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Extragradient method for convex minimization problem

Abstract

In this paper, we introduce and analyze a multi-step hybrid extragradient algorithm by combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method and the gradient-projection method (GPM) with regularization in the setting of infinite-dimensional Hilbert spaces. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a solution of the convex minimization problem (CMP) with constraints of several problems: finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to the unique solution of a hierarchical variational inequality problem (over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions and the CMP. The results presented in this paper improve and extend the corresponding results announced by many others.

MSC:49J30, 47H09, 47J20, 49M05.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and P C be the metric projection of H onto C. Let S:CH be a nonlinear mapping on C. We denote by Fix(S) the set of fixed points of S and by R the set of all real numbers. A mapping S:CH is called L-Lipschitz-continuous (or L-Lipschitzian) if there exists a constant L0 such that

SxSyLxy,x,yC.

In particular, if L=1 then S is called a nonexpansive mapping; if L[0,1) then S is called a contraction.

A mapping A:CH is said to be ζ-inverse-strongly monotone if there exists ζ>0 such that

AxAy,xyζ A x A y 2 ,x,yC.

It is clear that every inverse-strongly monotone mapping is a monotone and Lipschitz-continuous mapping. A mapping T:CC is said to be ξ-strictly pseudocontractive if there exists ξ[0,1) such that

T x T y 2 x y 2 +ξ ( I T ) x ( I T ) y 2 ,x,yC.

In this case, we also say that T is a ξ-strict pseudocontraction. In particular, whenever ξ=0, T becomes a nonexpansive mapping from C into itself.

Let A:CH be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point xC such that

Ax,yx0,yC.
(1.1)

The solution set of VIP (1.1) is denoted by VI(C,A).

The VIP (1.1) was first discussed by Lions [1]. There are many applications of VIP (1.1) in various fields; see e.g., [25]. It is well known that, if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (1.1) in Euclidean space  R n :

{ y n = P C ( x n τ A x n ) , x n + 1 = P C ( x n τ A y n ) , n 0 ,

with τ>0 a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see e.g., [719] and references therein, to name but a few.

Consider the following constrained convex minimization problem (CMP):

minimize  { f ( x ) : x C } ,
(1.2)

where f:CR is a real-valued convex functional. We denote by Γ the solution set of the CMP (1.2). If f is Fréchet differentiable, then the gradient-projection method (GPM) generates a sequence { x n } via the recursive formula

x n + 1 = P C ( x n λ f ( x n ) ) ,n0,
(1.3)

or more generally,

x n + 1 = P C ( x n λ n f ( x n ) ) ,n0,
(1.4)

where in both (1.3) and (1.4), the initial guess x 0 is taken from C arbitrarily, the parameters, λ or λ n , are positive real numbers, and P C is the metric projection from H onto C. The convergence of the algorithms (1.3) and (1.4) depends on the behavior of the gradient f. As a matter of fact, it is well known that if f is strongly monotone and Lipschitzian; namely, there are constants η,L>0 satisfying the properties

f ( x ) f ( y ) , x y η x y 2 and f ( x ) f ( y ) Lxy,x,yC,

then, for 0<λ<2η/ L 2 , the operator T:= P C (Iλf) is a contraction; hence, the sequence { x n } defined by (1.3) converges in norm to the unique solution of the CMP (1.2). More generally, if the sequence { λ n } is chosen to satisfy the property 0< lim inf n λ n lim sup n λ n <2η/ L 2 , then the sequence { x n } defined by (1.4) converges in norm to the unique minimizer of the CMP (1.2). However, if the gradient f fails to be strongly monotone, the operator T defined by T:= P C (Iλf) would fail to be contractive; consequently, the sequence { x n } generated by (1.3) may fail to converge strongly (see Section 4 in Xu [20]).

Theorem 1.1 (see [[20], Theorem 5.2])

Assume the CMP (1.2) is consistent and let Γ denote its solution set. Assume the gradient f satisfies the Lipschitz condition with constant L>0. Let V:CC be a ρ-contraction with coefficient ρ[0,1). Let a sequence { x n } be generated by the following hybrid gradient-projection algorithm (GPA):

x n + 1 = α n V x n +(1 α n ) P C ( x n λ n f ( x n ) ) ,n0.
(1.5)

Assume the sequence { λ n } satisfies the condition 0< lim inf n λ n lim sup n λ n <2/L, and, in addition, the following conditions are satisfied for { λ n } and { α n }[0,1]: (i)  α n 0, (ii)  n = 0 α n =, (iii)  n = 0 | α n + 1 α n |<, and (iv)  n = 0 | λ n + 1 λ n |<. Then the sequence { x n } converges in norm to a minimizer of CMP (1.2), which is also the unique solution x Γ to the VIP

( I V ) x , x x 0,xΓ.
(1.6)

In other words, x is the unique fixed point of the contraction P Γ V, x = P Γ V x .

On the other hand, let S and T be two nonexpansive mappings. In 2009, Yao et al. [21] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T, which is a solution to the VIP for monotone mapping IS; namely, find x ˜ Fix(T) such that

( I S ) x ˜ , p x ˜ 0,pFix(T).
(1.7)

The solution set of the hierarchical VIP (1.7) is denoted by Λ. It is not hard to check that solving the hierarchical VIP (1.7) is equivalent to the fixed point problem of the composite mapping P Fix ( T ) S, i.e., find x ˜ C such that x ˜ = P Fix ( T ) S x ˜ . The authors [21] introduced and analyzed the following iterative algorithm for solving the HVIP (1.7):

{ y n = β n S x n + ( 1 β n ) x n , x n + 1 = α n V x n + ( 1 α n ) T y n , n 0 .
(1.8)

Theorem 1.2 (see [[21], Theorem 3.2])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S and T be two nonexpansive mappings of C into itself. Let V:CC be a fixed contraction with α(0,1). Let { α n } and { β n } be two sequences in (0,1). For any given x 0 C, let { x n } be the sequence generated by (1.8). Assume that the sequence { x n } is bounded and that

  1. (i)

    n = 0 α n =;

  2. (ii)

    lim n 1 α n | 1 β n 1 β n 1 |=0, lim n 1 β n |1 α n 1 α n |=0;

  3. (iii)

    lim n β n =0, lim n α n β n =0 and lim n β n 2 α n =0;

  4. (iv)

    Fix(T)intC;

  5. (v)

    there exists a constant k>0 such that xTxkDist(x,Fix(T)) for each xC, where Dist(x,Fix(T))= inf y Fix ( T ) xy.

Then { x n } converges strongly to x ˜ = P Λ V x ˜ which solves the HVIP

( I S ) x ˜ , p x ˜ 0,pFix(T).

Furthermore, let φ:CR be a real-valued function, A:HH be a nonlinear mapping and Θ:C×CR be a bifunction. In 2008, Peng and Yao [9] introduced the generalized mixed equilibrium problem (GMEP) of finding xC such that

Θ(x,y)+φ(y)φ(x)+Ax,yx0,yC.
(1.9)

We denote the set of solutions of GMEP (1.9) by GMEP(Θ,φ,A). The GMEP (1.9) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see e.g., [7, 10, 15, 17, 19, 2224]. In particular, if φ=0, then GMEP (1.9) reduces to the generalized equilibrium problem (GEP) which is to find xC such that

Θ(x,y)+Ax,yx0,yC.

It was introduced and studied by Takahashi and Takahashi [25]. The set of solutions of GEP is denoted by GEP(Θ,A).

If A=0, then GMEP (1.9) reduces to the mixed equilibrium problem (MEP) which is to find xC such that

Θ(x,y)+φ(y)φ(x)0,yC.

It was considered and studied in [26]. The set of solutions of MEP is denoted by MEP(Θ,φ).

If φ=0, A=0, then GMEP (1.9) reduces to the equilibrium problem (EP) which is to find xC such that

Θ(x,y)0,yC.

It was considered and studied in [27, 28]. The set of solutions of EP is denoted by EP(Θ). It is worth to mention that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, etc.

It was assumed in [9] that Θ:C×CR is a bifunction satisfying conditions (A1)-(A4) and φ:CR is a lower semicontinuous and convex function with restriction (B1) or (B2), where

(A1) Θ(x,x)=0 for all xC;

(A2) Θ is monotone, i.e., Θ(x,y)+Θ(y,x)0 for any x,yC;

(A3) Θ is upper-hemicontinuous, i.e., for each x,y,zC,

lim sup t 0 + Θ ( t z + ( 1 t ) x , y ) Θ(x,y);

(A4) Θ(x,) is convex and lower semicontinuous for each xC;

(B1) for each xH and r>0, there exists a bounded subset D x C and y x C such that, for any zC D x ,

Θ(z, y x )+φ( y x )φ(z)+ 1 r y x z,zx<0;

(B2) C is a bounded set.

Given a positive number r>0. Let T r ( Θ , φ ) :HC be the solution set of the auxiliary mixed equilibrium problem, that is, for each xH,

T r ( Θ , φ ) (x):= { y C : Θ ( y , z ) + φ ( z ) φ ( y ) + 1 r y x , z y 0 , z C } .

In addition, let B be a single-valued mapping of C into H and R be a multivalued mapping with D(R)=C. Consider the following variational inclusion: find a point xC such that

0Bx+Rx.
(1.10)

We denote by I(B,R) the solution set of the variational inclusion (1.10). In particular, if B=R=0, then I(B,R)=C. If B=0, then problem (1.10) becomes the inclusion problem introduced by Rockafellar [29]. It is known that problem (1.10) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria, and game theory, etc. Let a set-valued mapping R:D(R)H 2 H be maximal monotone. We define the resolvent operator J R , λ :H D ( R ) ¯ associated with R and λ as follows:

J R , λ = ( I + λ R ) 1 ,xH,

where λ is a positive number.

In 1998, Huang [30] studied problem (1.10) in the case where R is maximal monotone and B is strongly monotone and Lipschitz-continuous with D(R)=C=H. Subsequently, Zeng et al. [31] further studied problem (1.10) in the case which is more general than Huang’s one [30]. Moreover, the authors [31] obtained the same strong convergence conclusion as in Huang’s result [30]. In addition, the authors also gave a geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [12, 24, 3234] and the references therein.

Motivated and inspired by the above facts, we introduce and analyze a multistep hybrid extragradient algorithm by combining Korpelevich’s extragradient method, the viscosity approximation method, thehybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method (GPM) with regularization in the setting of infinite-dimensional Hilbert spaces. It is proven that under appropriate assumptions the proposed algorithm converges strongly to a solution of the CMP (1.2) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to the unique solution of a hierarchical variational inequality problem (over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions, and the CMP (1.2). Our results represent the supplementation, improvement, extension, and development of the corresponding results in Xu [[20], Theorems 4.1 and 5.2] and Yao et al. [[21], Theorems 3.1 and 3.2].

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by , and , respectively. Let C be a nonempty closed convex subset of H. We write x n x to indicate that the sequence { x n } converges weakly to x and x n x to indicate that the sequence { x n } converges strongly to x. Moreover, we use ω w ( x n ) to denote the weak ω-limit set of the sequence { x n }, i.e.,

ω w ( x n ):= { x H : x n i x  for some subsequence  { x n i }  of  { x n } } .

The metric (or nearest point) projection from H onto C is the mapping P C :HC which assigns to each point xH the unique point P C xC satisfying the property

x P C x= inf y C xy=:d(x,C).

Definition 2.1 Let T be a nonlinear operator with the domain D(T)H and the range R(T)H. Then T is said to be

  1. (i)

    monotone if

    TxTy,xy0,x,yD(T);
  2. (ii)

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

    TxTy,xyη x y 2 ,x,yD(T);
  3. (iii)

    ν-inverse-strongly monotone if there exists a constant ν>0 such that

    TxTy,xyν T x T y 2 ,x,yD(T).

It is easy to see that the projection P C is 1-inverse-strongly monotone. Inverse-strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see e.g., [35]. It is obvious that if T is ν-inverse-strongly monotone, then T is monotone and 1 ν -Lipschitz-continuous. Moreover, we also have, for all u,vD(T) and λ>0,

( I λ T ) u ( I λ T ) v 2 = ( u v ) λ ( T u T v ) 2 = u v 2 2 λ T u T v , u v + λ 2 T u T v 2 u v 2 + λ ( λ 2 ν ) T u T v 2 .
(2.1)

So, if λ2ν, then IλT is a nonexpansive mapping.

Some important properties of projections are gathered in the following proposition.

Proposition 2.1 For given xH and zC:

  1. (i)

    z= P C xxz,yz0, yC;

  2. (ii)

    z= P C x x z 2 x y 2 y z 2 , yC;

  3. (iii)

    P C x P C y,xy P C x P C y 2 , yH.

Consequently, P C is nonexpansive and monotone.

Definition 2.2 A mapping T:HH is said to be

  1. (a)

    nonexpansive if

    TxTyxy,x,yH;
  2. (b)

    firmly nonexpansive if 2TI is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),

    xy,TxTy T x T y 2 ,x,yH;

alternatively, T is firmly nonexpansive if and only if T can be expressed as

T= 1 2 (I+S),

where S:HH is nonexpansive; projections are firmly nonexpansive.

It can easily be seen that if T is nonexpansive, then IT is monotone.

Next we list some elementary conclusions for the MEP.

Proposition 2.2 (see [26])

Assume that Θ:C×CR satisfies (A1)-(A4) and let φ:CR be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and xH, define a mapping T r ( Θ , φ ) :HC as follows:

T r ( Θ , φ ) (x)= { z C : Θ ( z , y ) + φ ( y ) φ ( z ) + 1 r y z , z x 0 , y C }

for all xH. Then the following hold:

  1. (i)

    for each xH, T r ( Θ , φ ) (x) is nonempty and single-valued;

  2. (ii)

    T r ( Θ , φ ) is firmly nonexpansive, that is, for any x,yH,

    T r ( Θ , φ ) x T r ( Θ , φ ) y 2 T r ( Θ , φ ) x T r ( Θ , φ ) y , x y ;
  3. (iii)

    Fix( T r ( Θ , φ ) )=MEP(Θ,φ);

  4. (iv)

    MEP(Θ,φ) is closed and convex;

  5. (v)

    T s ( Θ , φ ) x T t ( Θ , φ ) x 2 s t s T s ( Θ , φ ) x T t ( Θ , φ ) x, T s ( Θ , φ ) xx for all s,t>0 and xH.

Definition 2.3 A mapping T:HH is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,

T(1α)I+αS,

where α(0,1) and S:HH is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are 1 2 -averaged mappings.

Proposition 2.3 (see [36])

Let T:HH be a given mapping.

  1. (i)

    T is nonexpansive if and only if the complement IT is 1 2 -ism.

  2. (ii)

    If T is ν-ism, then for γ>0, γT is ν γ -ism.

  3. (iii)

    T is averaged if and only if the complement IT is ν-ism for some ν>1/2. Indeed, for α(0,1), T is α-averaged if and only if IT is 1 2 α -ism.

Proposition 2.4 (see [36, 37])

Let S,T,V:HH be given operators.

  1. (i)

    If T=(1α)S+αV for some α(0,1) and if S is averaged and V is nonexpansive, then T is averaged.

  2. (ii)

    T is firmly nonexpansive if and only if the complement IT is firmly nonexpansive.

  3. (iii)

    If T=(1α)S+αV for some α(0,1) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.

  4. (iv)

    The composite of finitely many averaged mappings is averaged. That is, if each of the mappings { T i } i = 1 N is averaged, then so is the composite T 1 T N . In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1 , α 2 (0,1), then the composite T 1 T 2 is α-averaged, where α= α 1 + α 2 α 1 α 2 .

  5. (v)

    If the mappings { T i } i = 1 N are averaged and have a common fixed point, then

    i = 1 N Fix( T i )=Fix( T 1 T N ).

The notation Fix(T) denotes the set of all fixed points of the mapping T, that is, Fix(T)={xH:Tx=x}.

We need some facts and tools in a real Hilbert space H which are listed as lemmas below.

Lemma 2.1 Let X be a real inner product space. Then we have the following inequality:

x + y 2 x 2 +2y,x+y,x,yX.

Lemma 2.2 Let H be a real Hilbert space. Then the following hold:

  1. (a)

    x y 2 = x 2 y 2 2xy,y for all x,yH;

  2. (b)

    λ x + μ y 2 =λ x 2 +μ y 2 λμ x y 2 for all x,yH and λ,μ[0,1] with λ+μ=1;

  3. (c)

    if { x n } is a sequence in H such that x n x, it follows that

    lim sup n x n y 2 = lim sup n x n x 2 + x y 2 ,yH.

It is clear that, in a real Hilbert space H, T:CC is ξ-strictly pseudocontractive if and only if the following inequality holds:

TxTy,xy x y 2 1 ξ 2 ( I T ) x ( I T ) y 2 ,x,yC.

This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then IT is 1 ξ 2 -inverse strongly monotone; for further details, we refer to [38] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.

Lemma 2.3 (see [[38], Proposition 2.1])

Let C be a nonempty closed convex subset of a real Hilbert space H and T:CC be a mapping.

  1. (i)

    If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition

    TxTy 1 + ξ 1 ξ xy,x,yC.
  2. (ii)

    If T is a ξ-strictly pseudocontractive mapping, then the mapping IT is semiclosed at 0, that is, if { x n } is a sequence in C such that x n x ˜ and (IT) x n 0, then (IT) x ˜ =0.

  3. (iii)

    If T is ξ-(quasi-)strict pseudocontraction, then the fixed point set Fix(T) of T is closed and convex so that the projection P Fix ( T ) is well defined.

Lemma 2.4 (see [11])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:CC be a ξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that (γ+δ)ξγ. Then

γ ( x y ) + δ ( T x T y ) (γ+δ)xy,x,yC.

Lemma 2.5 (see [[39], Demiclosedness principle])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with Fix(S). Then IS is demiclosed. That is, whenever { x n } is a sequence in C weakly converging to some xC and the sequence {(IS) x n } strongly converges to some y, it follows that (IS)x=y. Here I is the identity operator of H.

Lemma 2.6 Let A:CH be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition  2.1(i)) implies

uVI(C,A)u= P C (uλAu),λ>0.

Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ>0. Associated with a nonexpansive mapping T:CH, we define the mapping T λ :CH by

T λ x:=TxλμF(Tx),xC,

where F:HH is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the conditions:

FxFyκxyandFxFy,xyη x y 2

for all x,yH.

Lemma 2.7 (see [[40], Lemma 3.1])

T λ is a contraction provided 0<μ< 2 η κ 2 ; that is,

T λ x T λ y (1λτ)xy,x,yC,

where τ=1 1 μ ( 2 η μ κ 2 ) (0,1].

Remark 2.1 (i) Since F is κ-Lipschitzian and η-strongly monotone on H, we get 0<ηκ. Hence, whenever 0<μ< 2 η κ 2 , we have τ=1 1 μ ( 2 η μ κ 2 ) (0,1].

(ii) In Lemma 2.7, put F= 1 2 I and μ=2. Then we know that κ=η= 1 2 , 0<μ=2< 2 η κ 2 =4, and τ=1.

Lemma 2.8 (see [41])

Let { a n } be a sequence of nonnegative real numbers satisfying the property:

a n + 1 (1 s n ) a n + s n b n + t n ,n0,

where { s n }(0,1] and { b n } are such that:

  1. (i)

    n = 0 s n =;

  2. (ii)

    either lim sup n b n 0 or n = 0 | s n b n |<;

  3. (iii)

    n = 0 t n < where t n 0, for all n0.

Then lim n a n =0.

Recall that a set-valued mapping T:D(T)H 2 H is called monotone if for all x,yD(T), fTx and gTy imply

fg,xy0.

A set-valued mapping T is called maximal monotone if T is monotone and (I+λT)D(T)=H for each λ>0, where I is the identity mapping of H. We denote by G(T) the graph of T. It is well known that a monotone mapping T is maximal if and only if, for (x,f)H×H, fg,xy0 for every (y,g)G(T) implies fTx. Next we provide an example to illustrate the concept of a maximal monotone mapping.

Let A:CH be a monotone, k-Lipschitz-continuous mapping and let N C v be the normal cone to C at vC, i.e.,

N C v= { u H : v p , u 0 , p C } .

Define

T ˜ v= { A v + N C v , if  v C , , if  v C .

Then T ˜ is maximal monotone (see [29]) such that

0 T ˜ vvVI(C,A).
(2.2)

Let R:D(R)H 2 H be a maximal monotone mapping. Let λ,μ>0 be two positive numbers.

Lemma 2.9 (see [42])

We have the resolvent identity

J R , λ x= J R , μ ( μ λ x + ( 1 μ λ ) J R , λ x ) ,xH.

Remark 2.2 For λ,μ>0, we have the following relation:

J R , λ x J R , μ yxy+|λμ| ( 1 λ J R , λ x y + 1 μ x J R , μ y ) ,x,yH.
(2.3)

In terms of Huang [30] (see also [31]), we have the following property for the resolvent operator J R , λ :H D ( R ) ¯ .

Lemma 2.10 J R , λ is single-valued and firmly nonexpansive, i.e.,

J R , λ x J R , λ y,xy J R , λ x J R , λ y 2 ,x,yH.

Consequently, J R , λ is nonexpansive and monotone.

Lemma 2.11 (see [12])

Let R be a maximal monotone mapping with D(R)=C. Then for any given λ>0, uC is a solution of problem (1.5) if and only if uC satisfies

u= J R , λ (uλBu).

Lemma 2.12 (see [31])

Let R be a maximal monotone mapping with D(R)=C and let B:CH be a strongly monotone, continuous, and single-valued mapping. Then for each zH, the equation z(B+λR)x has a unique solution x λ for λ>0.

Lemma 2.13 (see [12])

Let R be a maximal monotone mapping with D(R)=C and B:CH be a monotone, continuous and single-valued mapping. Then (I+λ(R+B))C=H for each λ>0. In this case, R+B is maximal monotone.

3 Main results

In this section, we will introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of the CMP (1.2) with constraints of several problems: finitely many GMEPs and finitely many variational inclusions and the fixed point problem of a strict pseudocontraction in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method [43], Mann’s iteration method and the gradient-projection method (GPM) with regularization. Under appropriate assumptions, we prove the strong convergence of the proposed algorithm to a solution of the CMP (1.2), which is also the unique solution of a hierarchical variational inequality problem (HVIP).

Let C be a nonempty closed convex subset of a real Hilbert space H and f:CR be a convex functional with L-Lipschitz-continuous gradient f. We denote by Γ the solution set of the CMP (1.2). Then the CMP (1.2) is generally ill-posed. Consider the following Tikhonov regularization problem:

min x C f α (x):=f(x)+ 1 2 α x 2 ,

where α>0 is the regularization parameter. Hence, we have

f α =f+αI.

We are now in a position to state and prove the main result in this paper.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two positive integers. Let f:CR be a convex functional with L-Lipschitz-continuous gradient f. Let Θ k be a bifunction from C×C to R satisfying (A1)-(A4) and φ k :CR{+} be a proper lower semicontinuous and convex function with restriction (B1) or (B2), where k{1,2,,M}. Let R i :C 2 H be a maximal monotone mapping and let A k :HH and B i :CH be μ k -inverse-strongly monotone and η i -inverse-strongly monotone, respectively, where k{1,2,,M}, i{1,2,,N}. Let T:CC be a ξ-strictly pseudocontractive mapping, S:HH be a nonexpansive mapping and V:HH be a ρ-contraction with coefficient ρ[0,1). Let F:HH be κ-Lipschitzian and η-strongly monotone with positive constants κ,η>0 such that 0γ<τ and 0<μ< 2 η κ 2 where τ=1 1 μ ( 2 η μ κ 2 ) . Assume that Ω:= k = 1 M GMEP( Θ k , φ k , A k ) i = 1 N I( B i , R i )Fix(T)Γ. Let { λ n }[a,b](0, 2 L ), { α n }(0,) with n = 0 α n <, { ϵ n },{ δ n },{ β n },{ γ n },{ σ n }(0,1) with β n + γ n + σ n =1, and { λ i , n }[ a i , b i ](0,2 η i ), { r k , n }[ c k , d k ](0,2 μ k ) where i{1,2,,N} and k{1,2,,M}. For arbitrarily given x 0 H, let { x n } be a sequence generated by

{ u n = T r M , n ( Θ M , φ M ) ( I r M , n A M ) T r M 1 , n ( Θ M 1 , φ M 1 ) ( I r M 1 , n A M 1 ) T r 1 , n ( Θ 1 , φ 1 ) ( I r 1 , n A 1 ) x n , v n = J R N , λ N , n ( I λ N , n B N ) J R N 1 , λ N 1 , n ( I λ N 1 , n B N 1 ) J R 1 , λ 1 , n ( I λ 1 , n B 1 ) u n , y n = β n x n + γ n P C ( I λ n f α n ) v n + σ n T P C ( I λ n f α n ) v n , x n + 1 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( I ϵ n μ F ) y n , n 0 ,
(3.1)

where f α n = α n I+f for all n0. Suppose that

(C1) lim n ϵ n =0, n = 0 ϵ n = and lim n 1 ϵ n |1 δ n 1 δ n |=0;

(C2) lim sup n δ n ϵ n <, lim n 1 ϵ n | 1 δ n 1 δ n 1 |=0 and lim n 1 δ n |1 ϵ n 1 ϵ n |=0;

(C3) lim n | β n β n 1 | ϵ n δ n =0 and lim n | γ n γ n 1 | ϵ n δ n =0;

(C4) lim n | λ i , n λ i , n 1 | ϵ n δ n =0 and lim n | r k , n r k , n 1 | ϵ n δ n =0 for i=1,2,,N and k=1,2,,M;

(C5) lim n | λ n λ n 1 | ϵ n δ n =0, lim n | λ n α n λ n 1 α n 1 | ϵ n δ n =0 and ( γ n + σ n )ξ γ n for all n0;

(C6) { β n }[c,d](0,1), lim n α n δ n =0 and lim inf n σ n >0.

Then we have:

  1. (i)

    lim n x n + 1 x n δ n =0;

  2. (ii)

    ω w ( x n )Ω;

  3. (iii)

    { x n } converges strongly to a minimizer x of the CMP (1.2), which is a unique solution x in Ω to the HVIP

    ( μ F γ S ) x , p x 0,pΩ.

Proof First of all, observe that

μ η τ μ η 1 1 μ ( 2 η μ κ 2 ) 1 μ ( 2 η μ κ 2 ) 1 μ η 1 2 μ η + μ 2 κ 2 1 2 μ η + μ 2 η 2 κ 2 η 2 κ η

and

( μ F γ S ) x ( μ F γ S ) y , x y = μ F x F y , x y γ S x S y , x y μ η x y 2 γ x y 2 = ( μ η γ ) x y 2 , x , y H .

Since 0γ<τ and κη, we know that μητ>γ and hence the mapping μFγS is (μηγ)-strongly monotone. Moreover, it is clear that the mapping μFγS is (μκ+γ)-Lipschitzian. Thus, there exists a unique solution x in Ω to the VIP

( μ F γ S ) x , p x 0,pΩ.

That is, { x }=VI(Ω,μFγS). Now, we put

Δ n k = T r k , n ( Θ k , φ k ) (I r k , n A k ) T r k 1 , n ( Θ k 1 , φ k 1 ) (I r k 1 , n A k 1 ) T r 1 , n ( Θ 1 , φ 1 ) (I r 1 , n A 1 ) x n

for all k{1,2,,M} and n1,

Λ n i = J R i , λ i , n (I λ i , n B i ) J R i 1 , λ i 1 , n (I λ i 1 , n B i 1 ) J R 1 , λ 1 , n (I λ 1 , n B 1 )

for all i{1,2,,N}, Δ n 0 =I and Λ n 0 =I, where I is the identity mapping on H. Then we have u n = Δ n M x n and v n = Λ n N u n .

In addition, we show that P C (Iλ f α ) is ν-averaged for each λ(0, 2 α + L ), where

ν= 2 + λ ( α + L ) 4 (0,1).

Indeed, the Lipschitz continuity of f implies that f is 1 L -ism (see [20] (also [44])); that is,

f ( x ) f ( y ) , x y 1 L f ( x ) f ( y ) 2 .

Observe that

( α + L ) f α ( x ) f α ( y ) , x y = ( α + L ) [ α x y 2 + f ( x ) f ( y ) , x y ] = α 2 x y 2 + α f ( x ) f ( y ) , x y + α L x y 2 + L f ( x ) f ( y ) , x y α 2 x y 2 + 2 α f ( x ) f ( y ) , x y + f ( x ) f ( y ) 2 = α ( x y ) + f ( x ) f ( y ) 2 = f α ( x ) f α ( y ) 2 .

Therefore, it follows that f α =f+αI is 1 α + L -ism. Thus, by Proposition 2.3(ii), λ f α is 1 λ ( α + L ) -ism. From Proposition 2.3(iii), the complement Iλ f α is λ ( α + L ) 2 -averaged. Consequently, noting that P C is 1 2 -averaged and utilizing Proposition 2.4(iv), we find that, for each λ(0, 2 α + L ), P C (Iλ f α ) is ν-averaged with

ν= 1 2 + λ ( α + L ) 2 1 2 λ ( α + L ) 2 = 2 + λ ( α + L ) 4 (0,1).

This shows that P C (Iλ f α ) is nonexpansive. Taking into account that { λ n }[a,b](0, 2 L ) and α n 0, we get

lim sup n 2 + λ n ( α n + L ) 4 2 + b L 4 <1.

Without loss of generality, we may assume that ν n := 2 + λ n ( α n + L ) 4 <1 for each n0. So, P C (I λ n f α n ) is nonexpansive for each n0. Similarly, since

lim sup n λ n ( α n + L ) 2 b L 2 <1,

it is well known that I λ n f α n is nonexpansive for each n0.

We divide the rest of the proof into several steps.

Step 1. We prove that { x n } is bounded.

Indeed, take a fixed pΩ arbitrarily. Utilizing (2.1) and Proposition 2.2(ii) we have

u n p = T r M , n ( Θ M , φ M ) ( I r M , n B M ) Δ n M 1 x n T r M , n ( Θ M , φ M ) ( I r M , n B M ) Δ n M 1 p ( I r M , n B M ) Δ n M 1 x n ( I r M , n B M ) Δ n M 1 p Δ n M 1 x n Δ n M 1 p Δ n 0 x n Δ n 0 p = x n p .
(3.2)

Utilizing (2.1) and Lemma 2.10 we have

v n p = J R N , λ N , n ( I λ N , n A N ) Λ n N 1 u n J R N , λ N , n ( I λ N , n A N ) Λ n N 1 p ( I λ N , n A N ) Λ n N 1 u n ( I λ N , n A N ) Λ n N 1 p Λ n N 1 u n Λ n N 1 p Λ n 0 u n Λ n 0 p = u n p .
(3.3)

Combining (3.2) and (3.3), we have

v n p x n p.
(3.4)

For simplicity, put t n = P C (I λ n f α n ) v n for each n0. Note that P C (Iλf)p=p for λ(0, 2 L ). Hence, from (3.4), it follows that

t n p = P C ( I λ n f α n ) v n P C ( I λ n f ) p P C ( I λ n f α n ) v n P C ( I λ n f α n ) p + P C ( I λ n f α n ) p P C ( I λ n f ) p v n p + ( I λ n f α n ) p ( I λ n f ) p = v n p + λ n α n p x n p + λ n α n p .
(3.5)

Since ( γ n + σ n )ξ γ n for all n0 and T is ξ-strictly pseudocontractive, utilizing Lemma 2.4 we obtain from (3.1) and (3.5)

y n p = β n x n + γ n t n + σ n T t n p = β n ( x n p ) + γ n ( t n p ) + σ n ( T t n p ) β n x n p + γ n ( t n p ) + σ n ( T t n p ) β n x n p + ( γ n + σ n ) t n p β n x n p + ( γ n + σ n ) [ x n p + λ n α n p ] β n x n p + ( γ n + δ n ) x n p + λ n α n p = x n p + λ n α n p .
(3.6)

Utilizing Lemma 2.7, we deduce from (3.1), (3.6), { λ n }[a,b](0, 2 L ), and 0γ<τ that, for all n0,

x n + 1 p = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( I ϵ n μ F ) y n p = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) ϵ n μ F p + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) ϵ n μ F p + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p = ϵ n δ n ( γ V x n μ F p ) + ( 1 δ n ) ( γ S x n μ F p ) + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p ϵ n [ δ n γ V x n μ F p + ( 1 δ n ) γ S x n μ F p ] + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p ϵ n [ δ n ( γ V x n V p + γ V p μ F p ) + ( 1 δ n ) ( γ S x n S p + γ S p μ F p ) ] + ( I ϵ n μ F ) y n ( I λ n μ F ) p ϵ n [ δ n ( γ ρ x n p + γ V p μ F p ) + ( 1 δ n ) ( γ x n p + γ S p μ F p ) ] + ( 1 ϵ n τ ) y n p ϵ n [ ( 1 δ n ( 1 ρ ) ) γ x n p + max { γ V p μ F p , γ S p μ F p } ] + ( 1 ϵ n τ ) [ x n p + λ n α n p ] = ϵ n ( 1 δ n ( 1 ρ ) ) γ x n p + λ n max { γ V p μ F p , γ S p μ F p } + ( 1 ϵ n τ ) x n p + λ n α n p ϵ n γ x n p + ϵ n max { γ V p μ F p , γ S p μ F p } + ( 1 ϵ n τ ) x n p = ( 1 ϵ n ( τ γ ) ) x n p + ϵ n max { γ V p μ F p , γ S p μ F p } + α n b p = ( 1 ϵ n ( τ γ ) ) x n p + ϵ n ( τ γ ) max { γ V p μ F p , γ S p μ F p } τ γ + α n b p max { x n p , γ V p μ F p τ γ , γ S p μ F p τ γ } + α n b p .

By induction, we get

x n + 1 pmax { x 0 p , γ V p μ F p τ γ , γ S p μ F p τ γ } + j = 0 n α n bp,n0.

Thus, { x n } is bounded (due to n = 0 α n <) and so are the sequences { t n }, { u n }, { v n } and { y n }.

Step 2. We prove that lim n x n + 1 x n δ n =0.

Indeed, utilizing (2.1) and (2.3), we obtain

v n + 1 v n = Λ n + 1 N u n + 1 Λ n N u n = J R N , λ N , n + 1 ( I λ N , n + 1 B N ) Λ n + 1 N 1 u n + 1 J R N , λ N , n ( I λ N , n B N ) Λ n N 1 u n J R N , λ N , n + 1 ( I λ N , n + 1 B N ) Λ n + 1 N 1 u n + 1 J R N , λ N , n + 1 ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 + J R N , λ N , n + 1 ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 J R N , λ N , n ( I λ N , n B N ) Λ n N 1 u n ( I λ N , n + 1 B N ) Λ n + 1 N 1 u n + 1 ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 + ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 ( I λ N , n B N ) Λ n N 1 u n + | λ N , n + 1 λ N , n | × ( 1 λ N , n + 1 J R N , λ N , n + 1 ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 ( I λ N , n B N ) Λ n N 1 u n + 1 λ N , n ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 J R N , λ N , n ( I λ N , n B N ) Λ n N 1 u n ) | λ N , n + 1 λ N , n | ( B N Λ n + 1 N 1 u n + 1 + M ˜ ) + Λ n + 1 N 1 u n + 1 Λ n N 1 u n | λ N , n + 1 λ N , n | ( B N Λ n + 1 N 1 u n + 1 + M ˜ ) + | λ N 1 , n + 1 λ N 1 , n | ( B N 1 Λ n + 1 N 2 u n + 1 + M ˜ ) + Λ n + 1 N 2 u n + 1 Λ n N 2 u n | λ N , n + 1 λ N , n | ( B N Λ n + 1 N 1 u n + 1 + M ˜ ) + | λ N 1 , n + 1 λ N 1 , n | ( B N 1 Λ n + 1 N 2 u n + 1 + M ˜ ) + + | λ 1 , n + 1 λ 1 , n | ( B 1 Λ n + 1 0 u n + 1 + M ˜ ) + Λ n + 1 0 u n + 1 Λ n 0 u n M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + u n + 1 u n ,
(3.7)

where

sup n 0 { 1 λ N , n + 1 J R N , λ N , n + 1 ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 ( I λ N , n B N ) Λ n N 1 u n + 1 λ N , n ( I λ N , n B N ) Λ n + 1 N 1 u n + 1 J R N , λ N , n ( I λ N , n B N ) Λ n N 1 u n } M ˜

for some M ˜ >0 and sup n 0 { i = 1 N B i Λ n + 1 i 1 u n + 1 + M ˜ } M ˜ 0 for some M ˜ 0 >0.

Utilizing Proposition 2.2(ii), (v), we deduce that

u n + 1 u n = Δ n + 1 M x n + 1 Δ n M x n = T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 T r M , n ( Θ M , φ M ) ( I r M , n A M ) Δ n M 1 x n T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 T r M , n ( Θ M , φ M ) ( I r M , n A M ) Δ n + 1 M 1 x n + 1 + T r M , n ( Θ M , φ M ) ( I r M , n A M ) Δ n + 1 M 1 x n + 1 T r M , n ( Θ M , φ M ) ( I r M , n A M ) Δ n M 1 x n T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 T r M , n ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 + T r M , n ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 T r M , n ( Θ M , φ M ) ( I r M , n A M ) Δ n + 1 M 1 x n + 1 + ( I r M , n A M ) Δ n + 1 M 1 x n + 1 ( I r M , n A M ) Δ n M 1 x n | r M , n + 1 r M , n | r M , n + 1 T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 + | r M , n + 1 r M , n | A M Δ n + 1 M 1 x n + 1 + Δ n + 1 M 1 x n + 1 Δ n M 1 x n = | r M , n + 1 r M , n | [ A M Δ n + 1 M 1 x n + 1 + 1 r M , n + 1 T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 ] + Δ n + 1 M 1 x n + 1 Δ n M 1 x n | r M , n + 1 r M , n | [ A M Δ n + 1 M 1 x n + 1 + 1 r M , n + 1 T r M , n + 1 ( Θ M , φ M ) ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 ( I r M , n + 1 A M ) Δ n + 1 M 1 x n + 1 ] + + | r 1 , n + 1 r 1 , n | [ A 1 Δ n + 1 0 x n + 1 + 1 r 1 , n + 1 T r 1 , n + 1 ( Θ 1 , φ 1 ) ( I r 1 , n + 1 A 1 ) Δ n + 1 0 x n + 1 ( I r 1 , n + 1 A 1 ) Δ n + 1 0 x n + 1 ] + Δ n + 1 0 x n + 1 Δ n 0 x n M ˜ 1 k = 1 M | r k , n + 1 r k , n | + x n + 1 x n ,
(3.8)

where M ˜ 1 >0 is a constant such that, for each n0,

k = 1 M [ A k Δ n + 1 k 1 x n + 1 + 1 r k , n + 1 T r k , n + 1 ( Θ k , φ k ) ( I r k , n + 1 A k ) Δ n + 1 k 1 x n + 1 ( I r k , n + 1 A k ) Δ n + 1 k 1 x n + 1 ] M ˜ 1 .

Furthermore, we define y n = β n x n +(1 β n ) w n for all n0. It follows that

w n + 1 w n = y n + 1 β n + 1 x n + 1 1 β n + 1 y n β n x n 1 β n = γ n + 1 t n + 1 + σ n + 1 T t n + 1 1 β n + 1 γ n t n + σ n T t n 1 β n = γ n + 1 ( t n + 1 t n ) + σ n + 1 ( T t n + 1 T t n ) 1 β n + 1 + ( γ n + 1 1 β n + 1 γ n 1 β n ) t n + ( σ n + 1 1 β n + 1 σ n 1 β n ) T t n .
(3.9)

Since ( γ n + σ n )ξ γ n for all n0, utilizing Lemma 2.4 and the nonexpansivity of P C (I λ n f α n ) we have

γ n + 1 ( t n + 1 t n ) + σ n + 1 ( T t n + 1 T t n ) ( γ n + 1 + σ n + 1 ) t n + 1 t n
(3.10)

and

t n + 1 t n = P C ( I λ n + 1 f α n + 1 ) v n + 1 P C ( I λ n f α n ) v n P C ( I λ n + 1 f α n + 1 ) v n + 1 P C ( I λ n + 1 f α n + 1 ) v n + P C ( I λ n + 1 f α n + 1 ) v n P C ( I λ n f α n ) v n v n + 1 v n + ( I λ n + 1 f α n + 1 ) v n ( I λ n f α n ) v n v n + 1 v n + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) .
(3.11)

Hence it follows from (3.7)-(3.11) that

w n + 1 w n γ n + 1 ( t n + 1 t n ) + σ n + 1 ( T t n + 1 T t n ) 1 β n + 1 + | γ n + 1 1 β n + 1 γ n 1 β n | t n + | σ n + 1 1 β n + 1 σ n 1 β n | T t n ( γ n + 1 + σ n + 1 ) 1 β n + 1 t n + 1 t n + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) = t n + 1 t n + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) v n + 1 v n + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + u n + 1 u n + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + M ˜ 1 k = 1 M | r k , n + 1 r k , n | + x n + 1 x n + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) .
(3.12)

In the meantime, a simple calculation shows that

y n + 1 y n = β n ( x n + 1 x n )+(1 β n )( w n + 1 w n )+( β n + 1 β n )( x n + 1 w n + 1 ).

So, it follows from (3.12) that

y n + 1 y n β n x n + 1 x n + ( 1 β n ) w n + 1 w n + | β n + 1 β n | x n + 1 w n + 1 β n x n + 1 x n + ( 1 β n ) [ M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + M ˜ 1 k = 1 M | r k , n + 1 r k , n | + x n + 1 x n + | γ n + 1 1 β n + 1 γ n 1 β n | ( t n + T t n ) + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) ] + | β n + 1 β n | x n + 1 w n + 1 x n + 1 x n + M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + M ˜ 1 k = 1 M | r k , n + 1 r k , n | + | γ n + 1 γ n | ( 1 β n ) + γ n | β n + 1 β n | 1 β n + 1 ( t n + T t n ) + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) + | β n + 1 β n | x n + 1 w n + 1 x n + 1 x n + M ˜ 0 i = 1 N | λ i , n + 1 λ i , n | + M ˜ 1 k = 1 M | r k , n + 1 r k , n | + | γ n + 1 γ n | t n + T t n 1 d + | β n + 1 β n | ( x n + 1 w n + 1 + t n + T t n 1 d ) + | λ n + 1 α n + 1 λ n α n | v n + | λ n + 1 λ n | f ( v n ) x n + 1 x n + M ˜ 2 ( i = 1 N | λ i , n + 1 λ i , n | + k = 1 M | r k , n + 1 r k , n | + | γ n + 1 γ n | + | β n + 1 β n | + | λ n + 1 α n + 1 λ n α n | + | λ n + 1 λ n | ) ,
(3.13)

where sup n 0 { x n + 1 w n + 1 + t n + T t n 1 d + v n +f( v n )+ M ˜ 0 + M ˜ 1 } M ˜ 2 for some M ˜ 2 >0.

On the other hand, we define z n := δ n V x n +(1 δ n )S x n for all n0. Then it is well known that x n + 1 = ϵ n γ z n +(I ϵ n μF) y n for all n0. Simple calculations show that

{ z n + 1 z n = ( δ n + 1 δ n ) ( V x n S x n ) + δ n + 1 ( V x n + 1 V x n ) n + 1 z n = + ( 1 δ n + 1 ) ( S x n + 1 S x n ) , x n + 2 x n + 1 = ( ϵ n + 1 ϵ n ) ( γ z n μ F y n ) + ϵ n + 1 γ ( z n + 1 z n ) x n + 2 x n + 1 = + ( I λ n + 1 μ F ) y n + 1 ( I λ n + 1 μ F ) y n .

Since V is a ρ-contraction with coefficient ρ[0,1) and S is a nonexpansive mapping, we conclude that

z n + 1 z n | δ n + 1 δ n | V x n S x n + δ n + 1 V x n + 1 V x n + ( 1 δ n + 1 ) S x n + 1 S x n | δ n + 1 δ n | V x n S x n + δ n + 1 ρ x n + 1 x n + ( 1 δ n + 1 ) x n + 1 x n = ( 1 δ n + 1 ( 1 ρ ) ) x n + 1 x n + | δ n + 1 δ n | V x n S x n ,

which, together with (3.13) and 0γ<τ, implies that

x n + 2 x n + 1 | ϵ n + 1 ϵ n | γ z n μ F y n + ϵ n + 1 γ z n + 1 z n + ( I ϵ n + 1 μ F ) y n + 1 ( I ϵ n + 1 μ F ) y n | ϵ n + 1 ϵ n | γ z n μ F y n + ϵ n + 1 γ z n + 1 z n + ( 1 ϵ n + 1 τ ) y n + 1 y n | ϵ n + 1 ϵ n | γ z n μ F y n + ϵ n + 1 γ [ ( 1 δ n + 1 ( 1 ρ ) ) x n + 1 x n + | δ n + 1 δ n | V x n S x n ] + ( 1 ϵ n + 1 τ ) [ x n + 1 x n + M ˜ 2 ( i = 1 N | λ i , n + 1 λ i , n | + k = 1 M | r k , n + 1 r k , n | + | γ n + 1 γ n | + | β n + 1 β n | + | λ n + 1 α n + 1 λ n α n | + | λ n + 1 λ n | ) ] ( 1 ϵ n + 1 ( τ γ ) ) x n + 1 x n + | ϵ n + 1 ϵ n | γ z n μ F y n + | δ n + 1 δ n | V x n S x n + M ˜ 2 ( i = 1 N | λ i , n + 1 λ i , n | + k = 1 M | r k , n + 1 r k , n | + | γ n + 1 γ n | + | β n + 1 β n | + | λ n + 1 α n + 1 λ n α n | + | λ n + 1 λ n | ) ( 1 ϵ n + 1 ( τ γ ) ) x n + 1 x n + M ˜ 3 { i = 1 N | λ i , n + 1 λ i , n | + k = 1 M | r k , n + 1 r k , n | + | ϵ n + 1 ϵ n | + | δ n + 1 δ n | + | β n + 1 β n | + | γ n + 1 γ n | + | λ n + 1 α n + 1 λ n α n | + | λ n + 1 λ n | } ,

where sup n 0 {γ z n μF y n +V x n S x n + M ˜ 2 } M ˜ 3 for some M ˜ 3 >0. Consequently,

x n + 1 x n δ n ( 1 ϵ n ( τ γ ) ) x n x n 1 δ n + M ˜ 3 { i = 1 N | λ i , n λ i , n 1 | δ n + k = 1 M | r k , n r k , n 1 | δ n + | ϵ n ϵ n 1 | δ n + | δ n δ n 1 | δ n + | β n β n 1 | δ n + | γ n γ n 1 | δ n + | λ n + 1 α n + 1 λ n α n | δ n + | λ n + 1 λ n | δ n } = ( 1 ϵ n ( τ γ ) ) x n x n 1 δ n 1 + ( 1 ϵ n ( τ γ ) ) x n x n 1 ( 1 δ n 1 δ n 1 ) + M ˜ 3 { i = 1 N | λ i , n λ i , n 1 | δ n + k = 1 M | r k , n r k , n 1 | δ n + | ϵ n ϵ n 1 | δ n + | δ n δ n 1 | δ n + | β n β n 1 | δ n + | γ n γ n 1 | δ n + | λ n + 1 α n + 1 λ n α n | δ n + | λ n + 1 λ n | δ n } ( 1 ϵ n ( τ γ ) ) x n x n 1 δ n 1 + ϵ n ( τ γ ) M ˜ 4 τ γ { 1 ϵ n | 1 δ n 1 δ n 1 | + i = 1 N | λ i , n λ i , n 1 | ϵ n δ n + k = 1 M | r k , n r k , n 1 | ϵ n δ n + 1 δ n | 1 ϵ n 1 ϵ n | + 1 ϵ n | 1 δ n 1 δ n | + | β n β n 1 | ϵ n δ n + | γ n γ n 1 | ϵ n δ n + | λ n + 1 α n + 1 λ n α n | ϵ n δ n + | λ n + 1 λ n | ϵ n δ n } ,
(3.14)

where sup n 1 { x n x n 1 + M ˜ 3 } M ˜ 4 for some M ˜ 4 >0. From conditions (C1)-(C5) it follows that n = 0 ϵ n (τγ)= and

lim n M ˜ 4 τ γ { 1 ϵ n | 1 δ n 1 δ n 1 | + i = 1 N | λ i , n λ i , n 1 | ϵ n δ n + k = 1 M | r k , n r k , n 1 | ϵ n δ n + 1 δ n | 1 ϵ n 1 ϵ n | + 1 ϵ n | 1 δ n 1 δ n | + | β n β n 1 | ϵ n δ n + | γ n γ n 1 | ϵ n δ n + | λ n + 1 α n + 1 λ n α n | ϵ n δ n + | λ n + 1 λ n | ϵ n δ n } = 0 .

Thus, utilizing Lemma 2.8, we immediately conclude that

lim n x n + 1 x n δ n =0.

So, from δ n 0 it follows that

lim n x n + 1 x n =0.

Step 3. We prove that lim n x n u n =0, lim n x n v n =0, lim n v n t n =0 and lim n t n T t n =0.

Indeed, utilizing Lemmas 2.1 and 2.2(b), from (3.1), (3.4)-(3.5), and 0γ<τ we deduce that

y n p 2 = β n x n + γ n t n + σ n T t n p 2 = β n ( x n p ) + ( 1 β n ) ( γ n t n + σ n T t n 1 β n p ) 2 = β n x n p 2 + ( 1 β n ) γ n t n + σ n T t n 1 β n p 2 β n ( 1 β n ) γ n t n + σ n T t n 1 β n x n 2 = β n x n p 2 + ( 1 β n ) γ n ( t n p ) + σ n ( T t n p ) 1 β n 2 β n ( 1 β n ) y n x n 1 β n 2 β n x n p 2 + ( 1 β n ) ( γ n + σ n ) 2 t n p 2 ( 1 β n ) 2 β n 1 β n y n x n 2 = β n x n p 2 + ( 1 β n ) t n p 2 β n 1 β n y n x n 2 β n x n p 2 + ( 1 β n ) ( x n p + λ n α n p ) 2 β n 1 β n y n x n 2 β n ( x n p + λ n α n p ) 2 + ( 1 β n ) ( x n p + λ n α n p ) 2 β n 1 β n y n x n 2 = ( x n p + λ n α n p ) 2 β n 1 β n y n x n 2 ( x n p + α n b p ) 2 β n 1 β n y n x n 2 ,
(3.15)

and hence

x n + 1 p 2 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( I ϵ n μ F ) y n p 2 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) ϵ n μ F p + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p 2 = ϵ n [ δ n ( γ V x n μ F p ) + ( 1 δ n ) ( γ S x n μ F p ) ] + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p 2 = ϵ n [ δ n ( γ V x n γ V p ) + ( 1 δ n ) ( γ S x n γ S p ) ] + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p + ϵ n [ δ n ( γ V p μ F p ) + ( 1 δ n ) ( γ S p μ F p ) ] 2 ϵ n [ δ n ( γ V x n γ V p ) + ( 1 δ n ) ( γ S x n γ S p ) ] + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p [ ϵ n δ n ( γ V x n γ V p ) + ( 1 δ n ) ( γ S x n γ S p ) + ( I ϵ n μ F ) y n ( I ϵ n μ F ) p ] 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p [ ϵ n ( δ n γ ρ x n p + ( 1 δ n ) γ x n p ) + ( 1 ϵ n τ ) y n p ] 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ( 1 δ n ) ϵ n ( γ S p μ F p ) , x n + 1 p = [ ϵ n ( 1 δ n ( 1 ρ ) ) γ x n p + ( 1 ϵ n τ ) y n p ] 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p [ ϵ n γ x n p + ( 1 ϵ n τ ) y n p ] 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p = [ ϵ n τ γ τ x n p + ( 1 ϵ n τ ) y n p ] 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ϵ n γ 2 τ x n p 2 + ( 1 ϵ n τ ) y n p 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ϵ n γ 2 τ x n p 2 + ( 1 ϵ n τ ) [ ( x n p + α n b p ) 2 β n 1 β n y n x n 2 ] + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ϵ n γ 2 τ ( x n p + α n b p ) 2 + ( 1 ϵ n τ ) [ ( x n p + α n b p ) 2 β n 1 β n y n x n 2 ] + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p = ( 1 ϵ n τ 2 γ 2 τ ) ( x n p + α n b p ) 2 β n ( 1 ϵ n τ ) 1 β n y n x n 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ( x n p + α n b p ) 2 β n ( 1 ϵ n τ ) 1 β n y n x n 2 + 2 ϵ n δ n γ V p μ F p x n + 1 p + 2 ϵ n γ S p μ F p x n + 1 p ,
(3.16)

which, together with { β n }[c,d](0,1), immediately yields

c ( 1 ϵ n τ ) 1 c y n x n 2 β n ( 1 ϵ n τ ) 1 β n y n x n 2 ( x n p + α n b p ) 2 x n + 1 p 2 + 2 ϵ n δ n γ V p μ F p x n + 1 p + 2 ϵ n γ S p μ F p x n + 1 p ( x n x n + 1 + α n b p ) ( x n p + x n + 1 p + α n b p ) + 2 ϵ n δ n γ V p μ F p x n + 1 p + 2 ϵ n γ S p μ F p x n + 1 p .

Since ϵ n 0, δ n 0, α n 0, x n + 1 x n 0, and { x n } is bounded, we have

lim n y n x n =0.
(3.17)

Observe that

Δ n k x n p 2 = T r k , n ( Θ k , φ k ) ( I r k , n A k ) Δ n k 1 x n T r k , n ( Θ k , φ k ) ( I r k , n A k ) p 2 ( I r k , n A k ) Δ n k 1 x n ( I r k , n A k ) p 2 Δ n k 1 x n p 2 + r k , n ( r k , n 2 μ k ) A k Δ n k 1 x n A k p 2 x n p 2 + r k , n ( r k , n 2 μ k ) A k Δ n k 1 x n A k p 2
(3.18)

and

Λ n i u n p 2 = J R i , λ i , n ( I λ i , n B i ) Λ n i 1 u n J R i , λ i , n ( I λ i , n B i ) p 2 ( I λ i , n B i ) Λ n i 1 u n ( I λ i , n B i ) p 2 Λ n i 1 u n p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 u n p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 x n p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2
(3.19)

for i{1,2,,N} and k{1,2,,M}. Combining (3.5), (3.15), (3.18), and (3.19), we get

y n p 2 β n x n p 2 + ( 1 β n ) t n p 2 β n 1 β n y n x n 2 β n x n p 2 + ( 1 β n ) t n p 2 β n x n p 2 + ( 1 β n ) ( v n p + λ n α n p ) 2 β n x n p 2 + ( 1 β n ) ( v n p + α n b p ) 2 = β n x n p 2 + ( 1 β n ) [ v n p 2 + α n b p ( 2 v n p + α n b p ) ] β n x n p 2 + ( 1 β n ) v n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) Λ n i u n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) [ u n p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 ] + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) [ Δ n k x n p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 ] + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) [ x n p 2 + r k , n ( r k , n 2 μ k ) A k Δ n k 1 x n A k p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 ] + α n b p ( 2 v n p + α n b p ) = x n p 2 + ( 1 β n ) [ r k , n ( r k , n 2 μ k ) A k Δ n k 1 x n A k p 2 + λ i , n ( λ i , n 2 η i ) B i Λ n i 1 u n B i p 2 ] + α n b p ( 2 v n p + α n b p ) ,
(3.20)

which immediately leads to

( 1 β n ) [ r k , n ( 2 μ k r k , n ) A k Δ n k 1 x n A k p 2 + λ i , n ( 2 η i λ i , n ) B i Λ n i 1 u n B i p 2 ] x n p 2 y n p 2 + α n b p ( 2 v n p + α n b p ) x n y n ( x n p + y n p ) + α n b p ( 2 v n p + α n b p ) .

Since x n y n 0, α n 0, { β n }[c,d](0,1), { λ i , n }[ a i , b i ](0,2 η i ), { r k , n }[ c k , d k ](0,2 μ k ), i{1,2,,N}, k{1,2,,M}, and { v n }, { x n }, { y n } are bounded sequences, we have

lim n A k Δ n k 1 x n A k p =0and lim n B i Λ n i 1 u n B i p =0
(3.21)

for all k{1,2,,M} and i{1,2,,N}.

Furthermore, by Proposition 2.2(ii) and Lemma 2.2(a) we have

Δ n k x n p 2 = T r k , n ( Θ k , φ k ) ( I r k , n A k ) Δ n k 1 x n T r k , n ( Θ k , φ k ) ( I r k , n A k ) p 2 ( I r k , n A k ) Δ n k 1 x n ( I r k , n A k ) p , Δ n k x n p = 1 2 ( ( I r k , n A k ) Δ n k 1 x n ( I r k , n A k ) p 2 + Δ n k x n p 2 ( I r k , n A k ) Δ n k 1 x n ( I r k , n A k ) p ( Δ n k x n p ) 2 ) 1 2 ( Δ n k 1 x n p 2 + Δ n k x n p 2 Δ n k 1 x n Δ n k x n r k , n ( A k Δ n k 1 x n A k p ) 2 ) ,

which implies that

Δ n k x n p 2 Δ n k 1 x n p 2 Δ n k 1 x n Δ n k x n r k , n ( A k Δ n k 1 x n A k p ) 2 = Δ n k 1 x n p 2 Δ n k 1 x n Δ n k x n 2 r k , n 2 A k Δ n k 1 x n A k p 2 + 2 r k , n Δ n k 1 x n Δ n k x n , A k Δ n k 1 x n A k p Δ n k 1 x n p 2 Δ n k 1 x n Δ n k x n 2 + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p x n p 2 Δ n k 1 x n Δ n k x n 2 + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p .
(3.22)

By Lemma 2.2(a) and Lemma 2.10, we obtain

Λ n i u n p 2 = J R i , λ i , n ( I λ i , n B i ) Λ n i 1 u n J R i , λ i , n ( I λ i , n B i ) p 2 ( I λ i , n B i ) Λ n i 1 u n ( I λ i , n B i ) p , Λ n i u n p = 1 2 ( ( I λ i , n B i ) Λ n i 1 u n ( I λ i , n B i ) p 2 + Λ n i u n p 2 ( I λ i , n B i ) Λ n i 1 u n ( I λ i , n B i ) p ( Λ n i u n p ) 2 ) 1 2 ( Λ n i 1 u n p 2 + Λ n i u n p 2 Λ n i 1 u n Λ n i u n λ i , n ( B i Λ n i 1 u n B i p ) 2 ) 1 2 ( u n p 2 + Λ n i u n p 2 Λ n i 1 u n Λ n i u n λ i , n ( B i Λ n i 1 u n B i p ) 2 ) 1 2 ( x n p 2 + Λ n i u n p 2 Λ n i 1 u n Λ n i u n λ i , n ( B i Λ n i 1 u n B i p ) 2 ) ,

which immediately leads to

Λ n i u n p 2 x n p 2 Λ n i 1 u n Λ n i u n λ i , n ( B i Λ n i 1 u n B i p ) 2 = x n p 2 Λ n i 1 u n Λ n k u n 2 λ i , n 2 B i Λ n i 1 u n B i p 2 + 2 λ i , n Λ n i 1 u n Λ n i u n , B i Λ n i 1 u n B i p x n p 2 Λ n i 1 u n Λ n i u n 2 + 2 λ i , n Λ n i 1 u n Λ n i u n B i Λ n i 1 u n B i p .
(3.23)

Combining (3.20) and (3.23) we conclude that

y n p 2 β n x n p 2 + ( 1 β n ) v n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) Λ n i u n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) [ x n p 2 Λ n i 1 u n Λ n i u n 2 + 2 λ i , n Λ n i 1 u n Λ n i u n B i Λ n i 1 u n B i p ] + α n b p ( 2 v n p + α n b p ) x n p 2 ( 1 β n ) Λ n i 1 u n Λ n i u n 2 + 2 λ i , n Λ n i 1 u n Λ n i u n B i Λ n i 1 u n B i p + α n b p ( 2 v n p + α n b p ) ,

which yields

( 1 β n ) Λ n i 1 u n Λ n i u n 2 x n p 2 y n p 2 + 2 λ i , n Λ n i 1 u n Λ n i u n B i Λ n i 1 u n B i p + α n b p ( 2 v n p + α n b p ) x n y n ( x n p + y n p ) + 2 λ i , n Λ n i 1 u n Λ n i u n B i Λ n i 1 u n B i p + α n b p ( 2 v n p + α n b p ) .

Since { β n }[c,d](0,1), { λ i , n }[ a i , b i ](0,2 η i ), i=1,2,,N, and { u n }, { v n }, { x n } and { y n } are bounded sequences, we deduce from (3.17), (3.21), and α n 0 that

lim n Λ n i 1 u n Λ n i u n =0,i{1,2,,N}.
(3.24)

Also, combining (3.3), (3.20), and (3.22) we deduce that

y n p 2 β n x n p 2 + ( 1 β n ) v n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) u n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) Δ n k x n p 2 + α n b p ( 2 v n p + α n b p ) β n x n p 2 + ( 1 β n ) [ x n p 2 Δ n k 1 x n Δ n k x n 2 + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p ] + α n b p ( 2 v n p + α n b p ) x n p 2 ( 1 β n ) Δ n k 1 x n Δ n k x n 2 + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p + α n b p ( 2 v n p + α n b p ) ,

which yields

( 1 β n ) Δ n k 1 x n Δ n k x n 2 x n p 2 y n p 2 + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p + α n b p ( 2 v n p + α n b p ) x n y n ( x n p + y n p ) + 2 r k , n Δ n k 1 x n Δ n k x n A k Δ n k 1 x n A k p + α n b p ( 2 v n p + α n b p ) .

Since { β n }[c,d](0,1), { r k , n }[ c k , d k ](0,2 μ k ) for k=1,2,,M, and { v n }, { x n }, { y n } are bounded sequences, we deduce from (3.17), (3.21), and α n 0 that

lim n Δ n k 1 x n Δ n k x n =0,k{1,2,,M}.
(3.25)

Hence from (3.24) and (3.25) we get

x n u n = Δ n 0 x n Δ n M x n Δ n 0 x n Δ n 1 x n + Δ n 1 x n Δ n 2 x n + + Δ n M 1 x n Δ n M x n 0 as  n
(3.26)

and

u n v n = Λ n 0 u n Λ n N u n Λ n 0 u n Λ n 1 u n + Λ n 1 u n Λ n 2 u n + + Λ n N 1 u n Λ n N u n 0 as  n ,
(3.27)

respectively. Thus, from (3.26) and (3.27) we obtain

x n v n x n u n + u n v n 0 as  n .
(3.28)

On the other hand, note that Γ=VI(C,f). Then, utilizing Lemma 2.1 and the 1 L -inverse strong monotonicity of f, we deduce from (2.1) that

t n p 2 ( I λ n f α n ) v n ( I λ n f ) p 2 = v n p λ n ( f ( v n ) f ( p ) ) λ n α n v n 2 v n p λ n ( f ( v n ) f ( p ) ) 2 2 λ n α n v n , ( I λ n f α n ) v n ( I λ n f ) p v n p 2 + λ n ( λ n 2 L ) f ( v n ) f ( p ) 2 + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) .
(3.29)

Combining (3.4), (3.20), and (3.29) we obtain

y n p 2 β n x n p 2 + ( 1 β n ) t n p 2 β n x n p 2 + ( 1 β n ) [ v n p 2 + λ n ( λ n 2 L ) f ( v n ) f ( p ) 2 + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) ] β n x n p 2 + ( 1 β n ) [ x n p 2 + λ n ( λ n 2 L ) f ( v n ) f ( p ) 2 + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) ] x n p 2 + ( 1 β n ) λ n ( λ n 2 L ) f ( v n ) f ( p ) 2 + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) ,

which, together with { λ n }[a,b](0, 2 L ) and { β n }[c,d](0,1), leads to

( 1 d ) a ( 2 L b ) f ( v n ) f ( p ) 2 ( 1 β n ) λ n ( 2 L λ n ) f ( v n ) f ( p ) 2 x n p 2 y n p 2 + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) x n y n ( x n p + y n p ) + 2 α n b v n v n p λ n ( f α n ( v n ) f ( p ) ) .

Since { v n }, { x n }, and { y n } are bounded sequences, we deduce from (3.17) and α n 0 that

lim n f ( v n ) f ( p ) =0.

So, it is clear that

lim n f α n ( v n ) f ( p ) =0.
(3.30)

Again, utilizing Proposition 2.1(iii), from t n = P C (I λ n f α n ) v n and p= P C (I λ n f)p, we get

t n p 2 = P C ( I λ n f α n ) v n P C ( I λ n f ) p 2 ( I λ n f α n ) v n ( I λ n f ) p , t n p = 1 2 ( ( I λ n f α n ) v n ( I λ n f ) p 2 + t n p 2 ( I λ n f α n ) v n ( I λ n f ) p ( t n p ) 2 ) = 1 2 ( ( I λ n f α n ) v n ( I λ n f α n ) p λ n α n p 2 + t n p 2 ( I λ n f α n ) v n ( I λ n f ) p ( t n p ) 2 ) = 1 2 ( ( I λ n f α n ) v n ( I λ n f α n ) p 2 2 λ n α n p , ( I λ n f α n ) v n ( I λ n f ) p + t n p 2 ( I λ n f α n ) v n ( I λ n f ) p ( t n p ) 2 ) 1 2 ( v n p 2 2 λ n α n p , ( I λ n f α n ) v n ( I λ n f ) p + t n p 2 ( I λ n f α n ) v n ( I λ n f ) p ( t n p ) 2 ) 1 2 ( v n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p + t n p 2 v n t n λ n ( f α n ( v n ) f ( p ) ) 2 ) ,

which immediately leads to

t n p 2 v n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p v n t n λ n ( f α n ( v n ) f ( p ) ) 2 .
(3.31)

Combining (3.4), (3.20), and (3.31) we obtain

y n p 2 β n x n p 2 + ( 1 β n ) t n p 2 β n x n p 2 + ( 1 β n ) [ v n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p v n t n λ n ( f α n ( v n ) f ( p ) ) 2 ] β n x n p 2 + ( 1 β n ) [ x n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p v n t n λ n ( f α n ( v n ) f ( p ) ) 2 ] x n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p ( 1 β n ) v n t n λ n ( f α n ( v n ) f ( p ) ) 2 ,

which immediately yields

( 1 d ) v n t n λ n ( f α n ( v n ) f ( p ) ) 2 ( 1 β n ) v n t n λ n ( f α n ( v n ) f ( p ) ) 2 x n p 2 y n p 2 + 2 λ n α n p ( I λ n f α n ) v n ( I λ n f ) p x n y n ( x n p + y n p ) + 2 α n b p ( I λ n f α n ) v n ( I λ n f ) p .

Since { v n }, { x n }, and { y n } are bounded sequences, we deduce from (3.17) and α n 0 that

lim n v n t n λ n ( f α n ( v n ) f ( p ) ) =0.
(3.32)

Observe that

v n t n v n t n λ n ( f α n ( v n ) f ( p ) ) + λ n f α n ( v n ) f ( p ) .

Thus, from (3.30) and (3.32) we have

lim n v n t n =0.
(3.33)

Taking into account that x n t n x n v n + v n t n , from (3.28) and (3.33) we get

lim n x n t n =0.
(3.34)

Utilizing the relation y n x n = γ n ( t n x n )+ σ n (T t n x n ), we have

σ n ( T t n t n ) = σ n ( T t n x n ) σ n ( t n x n ) = y n x n γ n ( t n x n ) σ n ( t n x n ) = y n x n ( 1 β n ) ( t n x n ) y n x n + ( 1 β n ) t n x n y n x n + t n x n ,

which, together with (3.17) and (3.34), implies that

lim n σ n ( T t n t n ) =0.

Since lim inf n σ n >0, we obtain

lim n t n T t n =0.
(3.35)

Step 4. We prove that ω w ( x n )Ω.

Indeed, since H is reflexive and { x n } is bounded, there exists at least a weak convergence subsequence of { x n }. Hence we know that ω w ( x n ). Now, take an arbitrary w ω w ( x n ). Then there exists a subsequence { x n i } of { x n } such that x n i w. From (3.24)-(3.26), (3.28), and (3.34) we have that u n i w, v n i w, t n i w, Λ n i m u n i w and Δ n i k x n i w, where m{1,2,,N} and k{1,2,,M}. Utilizing Lemma 2.3(ii), we deduce from t n i w and (3.35) that wFix(T). Next, we prove that w m = 1 N I( B m , R m ). As a matter of fact, since B m is η m -inverse-strongly monotone, B m is a monotone and Lipschitz-continuous mapping. It follows from Lemma 2.13 that R m + B m is maximal monotone. Let (v,g)G( R m + B m ), i.e., g B m v R m v. Again, since Λ n m u n = J R m , λ m , n (I λ m , n B m ) Λ n m 1 u n , n1, m{1,2,,N}, we have

Λ n m 1 u n λ m , n B m Λ n m 1 u n (I+ λ m , n R m ) Λ n m u n ,

that is,

1 λ m , n ( Λ n m 1 u n Λ n m u n λ m , n B m Λ n m 1 u n ) R m Λ n m u n .

In terms of the monotonicity of R m , we get

v Λ n m u n , g B m v 1 λ m , n ( Λ n m 1 u n Λ n m u n λ m , n B m Λ n m 1 u n ) 0

and hence

v Λ n m u n , g v Λ n m u n , B m v + 1 λ m , n ( Λ n m 1 u n Λ n m u n λ m , n B m Λ n m 1 u n ) = v Λ n m u n , B m v B m Λ n m u n + B m Λ n m u n B m Λ n m 1 u n + 1 λ m , n ( Λ n m 1 u n Λ n m u n ) v Λ n m u n , B m Λ n m u n B m Λ n m 1 u n + v Λ n m u n , 1 λ m , n ( Λ n m 1 u n Λ n m u n ) .

In particular,

v Λ n i m u n i , g v Λ n i m u n i , B m Λ n i m u n i B m Λ n i m 1 u n i + v Λ n i m u n i , 1 λ m , n i ( Λ n i m 1 u n i Λ n i m u n i ) .

Since Λ n m u n Λ n m 1 u n 0 (due to (3.24)) and B m Λ n m u n B m Λ n m 1 u n 0 (due to the Lipschitz-continuity of B m ), we conclude from Λ n i m u n i w and { λ i , n }[ a i , b i ](0,2 η i ) that

lim i v Λ n i m u n i , g =vw,g0.

It follows from the maximal monotonicity of B m + R m that 0( R m + B m )w, i.e., wI( B m , R m ). Therefore, w m = 1 N I( B m , R m ). Next we prove that w k = 1 M GMEP( Θ k , φ k , A k ). Since Δ n k x n = T r k , n ( Θ k , φ k ) (I r k , n A k ) Δ n k 1 x n , n1, k{1,2,,M}, we have

Θ k ( Δ n k x n , y ) + φ k ( y ) φ k ( Δ n k x n ) + A k Δ n k 1 x n , y Δ n k x n + 1 r k , n y Δ n k x n , Δ n k x n Δ n k 1 x n 0 .

By (A2), we have

φ k ( y ) φ k ( Δ n k x n ) + A k Δ n k 1 x n , y Δ n k x n + 1 r k , n y Δ n k x n , Δ n k x n Δ n k 1 x n Θ k ( y , Δ n k x n ) .

Let z t =ty+(1t)w for all t(0,1] and yC. This implies that z t C. Then we have

z t Δ n k x n , A k z t φ k ( Δ n k x n ) φ k ( z t ) + z t Δ n k x n , A k z t z t Δ n k x n , A k Δ n k 1 x n z t Δ n k x n , Δ n k x n Δ n k 1 x n r k , n + Θ k ( z t , Δ n k x n ) = φ k ( Δ n k x n ) φ k ( z t ) + z t Δ n k x n , A k z t A k Δ n k x n + z t Δ n k x n , A k Δ n k x n A k Δ n k 1 x n z t Δ n k x n , Δ n k x n Δ n k 1 x n r k , n + Θ k ( z t , Δ n k x n ) .
(3.36)

By (3.25), we have A k Δ n k x n A k Δ n k 1 x n 0 as n. Furthermore, by the monotonicity of A k , we obtain z t Δ n k x n , A k z t A k Δ n k x n 0. Then by (A4) we obtain

z t w, A k z t φ k (w) φ k ( z t )+ Θ k ( z t ,w).
(3.37)

Utilizing (A1), (A4), and (3.37), we obtain

0 = Θ k ( z t , z t ) + φ k ( z t ) φ k ( z t ) t Θ k ( z t , y ) + ( 1 t ) Θ k ( z t , w ) + t φ k ( y ) + ( 1 t ) φ k ( w ) φ k ( z t ) t [ Θ k ( z t , y ) + φ k ( y ) φ k ( z t ) ] + ( 1 t ) z t w , A k z t = t [ Θ k ( z t , y ) + φ k ( y ) φ k ( z t ) ] + ( 1 t ) t y w , A k z t ,

and hence

0 Θ k ( z t ,y)+ φ k (y) φ k ( z t )+(1t)yw, A k z t .

Letting t0, we have, for each yC,

0 Θ k (w,y)+ φ k (y) φ k (w)+yw, A k w.

This implies that wGMEP( Θ k , φ k , A k ) and hence w k = 1 M GMEP( Θ k , φ k , A k ). Thus, wΩ= n = 1 Fix( T n ) k = 1 M GMEP( Θ k , φ k , A k ) m = 1 N I( B m , R m ).

Furthermore, let us show that wΓ. In fact, define

T ˜ v= { f ( v ) + N C v , if  v C , , if  v C ,

where N C v={uH:vx,u0,xC}. Then T ˜ is maximal monotone and 0 T ˜ v if and only if vVI(C,f); see [29]. Let (v, v ˜ )G( T ˜ ). Then we have v ˜ T ˜ v=f(v)+ N C v, and hence v ˜ f(v) N C v. So, we have vx, v ˜ f(v)0 for all xC. On the other hand, from t n = P C ( v n λ n f α n ( v n )) and vC, we get v n λ n f α n ( v n ) t n , t n v0, and hence,

v t n , t n v n λ n + f α n ( v n ) 0.

Therefore, from v ˜ f(v) N C v and t n i C, we have

v t n i , v ˜ v t n i , f ( v ) v t n i , f ( v ) v t n i , t n i v n i λ n i + f α n i ( v n i ) = v t n i , f ( v ) v t n i , t n i v n i λ n i + f ( v n i ) α n i v t n i , v n i = v t n i , f ( v ) f ( t n i ) + v t n i , f ( t n i ) f ( v n i ) v t n i , t n i v n i λ n i α n i v t n i , v n i v t n i , f ( t n i ) f ( v n i ) v t n i , t n i v n i λ n i α n i v t n i , v n i .

Hence, it is easy to see that vw, v ˜ 0 as i. Since T ˜ is maximal monotone, we have w T ˜ 1 0, and hence wVI(C,f)=Γ. Consequently, w k = 1 M GMEP( Θ k , φ k , A k ) i = 1 N I( B i , R i )Fix(T)Γ=:Ω. This shows that ω w ( x n )Ω.

Step 5. We prove that x n x where { x }=VI(Ω,γSμF).

Indeed, take an arbitrary w ω w ( x n ). Then there exists a subsequence { x n i } of { x n } such that x n i w. Utilizing (3.16), we find that, for all pΩ,

x n + 1 p 2 ( 1 ϵ n τ 2 γ 2 τ ) ( x n p + α n b p ) 2 β n ( 1 ϵ n τ ) 1 β n y n x n 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ( x n p + α n b p ) 2 + 2 ϵ n δ n ( γ V p μ F p ) , x n + 1 p + 2 ϵ n ( 1 δ n ) ( γ S p μ F p ) , x n + 1 p ,

which implies that

( μ F γ S ) p , x n p ( μ F γ S ) p , x n x n + 1 + ( μ F γ S ) p , x n + 1 p ( μ F γ S ) p x n x n + 1 + ( x n p + α n b p ) 2 x n + 1 p 2 2 ϵ n ( 1 δ n ) + δ n 1 δ n ( γ V μ F ) p , x n + 1 p ( μ F γ S ) p x n x n + 1 + ( x n x n + 1 + α n b p ) ( x n p + x n + 1 p + α n b p ) 2 ϵ n ( 1 δ n ) + δ n 1 δ n ( γ V μ F ) p x n + 1 p .
(3.38)

Since α n 0, α n δ n 0, x n x n + 1 0, and

lim n x n x n + 1 + α n b p ϵ n = lim n x n x n + 1 + α n b p δ n δ n ϵ n =0,

from (3.38) we conclude that

( μ F γ S ) p , w p = lim i ( μ F γ S ) p , x n i p lim sup n ( μ F γ S ) p , x n p 0 , p Ω ,

that is,

( μ F γ S ) p , w p 0,pΩ.
(3.39)

Since μFγS is (μηγ)-strongly monotone and (μκ+γ)-Lipschitz-continuous, by Minty’s lemma [39] we know that (3.39) is equivalent to the VIP

( μ F γ S ) w , p w 0,pΩ.
(3.40)

This shows that wVI(Ω,μFγS). Taking into account { x }=VI(Ω,μFγS), we know that w= x . Thus, ω w ( x n )={ x }; that is, x n x .

Next we prove that lim n x n x =0. As a matter of fact, utilizing (3.16) with p= x , we get

x n + 1 x 2 ( 1 ϵ n τ 2 γ 2 τ ) ( x n x + α n b x ) 2 β n ( 1 ϵ n τ ) 1 β n y n x n 2 + 2 ϵ n δ n ( γ V x μ F x ) , x n + 1 x + 2 ϵ n ( 1 δ n ) ( γ S x μ F x ) , x n + 1 x ( 1 ϵ n τ 2 γ 2 τ ) ( x n x + α n b x ) 2 + 2 ϵ n δ n ( γ V μ F ) x x n + 1 x + 2 ϵ n ( 1 δ n ) ( γ S μ F ) x , x n + 1 x = ( 1 ϵ n τ 2 γ 2 τ ) [ x n x 2 + α n b x ( 2 x n x + α n b x ) ] + 2 ϵ n δ n ( γ V μ F ) x x n + 1 x + 2 ϵ n ( 1 δ n ) ( γ S μ F ) x , x n + 1 x ( 1 ϵ n τ 2 γ 2 τ ) x n x 2 + 2 ϵ n δ n ( γ V μ F ) x x n + 1 x + 2 ϵ n ( 1 δ n ) ( γ S μ F ) x , x n + 1 x + α n b x ( 2 x n x + α n b x ) = ( 1 ϵ n τ 2 γ 2 τ ) x n x 2 + ϵ n τ 2 γ 2 τ 2 τ τ 2 γ 2 [ δ n ( γ V μ F ) x x n + 1 x + ( 1 δ n ) ( γ S μ F ) x , x n + 1 x ] + α n b x ( 2 x n x + α n b x ) .
(3.41)

Since n = 0 α n <, n = 0 ϵ n =, and lim n (γSμF) x , x x n + 1 =0 (due to x n x ), we deduce that n = 0 α n b x (2 x n x + α n b x )<, n = 0 ϵ n τ 2 γ 2 τ =, and

lim n 2 τ τ 2 γ 2 [ δ n ( γ V μ F ) x x n + 1 x + ( 1 δ n ) ( γ S μ F ) x , x n + 1 x ] =0.

Therefore, applying Lemma 2.8 to (3.41) we infer that lim n x n x =0. This completes the proof. □

In Theorem 3.1, putting M=1 and N=2, we obtain the following.

Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let f:CR be a convex functional with L-Lipschitz-continuous gradient f. Let Θ 1 be a bifunction from C×C to R satisfying (A1)-(A4), φ 1 :CR{+} be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and A 1 :HH be μ 1 -inverse-strongly monotone. Let R i :C 2 H be a maximal monotone mapping and B i :CH be η i -inverse-strongly monotone, for i=1,2. Let T:CC be a ξ-strictly pseudocontractive mapping, S:HH be a nonexpansive mapping and V:HH be a ρ-contraction with coefficient ρ[0,1). Let F:HH be κ-Lipschitzian and η-strongly monotone with positive constants κ,η>0 such that 0γ<τ and 0<μ< 2 η κ 2 where τ=1 1 μ ( 2 η μ κ 2 ) . Assume that Ω:=GMEP( Θ 1 , φ 1 , A 1 )I( B 2 , R 2 )I( B 1 , R 1 )Fix(T)Γ. Let { λ n }[a,b](0, 2 L ), { α n }(0,) with n = 0 α n <, { ϵ n },{ δ n },{ β n },{ γ n },{ σ n }(0,1) with β n + γ n + σ n =1, and { r 1 , n }[ c 1 , d 1 ](0,2 μ 1 ), { λ i , n }[ a i , b i ](0,2 η i ) for i=1,2. For arbitrarily given x 0 H, let { x n } be a sequence generated by

{ Θ ( u n , y ) + φ ( y ) φ ( u n ) + A x n , y u n + 1 r n y u n , u n x n 0 , y C , v n = J R 2 , λ 2 , n ( I λ 2 , n B 2 ) J R 1 , λ 1 , n ( I λ 1 , n B 1 ) u n , y n = β n x n + γ n P C ( I λ n f α n ) v n + σ n T P C ( I λ n f α n ) v n , x n + 1 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( I ϵ n μ F ) y n , n 0 ,
(3.42)

where f α n = α n I+f for all n0. Suppose that

(C1) lim n ϵ n =0, n = 0 ϵ n = and lim n 1 ϵ n |1 δ n 1 δ n |=0;

(C2) lim sup n δ n ϵ n <, lim n 1 ϵ n | 1 δ n 1 δ n 1 |=0 and lim n 1 δ n |1 ϵ n 1 ϵ n |=0;

(C3) lim n | β n β n 1 | ϵ n δ n =0 and lim n | γ n γ n 1 | ϵ n δ n =0;

(C4) lim n | r 1 , n r 1 , n 1 | ϵ n δ n =0 and lim n | λ i , n λ i , n 1 | ϵ n δ n =0 for i=1,2;

(C5) lim n | λ n λ n 1 | ϵ n δ n =0, lim n | λ n α n λ n 1 α n 1 | ϵ n δ n =0 and ( γ n + σ n )ξ γ n for all n0;

(C6) { β n }[c,d](0,1), lim n α n δ n =0 and lim inf n σ n >0.

Then we have:

  1. (i)

    lim n x n + 1 x n δ n =0;

  2. (ii)

    ω w ( x n )Ω;

  3. (iii)

    { x n } converges strongly to a minimizer x of the CMP (1.2), which is a unique solution x in Ω to the HVIP

    ( μ F γ S ) x , p x 0,pΩ.

In Theorem 3.1, putting M=N=1, we obtain the following.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let f:CR be a convex functional with L-Lipschitz-continuous gradient f. Let Θ 1 be a bifunction from C×C to R satisfying (A1)-(A4), φ 1 :CR{+} be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and A 1 :HH be μ 1 -inverse-strongly monotone. Let R 1 :C 2 H be a maximal monotone mapping and B 1 :CH be η 1 -inverse-strongly monotone. Let T:CC be a ξ-strictly pseudocontractive mapping, S:HH be a nonexpansive mapping and V:HH be a ρ-contraction with coefficient ρ[0,1). Let F:HH be κ-Lipschitzian and η-strongly monotone with positive constants κ,η>0 such that 0γ<τ and 0<μ< 2 η κ 2 where τ=1 1 μ ( 2 η μ κ 2 ) . Assume that Ω:=GMEP( Θ 1 , φ 1 , A 1 )I( B 1 , R 1 )Fix(T)Γ. Let { λ n }[a,b](0, 2 L ), { α n }(0,) with n = 0 α n <, { ϵ n },{ δ n },{ β n },{ γ n },{ σ n }(0,1) with β n + γ n + σ n =1, and { r 1 , n }[ c 1 , d 1 ](0,2 μ 1 ), { λ 1 , n }[ a 1 , b 1 ](0,2 η 1 ). For arbitrarily given x 0 H, let { x n } be a sequence generated by

{ Θ ( u n , y ) + φ ( y ) φ ( u n ) + A x n , y u n + 1 r n y u n , u n x n 0 , y C , v n = J R 1 , λ 1 , n ( I λ 1 , n B 1 ) u n , y n = β n x n + γ n P C ( I λ n f α n ) v n + σ n T P C ( I λ n f α n ) v n , x n + 1 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( I ϵ n μ F ) y n , n 0 ,
(3.43)

where f α n = α n I+f for all n0. Suppose that

(C1) lim n ϵ n =0, n = 0 ϵ n = and lim n 1 ϵ n |1 δ n 1 δ n |=0;

(C2) lim sup n δ n ϵ n <, lim n 1 ϵ n | 1 δ n 1 δ n 1 |=0 and lim n 1 δ n |1 ϵ n 1 ϵ n |=0;

(C3) lim n | β n β n 1 | ϵ n δ n =0 and lim n | γ n γ n 1 | ϵ n δ n =0;

(C4) lim n | λ 1 , n λ 1 , n 1 | ϵ n δ n =0 and lim n | r 1 , n r 1 , n 1 | ϵ n δ n =0;

(C5) lim n | λ n λ n 1 | ϵ n δ n =0, lim n | λ n α n λ n 1 α n 1 | ϵ n δ n =0 and ( γ n + σ n )ξ γ n for all n0;

(C6) { β n }[c,d](0,1), lim n α n δ n =0 and lim inf n σ n >0.

Then we have:

  1. (i)

    lim n x n + 1 x n δ n =0;

  2. (ii)

    ω w ( x n )Ω;

  3. (iii)

    { x n } converges strongly to a minimizer x of the CMP (1.2), which is a unique solution x in Ω to the HVIP

    ( μ F γ S ) x , p x 0,pΩ.

In Theorem 3.1, putting M=N=1, μ=2, and F 1 2 I where I is the identity mapping on H, we obtain the following.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let f:CR be a convex functional with L-Lipschitz-continuous gradient f. Let Θ 1 be a bifunction from C×C to R satisfying (A1)-(A4), φ 1 :CR{+} be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and A 1 :HH be μ 1 -inverse-strongly monotone. Let R 1 :C 2 H be a maximal monotone mapping and B 1 :CH be η 1 -inverse-strongly monotone. Let T:CC be a ξ-strictly pseudocontractive mapping, S:HH be a nonexpansive mapping and V:HH be a ρ-contraction with coefficient ρ[0,1). Assume that Ω:=GMEP( Θ 1 , φ 1 , A 1 )I( B 1 , R 1 )Fix(T)Γ. Let 0γ<1, { λ n }[a,b](0, 2 L ), { α n }(0,) with n = 0 α n <, { ϵ n },{ δ n },{ β n },{ γ n },{ σ n }(0,1) with β n + γ n + σ n =1, and { r 1 , n }[ c 1 , d 1 ](0,2 μ 1 ), { λ 1 , n }[ a 1 , b 1 ](0,2 η 1 ). For arbitrarily given x 0 H, let { x n } be a sequence generated by

{ Θ ( u n , y ) + φ ( y ) φ ( u n ) + A x n , y u n + 1 r n y u n , u n x n 0 , y C , v n = J R 1 , λ 1 , n ( I λ 1 , n B 1 ) u n , y n = β n x n + γ n P C ( I λ n f α n ) v n + σ n T P C ( I λ n f α n ) v n , x n + 1 = ϵ n γ ( δ n V x n + ( 1 δ n ) S x n ) + ( 1 ϵ n ) y n , n 0 ,
(3.44)

where f α n = α n I+f for all n0. Suppose that

(C1) lim n ϵ n =0, n = 0 ϵ n = and lim n 1 ϵ n |1 δ n 1 δ n |=0;

(C2) lim sup n δ n ϵ n <, lim n 1 ϵ n | 1 δ n 1 δ n 1 |=0 and lim n 1 δ n |1 ϵ n 1 ϵ n |=0;

(C3) lim n | β n β n 1 | ϵ n δ n =0 and lim n | γ n γ n 1 | ϵ n δ n =0;

(C4) lim n | λ 1 , n λ 1 , n 1 | ϵ n δ n =0 and lim n | r 1 , n r 1 , n 1 | ϵ n δ n =0;

(C5) lim n | λ n λ n 1 | ϵ n δ n =0, lim n | λ n α n λ n 1 α n 1 | ϵ n δ n =0 and ( γ n + σ n )ξ γ n for all n0;

(C6) { β n }[c,d](0,1), lim n α n δ n =0 and lim inf n σ n >0.

Then we have:

  1. (i)

    lim n x n + 1 x n δ n =0;

  2. (ii)

    ω w ( x n )Ω;

  3. (iii)

    { x n } converges strongly to a minimizer x of the CMP (1.2), which is a unique solution x in Ω to the HVIP

    ( I γ S ) x , p x 0,pΩ.

Remark 3.1 It is obvious that our iterative scheme (3.1) is very different from Xu’s iterative one (1.4) and Yao et al.’s iterative one (1.8). Here, Xu’s iterative scheme in [[20], Theorem 5.2] is extended to develop our four-step iterative scheme (3.1) for the CMP (1.2) with constraints of finite many GMEPs, finite many variational inclusions and the fixed point problem of a strict pseudocontraction by combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method (GPM) with regularization. It is worth pointing out that under the lack of the assumptions similar to those in [[21], Theorem 3.2], e.g., { x n } is bounded, Fix(T)intC and xTxkDist(x,Fix(T)), xC for some k>0, the sequence { x n } generated by (3.1) converges strongly to a point x k = 1 M GMEP( Θ k , φ k , A k ) i = 1 N I( B i , R i )Fix(T)Γ=:Ω, which is a unique solution of the HVIP (over the fixed point set of strictly pseudocontractive mapping T), i.e., (μFγS) x ,p x 0, pΩ.

Remark 3.2 Our Theorem 3.1 improves, extends, supplements, and develops Xu [[20], Theorem 5.2] and Yao et al. [[21], Theorems 3.1 and 3.2] in the following aspects:

  1. (a)

    Our HVIP with the unique solution x Ω satisfying

    x = P k = 1 M GMEP ( Θ k , φ k , A k ) i = 1 N I ( B i , R i ) Fix ( T ) Γ ( I ( μ F γ S ) ) x

    is more general than the problem of finding a point x ˜ C satisfying x ˜ = P Fix ( T ) S x ˜ in [21] and very different from the problem of finding a point x Γ satisfying x = P Γ V x in [[20], Theorem 5.2]. It is worth emphasizing that the nonexpansive mapping T in [[21], Theorems 3.1 and 3.2] is extended to the strict pseudocontraction T and the CMP in [[20], Theorem 5.2] is generalized to the setting of finitely many GMEPs, finitely many variational inclusions and the fixed point problem of a strict pseudocontraction T.

  2. (b)

    Our four-step iterative scheme (3.1) for the CMP with constraints of several problems is more flexible, more advantageous and more subtle than Xu’s one-step iterative one (1.4) and than Yao et al.’s two-step iterative one (1.8) because it can be used to solve two kinds of problems, e.g., the HVIP (over the fixed point set of strictly pseudocontractive mapping T), and the problem of finding a common point of four sets: k = 1 M GMEP( Θ k , φ k , A k ), i = 1 N I( B i , R i ), Fix(T) and Γ. In addition, our Theorem 3.1 also drops the crucial requirements in [[21], Theorem 3.2(v)] that Fix(T)intC and xTxkDist(x,Fix(T)), xC for some k>0.

  3. (c)

    The techniques for the argument in our Theorem 3.1 are very different from [[21], Theorems 3.1 and 3.2] and from [[20], Theorem 5.2] because we make use of the properties of strictly pseudocontractive mappings (see Lemmas 2.3 and 2.4), the ones of resolvent operators and maximal monotone mappings (see Proposition 2.2, Remark 2.2, and Lemmas 2.9-2.13), the ones of averaged mappings (see Propositions 2.3 and 2.4), the inclusion problem 0 T ˜ v (vVI(C,A) for maximal monotone operator T ˜ ) (see (2.2)), and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (see Lemma 2.7).

  4. (d)

    Compared with the restrictions on the parameter sequences of [[21], Theorem 3.2] and [[20], Theorem 5.2], respectively, the hypotheses (C3)-(C6) in our Theorem 3.1 are added because our Theorem 3.1 involves the quite complex problem, i.e., the HVIP (over the fixed point set of strictly pseudocontractive mapping T) with constraints of several problems: CMP (1.2), finitely many GMEPs and finitely many variational inclusions.

References

  1. Lions JL: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris; 1969.

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.

    MATH  Google Scholar 

  4. Oden JT: Quantitative Methods on Nonlinear Mechanics. Prentice Hall, Englewood Cliffs; 1986.

    MATH  Google Scholar 

  5. Zeidler E: Nonlinear Functional Analysis and Its Applications. Springer, New York; 1985.

    Book  MATH  Google Scholar 

  6. Korpelevich GM: The extragradient method for finding saddle points and other problems. Matecon 1976, 12: 747–756.

    MathSciNet  MATH  Google Scholar 

  7. Cai G, Bu SQ: Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces. J. Comput. Appl. Math. 2013, 247: 34–52.

    Article  MathSciNet  MATH  Google Scholar 

  8. Zeng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 2006,10(5):1293–1303.

    MathSciNet  MATH  Google Scholar 

  9. Peng JW, Yao JC: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwan. J. Math. 2008, 12: 1401–1432.

    MathSciNet  MATH  Google Scholar 

  10. Ceng LC, Ansari QH, Schaible S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, system of generalized equilibrium problems and optimization problems. J. Glob. Optim. 2012, 53: 69–96. 10.1007/s10898-011-9703-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Yao Y, Liou YC, Kang SM: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 2010, 59: 3472–3480. 10.1016/j.camwa.2010.03.036

    Article  MathSciNet  MATH  Google Scholar 

  12. Ceng LC, Ansari QH, Wong MM, Yao JC: Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems. Fixed Point Theory 2012,13(2):403–422.

    MathSciNet  MATH  Google Scholar 

  13. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006, 128: 191–201. 10.1007/s10957-005-7564-z

    Article  MathSciNet  MATH  Google Scholar 

  14. Ceng LC, Hadjisavvas N, Wong NC: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 2010, 46: 635–646. 10.1007/s10898-009-9454-7

    Article  MathSciNet  MATH  Google Scholar 

  15. Ceng LC, Yao JC: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 2010, 72: 1922–1937. 10.1016/j.na.2009.09.033

    Article  MathSciNet  MATH  Google Scholar 

  16. Ceng LC, Wong MM: Relaxed extragradient method for finding a common element of systems of variational inequalities and fixed point problems. Taiwan. J. Math. 2013,17(2):701–724.

    MathSciNet  MATH  Google Scholar 

  17. Ceng LC, Liao CW, Pang CT, Wen CF: Convex minimization with constraints of systems of variational inequalities, mixed equilibrium, variational inequality, and fixed point problems. J. Appl. Math. 2014., 2014: Article ID 105928

    Google Scholar 

  18. Ceng LC, Ansari QH, Yao JC: Relaxed extragradient iterative methods for variational inequalities. Appl. Math. Comput. 2011, 218: 1112–1123. 10.1016/j.amc.2011.01.061

    Article  MathSciNet  MATH  Google Scholar 

  19. Ceng LC, Petrusel A: Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem. Taiwan. J. Math. 2012,16(2):445–478.

    MathSciNet  MATH  Google Scholar 

  20. Xu HK: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 2011, 150: 360–378. 10.1007/s10957-011-9837-z

    Article  MathSciNet  MATH  Google Scholar 

  21. Yao Y, Liou YC, Marino G: Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems. J. Appl. Math. Comput. 2009,31(1–2):433–445. 10.1007/s12190-008-0222-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Ceng LC, Guu SM, Yao JC: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 92

    Google Scholar 

  23. Ceng LC, Hu HY, Wong MM: Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed point problem of infinitely many nonexpansive mappings. Taiwan. J. Math. 2011,15(3):1341–1367.

    MathSciNet  MATH  Google Scholar 

  24. Ceng LC, Al-Homidan S: Algorithms of common solutions for generalized mixed equilibria, variational inclusions, and constrained convex minimization. Abstr. Appl. Anal. 2014., 2014: Article ID 132053

    Google Scholar 

  25. Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042

    Article  MathSciNet  MATH  Google Scholar 

  26. Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022

    Article  MathSciNet  MATH  Google Scholar 

  27. Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041

    Article  MathSciNet  MATH  Google Scholar 

  28. Ceng LC, Petrusel A, Yao JC: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. J. Optim. Theory Appl. 2009, 143: 37–58. 10.1007/s10957-009-9549-9

    Article  MathSciNet  MATH  Google Scholar 

  29. Rockafellar RT: Monotone operators and the proximal point algorithms. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang NJ: A new completely general class of variational inclusions with noncompact valued mappings. Comput. Math. Appl. 1998,35(10):9–14. 10.1016/S0898-1221(98)00067-4

    Article  MathSciNet  MATH  Google Scholar 

  31. Zeng LC, Guu SM, Yao JC: Characterization of H -monotone operators with applications to variational inclusions. Comput. Math. Appl. 2005,50(3–4):329–337. 10.1016/j.camwa.2005.06.001

    Article  MathSciNet  MATH  Google Scholar 

  32. Ceng LC, Guu SM, Yao JC: Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions. Comput. Math. Appl. 2008, 56: 978–987. 10.1016/j.camwa.2008.01.026

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang SS, Lee HWJ, Chan CK: Algorithms of common solutions for quasi-variational inclusions and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s10483-008-0502-y

    Article  MathSciNet  MATH  Google Scholar 

  34. Ceng LC, Guu SM, Yao JC: Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions. Comput. Math. Appl. 2008, 56: 942–952. 10.1016/j.camwa.2008.01.024

    Article  MathSciNet  MATH  Google Scholar 

  35. Han D, Lo HK: Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities. Eur. J. Oper. Res. 2004, 159: 529–544. 10.1016/S0377-2217(03)00423-5

    Article  MathSciNet  MATH  Google Scholar 

  36. 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

    Article  MathSciNet  MATH  Google Scholar 

  37. Combettes PL: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 2004,53(5–6):475–504. 10.1080/02331930412331327157

    Article  MathSciNet  MATH  Google Scholar 

  38. Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055

    Article  MathSciNet  MATH  Google Scholar 

  39. Goebel K, Kirk WA: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

    Book  MATH  Google Scholar 

  40. Xu HK, Kim TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 2003,119(1):185–201.

    Article  MathSciNet  MATH  Google Scholar 

  41. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 2002,66(1):240–256. 10.1112/S0024610702003332

    Article  MathSciNet  MATH  Google Scholar 

  42. Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Groningen; 1976.

    Book  MATH  Google Scholar 

  43. Yamada I: The hybrid steepest-descent method for the variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Batnariu D, Censor Y, Reich S. North-Holland, Amsterdam; 2001:473–504.

    Chapter  Google Scholar 

  44. Baillon JB, Haddad G: Quelques propriétés des opérateurs angle-bornés et n -cycliquement monotones. Isr. J. Math. 1977, 26: 137–150. 10.1007/BF03007664

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

LC Ceng was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and PhD Program Foundation of Ministry of Education of China (20123127110002). YC Liou was supported in part by the Grants NSC 101-2628-E-230-001-MY3 and NSC 103-2923-E-037-001-MY3. CF Wen was partially supported by the Grant NSC 103-2923-E-037-001-MY3.

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Ceng, LC., Liou, YC. & Wen, CF. Extragradient method for convex minimization problem. J Inequal Appl 2014, 444 (2014). https://doi.org/10.1186/1029-242X-2014-444

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