + μ n i B x n i .
(3.10)

Since ψ( x n i ) u n i 0 and ψ( x n i )ψ(z), we deduce that ψ(v)ψ(z),w0 by taking i in (3.10). Thus, z R 1 0 by the maximal ψ-monotonicity of R. Hence, zGVI(B,ψ,C). Therefore, zS. From (3.7), we obtain

lim sup n δ φ ( x ) ψ ( x ) , u n ψ ( x ) = lim i δ φ ( x ) ψ ( x ) , ψ ( x n i ) ψ ( x ) = δ φ ( x ) ψ ( x ) , ψ ( z ) ψ ( x ) 0 .

Note that

u n ψ ( x ) 2 α n ( δ φ ( x n ) ψ ( x ) ) + ( 1 α n ) y n , u n ψ ( x ) α n δ φ ( x n ) φ ( x ) , u n ψ ( x ) + α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) + ( 1 α n ) ψ ( x n ) μ n B x n ( ψ ( x ) μ n B x ) u n ψ ( x ) α n L δ x n x u n ψ ( x ) + α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) + ( 1 α n ) ψ ( x n ) ψ ( x ) u n ψ ( x ) α n ( δ L / γ ) ψ ( x n ) ψ ( x ) u n ψ ( x ) + α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) + ( 1 α n ) ψ ( x n ) ψ ( x ) u n ψ ( x ) = [ 1 ( 1 L δ / γ ) α n ] ψ ( x n ) ψ ( x ) u n ψ ( x ) + α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) = 1 ( 1 L δ / γ ) α n 2 ψ ( x n ) ψ ( x ) 2 + 1 2 u n ψ ( x ) 2 + α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) .

It follows that

u n ψ ( x ) 2 [ 1 ( 1 L δ / γ ) α n ] ψ ( x n ) ψ ( x ) 2 + 2 α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) .

Therefore,

ψ ( x n + 1 ) ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) u n ψ ( x ) 2 β n ψ ( x n ) ψ ( x ) 2 + ( 1 β n ) [ 1 ( 1 δ L / γ ) α n ] ψ ( x n ) ψ ( x ) 2 + 2 ( 1 β n ) α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) = [ 1 ( 1 δ L / γ ) ( 1 β n ) α n ] ψ ( x n ) ψ ( x ) 2 + 2 ( 1 β n ) α n δ φ ( x ) ψ ( x ) , u n ψ ( x ) = [ 1 ( 1 δ L / γ ) ( 1 β n ) α n ] ψ ( x n ) ψ ( x ) 2 + ( 1 δ L / γ ) ( 1 β n ) × α n ( 2 1 δ L / γ δ φ ( x ) ψ ( x ) , u n ψ ( x ) ) = ( 1 γ n ) ψ ( x n ) ψ ( x ) 2 + δ n γ n ,

where γ n =(1δL/γ)(1 β n ) α n and δ n = 2 1 δ L / γ δφ( x )ψ( x ), u n ψ( x ). It is easily seen that n γ n = and lim sup n δ n 0. We can therefore apply Lemma 2.4 to conclude that ψ( x n )ψ( x ) and x n x . This completes the proof. □

References

  1. Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692

    MATH  MathSciNet  Article  Google Scholar 

  2. Censor Y, Bortfeld T, Martin B, Trofimov A: A unified approach for inversion problems in intensity modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001

    Article  Google Scholar 

  3. Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017

    MATH  MathSciNet  Article  Google Scholar 

  4. Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2004, 20: 103–120. 10.1088/0266-5611/20/1/006

    MATH  MathSciNet  Article  Google Scholar 

  5. Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014

    MATH  Article  Google Scholar 

  6. Qu B, Xiu N: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 2005, 21: 1655–1665. 10.1088/0266-5611/21/5/009

    MATH  MathSciNet  Article  Google Scholar 

  7. Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017

    MATH  Article  Google Scholar 

  8. Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007

    MATH  Article  Google Scholar 

  9. Dang Y, Gao Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 2011., 27: Article ID 015007

    Google Scholar 

  10. Wang F, Xu HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010. doi:10.1155/2010/102085

    Google Scholar 

  11. Xu HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 2010., 26: Article ID 105018

    Google Scholar 

  12. Yao Y, Kim TH, Chebbi S, Xu HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 2012, 13(3):383–396.

    MATH  MathSciNet  Google Scholar 

  13. Yao Y, Liou YC, Shahzad N: A strongly convergent method for the split feasibility problem. Abstr. Appl. Anal. 2012., 2012: Article ID 125046

    Google Scholar 

  14. Stampacchia G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 1964, 258: 4413–4416.

    MATH  MathSciNet  Google Scholar 

  15. Korpelevich GM: An extragradient method for finding saddle points and for other problems. Èkon. Mat. Metody 1976, 12: 747–756.

    MATH  Google Scholar 

  16. Glowinski R: Numerical Methods for Nonlinear Variational Problems. Springer, New York; 1984.

    MATH  Book  Google Scholar 

  17. Iusem AN: An iterative algorithm for the variational inequality problem. Comput. Appl. Math. 1994, 13: 103–114.

    MATH  MathSciNet  Google Scholar 

  18. Noor MA: Some development in general variational inequalities. Appl. Math. Comput. 2004, 152: 199–277. 10.1016/S0096-3003(03)00558-7

    MATH  MathSciNet  Article  Google Scholar 

  19. Facchinei F, Pang JS Springer Series in Operations Research I. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.

    Google Scholar 

  20. Facchinei F, Pang JS Springer Series in Operations Research II. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.

    Google Scholar 

  21. Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 2003, 119: 185–201.

    MATH  MathSciNet  Article  Google Scholar 

  22. Yao JC: Variational inequalities with generalized monotone operators. Math. Oper. Res. 1994, 19: 691–705. 10.1287/moor.19.3.691

    MATH  MathSciNet  Article  Google Scholar 

  23. Ceng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 2006, 10: 1293–1303.

    MathSciNet  Google Scholar 

  24. Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032

    MATH  MathSciNet  Article  Google Scholar 

  25. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.

    MATH  MathSciNet  Google Scholar 

  26. Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.

    MATH  MathSciNet  Google Scholar 

  27. Yao Y, Cho YJ, Liou YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042

    MATH  MathSciNet  Article  Google Scholar 

  28. Qin X, Cho SY, Kang SM: Some results on fixed points of asymptotically strict quasi- ϕ -pseudocontractions in the intermediate sense. Fixed Point Theory Appl. 2012., 2012: Article ID 143

    Google Scholar 

  29. Zegeye H, Shahzad N, Alghamdi MA: Strong convergence theorems for a common point of solution of variational inequality, solutions of equilibrium and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 119. doi:10.1186/1687–1812–2012–119

    Google Scholar 

  30. Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasi- ψ -nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58. doi:10.1186/1687–1812–2012–58

    Google Scholar 

  31. Browder FE: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. Ration. Mech. Anal. 1967, 24: 82–90.

    MATH  MathSciNet  Article  Google Scholar 

  32. Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0

    MATH  Article  Google Scholar 

  33. Geobel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

    Chapter  Google Scholar 

  34. Lions PL: Approximation de points fixes de contractions. C. R. Hebd. Séances Acad. Sci., Sér. A, Sci. Math. 1977, 284: 1357–1359.

    MATH  Google Scholar 

  35. Opial Z: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 595–597.

    MathSciNet  Article  Google Scholar 

  36. Wittmann R: Approximation of fixed points of non-expansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119

    MATH  MathSciNet  Article  Google Scholar 

  37. Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615

    MATH  MathSciNet  Article  Google Scholar 

  38. Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059

    MATH  MathSciNet  Article  Google Scholar 

  39. Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 2005: 103–123.

    MATH  Article  Google Scholar 

  40. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 2: 1–17.

    Google Scholar 

  41. Zhang LJ, Chen JM, Hou ZB: Viscosity approximation methods for nonexpansive mappings and generalized variational inequalities. Acta Math. Sin. 2010, 53: 691–6988.

    MATH  MathSciNet  Google Scholar 

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Acknowledgements

Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Rudong Chen was supported in part by NSFC 11071279. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

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Yao, Y., Chen, R. & Liou, YC. Affine algorithms for the split variational inequality and equilibrium problems. Fixed Point Theory Appl 2013, 140 (2013). https://doi.org/10.1186/1687-1812-2013-140

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Keywords

  • affine algorithm
  • split method
  • variational inequality
  • equilibrium problem