Algorithms for finding minimum norm solution of equilibrium and fixed point problems for

Research Article

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for

nonexpansive semigroups in Hilbert spaces

Yaqiang Liu^a, Shin Min Kang^b,∗, Youli Yu^c, Lijun Zhu^d

aSchool of Management, Tianjin Polytechnic University, Tianjin 300387, China.

bDepartment of Mathematics and the RINS, Gyeongsang National University, Jinju 52828, Korea.

cSchool of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China.

dSchool of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China.

Communicated by R. Saadati

Abstract

In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup {T(s)}s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common elementx^∗ of the set of the equilibrium points and the set of common fixed points of{T(s)}_s≥0. Such common elementx^∗ is the unique solution of some variational inequality, which is the optimality condition for some minimization problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of the equilibrium points and the set of common fixed points of {T(s)}_s≥0. The results obtained in the present paper improve and extend the corresponding results by Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, J. Optim. Theory. Appl., 146 (2010), 491–509]

and many others. c2016 All rights reserved.

Keywords: Equilibrium problem, variational inequality, fixed point, nonexpansive semigroup, algorithm, minimum norm.

2010 MSC: 47J05, 47J25, 47H09.

1. Introduction

LetH be a real Hilbert space with inner producth·,·iand normk · k, respectively. LetC be a nonempty

∗Corresponding author

Email addresses: [email protected](Yaqiang Liu),[email protected](Shin Min Kang),[email protected](Youli Yu),[email protected](Lijun Zhu)

Received 2016-01-07

closed convex subset of H. Recall that a mapping f : C → H be a ρ-contraction; that is, there exists a constantρ ∈ [0,1) such that kf(x)−f(y)k ≤ ρkx−yk for all x, y ∈C. A mapping T :C →C is said to be nonexpansive ifkT x−T yk ≤ kx−yk for all x, y ∈C. Denote the set of fixed points of T by F ix(T).

Let A be a strongly positive bounded linear operator on H, i.e., there exists a constant ¯γ > 0 such that hAx, xi ≥¯γkxk² for all x∈H.

Iterative methods for nonexpansive mappings are widely used to solve convex minimization problems.

A typical problem is to minimize a function over the set of fixed points of a nonexpansive mappingT,

x∈F ix(Tmin )

1

2hAx, xi − hx, bi. (1.1)

In [20], Xu proved that the sequence {x_n}defined by

xn+1=αnb+ (1−αnA)T xn, n≥0,

strongly converges to the unique solution of (1.1) under certain conditions. Recently, Marino and Xu [11]

introduced the viscosity approximation method

x_n+1 =α_nγf(x_n) + (1−α_nA)T x_n, n≥0,

and proved that the sequence{x_n} converges strongly to the unique solution of the variational inequality h(A−γf)z, x−zi ≥0, ∀x∈F ix(T),

which is the optimality condition for the minimization problem

x∈F ix(Tmin )

1

2hAx, xi −h(x), whereh is a potential function forγf (i.e.,h⁰=γf on H).

Recall also that a mapping B :C → H is called α-inverse-strongly monotone if there exists a positive real numberα such that

hBx−By, x−yi ≥αkBx−Byk², ∀x, y∈C.

It is clear that any α-inverse-strongly monotone mapping is monotone (that is, hBx−By, x−yi is non-negative) and _α¹-Lipschitz continuous.

LetB :C →H be a nonlinear mapping andF :C×C →Rbe a bifunction. We concerned equilibrium problem is to find z∈C such that

F(z, y) +hBz, y−zi ≥0, ∀y ∈C. (1.2)

The solution set of (1.2) is denoted by Ω. If B = 0, then (1.2) reduces to the following equilibrium problem of finding z∈C such that

F(z, y)≥0, ∀y∈C. (1.3)

The equilibrium problem and the variational inequality problem have been investigated by many au- thors. To see related works, we refer the reader to [1–5, 7–15, 17, 19–28] and the references therein. The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others.

For solving equilibrium problem (1.2), Moudafi [12] introduced an iterative algorithm and proved a weak convergence theorem. Further, Takahashi and Takahashi [19] introduced another iterative algorithm for finding an element of Ω∩F ix(S) and they obtained a strong convergence result. Recently, Plubtieng and Punpaeng [15] introduced the following iterative method to find an equilibrium point ofF, which is also a fixed point of a nonexpansive mappingT :H→H,

(F(un, y) +_r¹

nhx−un, un−xni ≥0, ∀y∈H,

x_n+1 =α_nf(x_n) + (I−α_nA)T u_n, n≥0. (1.4)

They proved that, with suitable conditions, both the sequences {x_n} and {u_n} defined by (1.4) are strongly convergent to the unique solutionz∈F ix(T)∩Ω of the variational inequalityh(A−γf)z, x−zi ≥0 for all x ∈ F ix(T) ∩ Ω, which is the optimality condition for the minimization problem min_{x∈F ix(T)∩Ω} ¹₂hAx, xi −h(x), where h is a potential function forγf.

In this paper, we focus on nonexpansive semigroup {T(s)}s≥0. Recall that a family S := {T(s)}s≥0

of mappings ofC into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:

(S1) T(0)x=xfor all x∈C;

(S2) T(s+t) =T(s)T(t) for all s, t≥0;

(S3) kT(s)x−T(s)yk ≤ kx−yk for all x, y∈C and s≥0;

(S4) for allx∈H,s→T(s)x is continuous.

We denote byF ix(T(s)) the set of fixed points ofT(s) and byF ix(S) the set of all common fixed points ofS, i.e., F ix(S) =T

s≥0F ix(T(s)). It is known thatF ix(S) is closed and convex.

Very recently, Cianciaruso et al. [5] introduced the following implicit and explicit schemes for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup in Hilbert spaces:

Implicit algorithm:

(G(u_t, y) +_r¹

thy−u_t, u_t−x_ti ≥0, ∀y ∈H, xt=tγf(xt) + (I−tA)_λ¹

t

Rλt

0 T(s)utds, (1.5)

and

Explicit algorithm:

(F(u_n, y) +_r¹

nhy−u_n, u_n−x_ni ≥0, ∀y∈H, xn+1 =αnγf(xn) + (I−αnA)_λ¹

n

Rλn

0 T(s)unds, n≥0. (1.6) They proved that the above both approaches (1.5) and (1.6) have strong convergence. (Note that the integral mentioned in the present paper is the usual integral, for example, we can compute R_t

0T(s)xdsas limn→∞Pn

i=1 t

nT(_n^ti)x).

The following interesting problem arises: can one construct some more general algorithms which unify the above algorithms?

On the other hand, we also notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H1 and a bounded linear operator R : H1 → H2, where H2 is another Hilbert space. The C-constrained pseudoinverse of R, R^†_C, is then defined as the minimum-norm solution of the constrained minimization problem

R^†_C(b) := arg min

x∈CkRx−bk, which is equivalent to the fixed point problem

x=P_C(x−λR^∗(Rx−b)),

whereP_C is the metric projection fromH₁ onto C,R^∗ is the adjoint of R,λ >0 is a constant, andb∈H₂ is such that P_R(C)(b)∈R(C).

It is therefore another interesting problem to invent some algorithms that can generate schemes which converge strongly to the minimum-norm solution of a given problem.

In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup {T(s)}_s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common elementx^∗ of the set of the equilibrium points and the set of common fixed points of{T(s)}_s≥0. Such common elementx^∗ is the unique solution of some variational inequality, which is the optimality condition for some minimization

problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of the equilibrium points and the set of common fixed points of {T(s)}_s≥0.

The results contained in the present paper improve and extend the corresponding results by Cianciaruso et al. [5] and many others.

2. Preliminaries

LetCbe a nonempty closed convex subset of a real Hilbert space H. Throughout this paper, we assume that a bifunction F :C×C→Rsatisfies the following conditions:

(H1) F(x, x) = 0 for allx∈C;

(H2) F is monotone, i.e.,F(x, y) +F(y, x)≤0 for allx, y∈C;

(H3) for each x, y, z∈C, limt↓0F(tz+ (1−t)x, y)≤F(x, y);

(H4) for each x∈C,y7→F(x, y) is convex and lower semicontinuous.

The metric (or nearest point) projection from H onto C is the mapping P_C :H →C which assigns to each pointx∈C the unique pointP_Cx∈C satisfying the property

kx−P_Cxk= inf

y∈Ckx−yk=:d(x, C).

It is well known that P_C is a nonexpansive mapping and satisfies

hx−y, PCx−PCyi ≥ kP_Cx−PCyk², ∀x, y∈H.

Moreover, PC is characterized by the following properties:

hx−P_Cx, y−P_Cxi ≤0, (2.1)

and

kx−yk² ≥ kx−P_Cxk²+ky−P_Cxk² for all x∈H and y∈C.

We need the following lemmas to prove our main results.

Lemma 2.1 ([8]). Let C be a nonempty closed convex subset of a real Hilbert spaceH. LetF :C×C →R be a bifunction which satisfies conditions (H1)–(H4). Letr > 0 and x∈C. Then, there exists z∈C such that

F(z, y) +1

rhy−z, z−xi ≥0, ∀y∈C.

Further, if Sr(x) ={z∈C :F(z, y) +¹_rhy−z, z−xi ≥0,∀y∈C}, then the following hold:

(i) Sr is single-valued and Sr is firmly nonexpansive, i.e., for any x, y ∈ H, kS_rx−Sryk² ≤ hS_rx− S_ry, x−yi;

(ii) SF (as the set of all z∈C holding (1.3)) is closed and convex and SF =F ix(Sr).

Lemma 2.2 ([13]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping B :C→H be α-inverse strongly monotone and r >0 be a constant. Then, we have

k(I −rB)x−(I−rB)yk²≤ kx−yk²+r(r−2α)kBx−Byk², ∀x, y∈C.

In particular, if 0≤r≤2α, then I−rB is nonexpansive.

Lemma 2.3([18]). LetCbe a nonempty bounded closed convex subset of a Hilbert spaceH and let{T(s)}_s≥0 be a nonexpansive semigroup on C. Then, for everyh≥0,

t→∞lim sup

x∈C

1 t

Z t

0

T(s)xds−T(h)1 t

Z t

0

T(s)xds

= 0.

Lemma 2.4 ([9]). Let C be a closed convex subset of a real Hilbert space H and let S : C → C be a nonexpansive mapping. Then, the mapping I −S is demiclosed. That is, if {x_n} is a sequence in C such thatxn→x^∗ weakly and (I−S)xn→y strongly, then (I−S)x^∗ =y.

Lemma 2.5 ([20]). Assume {a_n} is a sequence of nonnegative real numbers such that a_n+1 ≤(1−γ_n)a_n+δ_nγ_n,

where {γ_n} is a sequence in (0,1) and{δ_n} is a sequence such that (1) P∞

n=1γ_n=∞;

(2) lim sup_n→∞δ_n≤0 or P∞

n=1|δ_nγ_n|<∞.

Thenlimn→∞an= 0.

3. Main results

In this section we will show our main results.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert spaceH. Let S ={T(s)}s≥0 be a nonexpansive semigroup on C. Let f :C →H be a ρ-contraction (possibly non-self) with ρ ∈[0,1). Let A be a strongly positive linear bounded self-adjoint operator onH with coefficient γ >¯ 0. Let B :C→H be an α-inverse strongly monotone mapping. Let {r_t}0<t<1 be a continuous net of positive real numbers such that rt ∈[a, b]⊂(0,2α). Let {λ_t}0<t<1 be a continuous net of positive real numbers such that limt→0λt= +∞. Let γ and β be two real numbers such that 0 < γ < γ/ρ¯ and β ∈ [0,1). Suppose that the function F : C×C → R satisfies (H1)-(H4) and F ix(S)∩Ω 6= ∅. Let the nets {x_t} and {u_t} be defined by the following implicit scheme:

(F(u_t, y) +hBx_t, y−u_ti+_r¹

thy−u_t, u_t−x_ti ≥0, ∀y∈C, xt=PC[tγf(xt) +βxt+ ((1−β)I−tA)_λ¹

t

Rλt

0 T(s)utds]. (3.1)

Then the nets {x_t} and {u_t} defined by (3.1) strongly converge to x^∗ ∈F ix(S)∩Ω as t→0 and x^∗ is the unique solution of the following variational inequality:

x^∗ ∈F ix(S)∩Ω, h(γf −A)x^∗, x−x^∗i ≥0, ∀x∈F ix(S)∩Ω. (3.2) In particular, if we take f = 0 and A=I, then the nets {x_t} and {u_t} defined by (3.1)reduces to

(F(ut, y) +hBx_t, y−uti+_r¹

thy−ut, ut−xti ≥0, ∀y∈C, xt=P_C[βxt+ (1−β−t)_λ¹

t

R_λt

0 T(s)utds]. (3.3)

In this case, the nets {x_t}and {u_t} defined by (3.3) converge in norm to the minimum norm element x^∗ of F ix(S)∩Ω, namely, the point x^∗ is the unique solution to the minimization problem:

x^∗ = arg min

x∈F ix(S)∩Ωkxk.

Proof. First, we note that the nets {x_t} and {u_t} defined by (3.1) are well-defined. As a matter of fact, from Lemma 2.1, we have ut=Srt(xt−rtBxt). Now we define a mapping

Gx:=P_C

tγf(x) +βx+ ((1−β)I−tA)1 λt

Z λt

0

T(s)S_rt(x−r_tBx)ds

.

Since S_rt and (I−r_tB) are nonexpansive, we have kGx−Gyk ≤

tγ(f(x)−f(y)) +β(x−y) + ((1−β)I −tA) 1

λt

Z λt

0

T(s)[S_rt(x−r_tBx)−S_rt(y−r_tBy)]ds

≤tγkf(x)−f(y)k+βkx−yk +

((1−β)I−tA) 1 λ_t

Z _λt

0

T(s)[S_rt(x−r_tBx)−S_rt(y−r_tBy)]ds

≤tγρkx−yk+βkx−yk+ (1−β−t¯γ)kx−yk

= (1−(¯γ−γρ)t)kx−yk.

This implies that the mappingG is a contraction and so it has a unique fixed point. Hence, the nets {x_t} and {u_t} defined by (3.1) are well-defined.

Let p∈F ix(S)∩Ω. It is clear that p=S_rt(p−r_tBp) for all t∈(0,1). From Lemma 2.2, we have ku_t−pk² =kS_rt(xt−rtBxt)−Srt(p−rtBp)k²

≤ kx_t−rtBxt−(p−rtBp)k²

≤ kx_t−pk²+r_t(r_t−2α)kBx_t−Bpk²

≤ kx_t−pk².

(3.4)

So, we have

ku_t−pk ≤ kx_t−pk.

Then, we obtain kx_t−pk=

P_C

tγf(x_t) +βx_t+ ((1−β)I −tA) 1 λt

Z λt

0

T(s)u_tds

−p

≤

t(γf(xt)−Ap) +β(xt−p) + ((1−β)I−tA) 1

λt

Z λt

0

T(s)utds−p

≤tkγf(x_t)−Apk+βkx_t−pk+ (1−β−γt)¯ 1 λ_t

Z _λt

0

kT(s)u_t−T(s)pkds

≤tγkf(x_t)−f(p)k+tkγf(p)−Apk+βkx_t−pk+ (1−β−γt)ku¯ _t−pk

≤tγρkx_t−pk+tkγf(p)−Apk+βkx_t−pk+ (1−β−¯γt)kx_t−pk.

Hence

kx_t−pk ≤ 1

¯

γ−γρkγf(p)−Apk, which implies that the net{x_t}is bounded and so is the net{u_t}.

SetR:= _γ−γρ¯¹ kγf(p)−Apk. It is clear that{x_t} ⊂B(p, R) and{u_t} ⊂B(p, R). Notice that

1 λ_t

Z λt

0

T(s)u_tds−p

≤ ku_t−pk ≤ kx_t−pk ≤R.

Moreover, we observe that ifx∈B(p, R) then

kT(s)x−pk=kT(s)x−T(s)pk ≤ kx−pk ≤R, i.e., B(p, R) is T(s)-invariant for all s.

Set yt = tγf(xt) +βxt+ ((1−β)I −tA)_λ¹

t

Rλt

0 T(s)utds. It follows that xt = PC[yt]. By using the property of the metric projection (2.1), we have

kx_t−pk²=hx_t−y_t, x_t−pi+hy_t−p, x_t−pi

≤ hy_t−p, x_t−pi

=thγf(xt)−Ap, xt−pi+βkx_t−pk² +

((1−β)I−tA)1 λt

Z λt

0

(T(s)ut−p)ds, xt−p

≤tkγf(xt)−Apkkx_t−pk+βkx_t−pk²+ (1−β−t¯γ)ku_t−pkkx_t−pk.

(3.5)

This implies that

kx_t−pk ≤ t

1−βkγf(xt)−Apk+ku_t−pk.

Hence,

kx_t−pk² ≤ ku_t−pk²+t t

(1−β)²kγf(x_t)−Apk²+ 2 1

1−βkγf(x_t)−Apkku_t−pk

≤ ku_t−pk²+tM

≤ kx_t−pk²+rt(rt−2α)kBx_t−Bpk²+tM,

(3.6)

where

M := sup

0<t<1

t

(1−β)²kγf(x_t)−Apk²+ 2

1−βkγf(x_t)−Apkku_t−pk,2r_tkx_t−u_tk

.

It follows that

rt(2α−rt)kBx_t−Bpk² ≤tM →0.

Since limt→0rt=r∈(0,2α), we derive

t→0limkBx_t−Bpk= 0. (3.7)

From Lemmas 2.1, 2.2 and (3.1), we obtain

ku_t−pk² =kS_rt(x_t−r_tBx_t)−S_rt(p−r_tBp)k²

≤ h(x_t−r_tBx_t)−(p−r_tBp), u_t−pi

= 1

2 k(x_t−rtBxt)−(p−rtBp)k²+ku_t−pk²− k(x_t−p)−rt(Bxt−Bp)−(ut−p)k²

≤ 1

2 kx_t−pk²+ku_t−pk²− k(x_t−ut)−rt(Bxt−Bp)k²

= 1

2 kx_t−pk²+ku_t−pk²− kx_t−u_tk²+ 2r_thx_t−u_t, Bx_t−Bpi −r_t²kBx_t−Bpk² ,

which implies that

ku_t−pk²≤ kx_t−pk²− kx_t−u_tk²+ 2r_thx_t−u_t, Bx_t−Bpi −r²_tkBx_t−Bpk²

≤ kx_t−pk²− kx_t−utk²+ 2rtkx_t−utkkBx_t−Bpk

≤ kx_t−pk²− kx_t−u_tk²+MkBx_t−Bpk.

(3.8)

By (3.6) and (3.8), we have

kx_t−pk² ≤ kx_t−pk²− kx_t−utk²+ (kBx_t−Bpk+t)M.

It follows that

kx_t−u_tk² ≤(kBx_t−Bpk+t)M.

This together with (3.7) implies that

t→0limkx_t−u_tk= 0.

From (3.1), we deduce

kT(τ)x_t−x_tk=kP_C[T(τ)x_t]−P_C[y_t]k

≤ kT(τ)x_t−y_tk

≤

T(τ)x_t−T(τ) 1 λ_t

Z λt

0

T(s)u_tds

+

T(τ)1 λ_t

Z λt

0

T(s)u_tds− 1 λ_t

Z λt

0

T(s)u_tds

+

1 λt

Z λt

0

T(s)utds−yt

≤

xt− 1 λ_t

Z λt

0

T(s)utds

+

T(τ) 1 λ_t

Z λt

0

T(s)utds− 1 λ_t

Z λt

0

T(s)utds

+

1 λt

Z λt

0

T(s)u_tds−y_t .

Note that

y_t− 1 λ_t

Z _λt

0

T(s)u_tds

≤t

γf(x_t)− A λ_t

Z _λt

0

T(s)u_tds

+β

x_t− 1 λ_t

Z _λt

0

T(s)u_tds .

Since

xt− 1 λt

Z λt

0

T(s)utds

≤

tγf(xt) +βxt−(βI+tA) 1 λt

Z λt

0

T(s)utds

≤t

γf(xt)− A λ_t

Z λt

0

T(s)utds

+β

xt− 1 λ_t

Z λt

0

T(s)utds ,

we obtain

x_t− 1 λ_t

Z λt

0

T(s)u_tds

≤ t 1−β

γf(x_t)− A λ_t

Z λt

0

T(s)u_tds .

Therefore,

kT(τ)xt−xtk ≤ 2t 1−β

γf(xt)− A λt

Z λt

0

T(s)utds

+

T(τ) 1 λt

Z λt

0

T(s)utds− 1 λt

Z λt

0

T(s)utds .

From Lemma 2.3, we deduce for all 0≤τ <∞

limt→0kT(τ)xt−xtk= 0. (3.9)

From (3.5), we have

kx_t−pk²≤ hy_t−p, xt−pi

=thγf(x_t)−Ap, x_t−pi+βkx_t−pk² +

((1−β)I−tA) 1

λ_t Z λt

0

T(s)u_tds−p

, x_t−p

≤βkx_t−pk²+ (1−β−¯γt)ku_t−pkkx_t−pk +tγhf(xt)−f(p), xt−pi+thγf(p)−Ap, xt−pi

≤[1−(¯γ−γρ)t]kx_t−pk²+thγf(p)−Ap, x_t−pi.

Therefore,

kx_t−pk² ≤ 1

¯

γ−γρhγf(p)−Ap, xt−pi, ∀p∈F ix(S)∩Ω.

From this inequality, we have immediately thatω_w(x_t) =ω_s(x_t), whereω_w(x_t) andω_s(x_t) denote the set of weak and strong cluster points of{x_t}, respectively.

Let {t_n} ⊂(0,1) be a sequence such that tn → 0 as n→ ∞. Put xn := xtn,un := utn, rn := rtn and λ_n:=λ_tn. Since{x_n}is bounded, without loss of generality, we may assume that{x_n}converges weakly to a pointx^∗ ∈C. Also yn→x^∗ weakly. Noticing (3.9) we can use Lemma 2.4 to get x^∗ ∈F ix(S).

Now we show x^∗ ∈Ω. Sinceun=Srn(xn−rnBxn), for anyy∈C we have F(u_n, y) + 1

r_nhy−u_n, u_n−(x_n−r_nBx_n)i ≥0.

From the monotonicity of F, we have 1

r_nhy−u_n, u_n−(x_n−r_nBx_n)i ≥F(y, u_n), ∀y∈C.

Hence,

y−uni,uni−xni

r_ni +Bxni

≥F(y, uni), ∀y∈C. (3.10) Putz_t=ty+ (1−t)x^∗ for all t∈(0,1] andy∈C. Then, we have z_t∈C. So, from (3.10) we have

hz_t−uni, Bzti ≥ hz_t−uni, Bzti −

zt−uni,uni−xni

rni

+Bxni

+F(zt, uni)

=hz_t−u_ni, Bz_t−Bu_nii+hz_t−u_ni, Bu_ni−Bx_nii

−

z_t−u_ni,u_ni−x_ni r_ni

+F(z_t, u_ni).

(3.11)

Note thatkBu_ni−Bx_nik ≤ _α¹ku_ni−x_nik → 0. Further, from monotonicity ofB, we havehz_t−u_ni, Bz_t− Bunii ≥0. Lettingi→ ∞ in (3.11), we have

hz_t−x^∗, Bz_ti ≥F(z_t, x^∗). (3.12) From (H1), (H4) and (3.12), we also have

0 =F(z_t, z_t)≤tF(z_t, y) + (1−t)F(z_t, x^∗)

≤tF(z_t, y) + (1−t)hz_t−x^∗, Bz_ti

=tF(zt, y) + (1−t)thy−x^∗, Bzti and hence

0≤F(z_t, y) + (1−t)hBz_t, y−x^∗i. (3.13) Lettingt→0 in (3.13), we have, for eachy ∈C,

0≤F(x^∗, y) +hy−x^∗, Bx^∗i.

This implies thatx^∗ ∈Ω. We can rewrite (3.1) as (A−γf)xt=−1

t((1−β)I −tA)

xt− 1 λ_t

Z λt

0

T(s)utds

+1

t(xt−yt).

Therefore,

h(A−γf)x_t, x_t−pi=−1−β t

1 λ_t

Z λt

0

h(I−T(s)S_rt(I −r_tB))x_t−(I−T(s)S_rt(I−r_tB))p, x_t−pids

+ 1 λt

A

Z λt

0

[x_t−T(s)u_t]ds, x_t−p

+1

thx_t−y_t, x_t−pi.

Noting thatI−T(s)S_rt(I−r_tB) is monotone and hx_t−y_t, x_t−pi ≤0, so h(A−γf)xt, xt−pi ≤ 1

λ_t

A Z λt

0

[xt−T(s)ut]ds, xt−p

=

Ax_t− A λ_t

Z λt

0

T(s)u_tds, x_t−p

≤ kAk

xt− 1 λt

Z λt

0

T(s)utds

kx_t−pk

≤ t 1−βkAk

γf(xt)−A1 λ_t

Z λt

0

T(s)utds

kx_t−pk.

Taking the limit throught:=tni →0, we have h(A−γf)x^∗, x^∗−pi= lim

i→∞h(A−γf)x_ni, x_ni−pi ≤0.

Since the solution of the variational inequality (3.2) is unique, henceωw(xt) =ωs(xt) is singleton. Therefore, x_t→x^∗.

In particular, if we take f = 0 and A =I, then it follows that x_t → x^∗ =P_{F ix(S)∩Ω}(0), which implies thatx^∗ is the minimum norm fixed point of T. As a matter of fact, by (3.2), we deduce

hx^∗, x^∗−xi ≤0, ∀x∈F ix(S)∩Ω, that is,

kx^∗k² ≤ hx^∗, xi ≤ kx^∗kkxk, ∀x∈F ix(S)∩Ω.

Therefore, the pointx^∗ is the unique solution to the minimization problem x^∗ = arg min

x∈F ix(S)∩Ωkxk.

This completes the proof.

Next we introduce an explicit algorithm to find a solution of minimization problem (1.1). This scheme is obtained by discretizing the implicit scheme (3.1). We will show the strong convergence of this algorithm.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert spaceH. Let S ={T(s)}_s≥0 be a nonexpansive semigroup on C. Let f :C →H be a ρ-contraction (possibly non-self) with ρ ∈[0,1). Let A be a strongly positive linear bounded self-adjoint operator on H with coefficient γ >¯ 0. Let B : C → H be an α-inverse strongly monotone mapping. Let γ and β be two real numbers such that 0 < γ <¯γ/ρ and β ∈[0,1). Suppose that the function F :C×C → R satisfies (H1)-(H4) and F ix(S)∩Ω 6=∅. Let {x_n} and {u_n} be defined by the following explicit algorithm:

(F(u_n, y) +hBx_n, y−u_ni+_r¹

nhy−u_n, u_n−x_ni ≥0, ∀y∈C,

x_n+1=P_C[α_nγf(x_n) +βx_n+ ((1−β)I−α_nA)_λ¹

n

Rλn

0 T(s)u_nds], n≥0, (3.14) where {α_n} is real number sequence in [0,1] and {λ_n}, {r_n} are two sequences of positive real numbers.

Suppose that the following conditions are satisfied:

(i) limn→∞α_n= 0,P∞

n=1α_n=∞ and P∞

n=0|α_n−αn−1|<∞;

(ii) limn→∞λ_n=∞ and limn→∞ |λ_n−λn−1| λnαn = 0;

(iii) rn∈[a, b]⊂(0,2α) and P∞

n=0|r_n+1−rn|<∞.

Then the sequences {x_n} and {u_n} defined by (3.14) strongly converge to x^∗ ∈ F ix(S)∩Ω and x^∗ is the unique solution of the variational inequality (3.2).

In particular, if we take f = 0 and A =I, then the sequences {x_n} and {u_n} defined by (3.14) reduces

to (

F(un, y) +hBx_n, y−uni+_r¹

nhy−un, un−xni ≥0, ∀y ∈C, xn+1=P_C[βxn+ (1−αn−β)_λ¹

n

R_λn

0 T(s)unds], n≥0. (3.15)

In this case, the sequences{x_n}and{u_n}defined by (3.15)converge in norm to the minimum norm element x^∗ of F ix(S)∩Ω.

Proof. Take p∈F ix(S)∩Ω, we have

kx_n+1−pk ≤α_nkγf(x_n)−Apk+βkx_n−pk+ (1−β−γα¯ _n) 1 λn

Z λn

0

kT(s)u_n−T(s)pkds

≤α_nγρkx_n−pk+α_nkγf(p)−Apk+βkx_n−pk+ (1−β−α_n¯γ)ku_n−pk.

(3.16) From Lemma 2.2, we have

ku_n−pk² =kS_rn(x_n−r_nBx_n)−S_rn(p−r_nBp)k²

≤ kx_n−rnBxn−(p−rnBp)k²

≤ kx_n−pk²+r_n(r_n−2α)kBx_n−Bpk²

≤ kx_n−pk².

(3.17)

So, we have

ku_n−pk ≤ kx_n−pk. (3.18)

By (3.16) and (3.18), we derive

kx_n+1−pk ≤[1−(¯γ−ργ)αn]kx_n−pk+αnkγf(p)−Apk.

Using induction, it follows that

kx_n−pk ≤max

kx₀−pk,kγf(p)−Apk

¯ γ−γρ

.

Sety_n= _λ¹

n

Rλn

0 T(s)u_nds for all n≥0. From (3.14), we get

kx_n+1−x_nk ≤ kα_nγf(x_n) +βx_n+ ((1−β)I−α_nA)y_n

−αn−1γf(xn−1)−βxn−1−((1−β)I−αn−1A)yn−1k

=kγα_n(f(xn)−f(xn−1)) +γ(αn−αn−1)f(xn−1) +β(xn−xn−1) + ((1−β)I−αnA)(yn−yn−1) + (αn−1−αn)Ayn−1k

≤γα_nkf(x_n)−f(xn−1)k+|α_n−αn−1|(kγf(xn−1)k+kAy_n−1k) +βkx_n−xn−1k+ (1−β−α_n¯γ)ky_n−yn−1k,

and

ky_n−yn−1k=

1 λ_n

Z λn

0

[T(s)un−T(s)un−1]ds+ 1

λ_n − 1 λn−1

Z λn−1

0

T(s)un−1ds

+ 1 λn

Z λn

λn−1

T(s)un−1ds

=

1 λ_n

Z λn

0

[T(s)u_n−T(s)un−1]ds+ 1

λ_n − 1 λn−1

Z λn−1

0

[T(s)un−1−T(s)p]ds + 1

λn

Z λn

λn−1

[T(s)un−1−T(s)p]ds

≤ 1 λ_n

Z λn

0

kT(s)u_n−T(s)un−1kds+ 1 λ_n

Z λn

λn−1

[T(s)un−1−T(s)p]ds

+

1 λn

− 1 λn−1

Z λn−1

0

kT(s)un−1−T(s)pkds

≤ ku_n−un−1k+2|λ_n−λn−1| λn

kun−1−pk.

Next, we estimateku_n+1−unk. From (3.14), we have F(u_n, y) +hBx_n, y−u_ni+ 1

r_nhy−u_n, u_n−x_ni ≥0, ∀y∈C, (3.19) and

F(un+1, y) +hBx_n+1, y−un+1i+ 1 rn+1

hy−un+1, un+1−xn+1i ≥0, ∀y∈C. (3.20) Puttingy=un+1 in (3.19) andy=un in (3.20), we have

F(un, un+1) +hBx_n, un+1−uni+ 1 rn

hu_n+1−un, un−xni ≥0, and

F(un+1, un) +hBx_n+1, un−un+1i+ 1 rn+1

hu_n−un+1, un+1−xn+1i ≥0.

From the monotonicity of F, we have

F(u_n, u_n+1) +F(u_n+1, u_n)≤0.

Then,

hBx_n−Bxn+1, un+1−uni+

un+1−un,un−xn

rn

−un+1−xn+1

rn+1

≥0, and hence

un+1−un, un−un+1+un+1−xn− rn

r_n+1(un+1−xn+1)

+rnhBx_n−Bxn+1, un+1−uni ≥0.

It follows that

ku_n+1−u_nk²≤

u_n+1−u_n,r_n+1−r_n

r_n+1 (u_n+1−x_n+1) +x_n+1−x_n

+rnhBx_n−Bxn+1, un+1−uni

=h(I−rnB)xn+1−(I−rnB)xn, un+1−uni +

u_n+1−u_n,r_n+1−r_n rn+1

(u_n+1−x_n+1)

≤ k(I−r_nB)x_n+1−(I−r_nB)x_nkku_n+1−u_nk +

rn+1−rn

r_n+1

ku_n+1−unkku_n+1−xn+1k,

that is,

ku_n+1−unk ≤ k(I−rnB)xn+1−(I−rnB)xnk+

rn+1−rn

rn+1

ku_n+1−xn+1k

≤ kx_n+1−xnk+

rn+1−rn

r_n+1

ku_n+1−xn+1k

≤ kx_n+1−x_nk+

r_n+1−r_n a

ku_n+1−x_n+1k.

Therefore,

ky_n−yn−1k ≤ kx_n−xn−1k+

r_n−rn−1

a

ku_n−x_nk+2|λ_n−λn−1|

λ_n ku_n−1−pk, and hence

kx_n+1−x_nk ≤γα_nρkx_n−xn−1k+|α_n−αn−1|(kγf(xn−1)k+kAy_n−1k) +βkx_n−xn−1k+ (1−β−αnγ)¯

kx_n−xn−1k +

r_n−rn−1

a

ku_n−x_nk+2|λ_n−λn−1|

λ_n ku_n−1−pk

≤[1−(¯γ−γρ)α_n]kx_n−xn−1k+M

|α_n−αn−1| +

r_n−rn−1

a

+|λ_n−λn−1| λn

,

(3.21)

whereM >0 is a constant such that sup

n

{kγf(xn−1)k+kAy_n−1k,2r_nku_n−x_nk,2ku_n−1−pk} ≤M.

From Lemma 2.5 and (3.21), we derive

n→∞lim kx_n+1−xnk= 0.

It follows that

n→∞lim ku_n+1−unk= lim

n→∞ky_n+1−ynk= 0.

Note that

kx_n−y_nk ≤ kx_n+1−x_nk+kx_n+1−y_nk

≤ kx_n+1−x_nk+α_nγkf(x_n)k+βkx_n−y_nk+α_nkAy_nk, that is,

kx_n−y_nk ≤ 1

1−β(kx_n+1−x_nk+α_nγkf(x_n)k+α_nkAy_nk)

→0 (as n→ ∞).

Setvn=αnγf(xn) +βxn+ ((1−β)I−αnA)_λ¹

n

Rλn

0 T(s)unds. It follows that xn+1=PC[vn] for all n≥0.

By using the property of the metric projection (2.1) and (3.14), we have kx_n+1−pk² =hx_n+1−p, xn+1−pi

=hx_n+1−vn, xn+1−pi+hv_n−p, xn+1−pi

≤ hv_n−p, x_n+1−pi

=α_nhγf(x_n)−Ap, x_n+1−pi+βhx_n−p, x_n+1−pi

+h((1−β)I−α_nA)(y_n−p), x_n+1−pi

≤α_nkγf(x_n)−Apkkx_n+1−pk+βkx_n−pkkx_n+1−pk

+ (1−β−¯γαn)ky_n−pkkx_n+1−pk (3.22)

≤αnkγf(xn)−Apkkx_n+1−pk+βkx_n−pkkx_n+1−pk + (1−β)ky_n−pkkx_n+1−pk

≤α_nkγf(x_n)−Apkkx_n+1−pk+β

2(kx_n−pk²+kx_n+1−pk²) +1−β

2 (ky_n−pk²+kx_n+1−pk²), which implies that

kx_n+1−pk² ≤2αnkγf(xn)−Apkkx_n+1−pk+βkx_n−pk²+ (1−β)ky_n−pk²

= 2αnkγf(xn)−Apkkx_n+1−pk+βkx_n−pk²+ (1−β)

1 λn

Z λn

0

[T(s)un−T(s)p]ds

2

≤2αnkγf(xn)−Apkkx_n+1−pk+βkx_n−pk²+ (1−β)ku_n−pk² (3.23)

≤2α_nkγf(x_n)−Apkkx_n+1−pk+βkx_n−pk² + (1−β)(kx_n−pk²+rn(rn−2α)kBx_n−Bpk²).

Hence,

r_n(2α−r_n)(1−β)kBx_n−Bpk² ≤2α_nkγf(x_n)−Apkkx_n+1−pk +kx_n−pk²− kx_n+1−pk²

≤2α_nkγf(x_n)−Apkkx_n+1−pk

+kx_n−x_n+1k(kx_n−pk+kx_n+1−pk).

It follows that

n→∞lim kBx_n−Bpk= 0.

From Lemmas 2.1 and 2.2, we obtain

ku_n−pk² =kS_rn(xn−rnBxn)−Srn(p−rnBp)k²

≤ h(x_n−rnBxn)−(p−rnBp), un−pi

= 1

2 k(x_n−r_nBx_n)−(p−r_nBp)k²+ku_n−pk²

− k(x_n−p)−rn(Bxn−Bp)−(un−p)k²

≤ 1

2 kx_n−pk²+ku_n−pk²− k(x_n−un)−rn(Bxn−Bp)k²

= 1

2 kx_n−pk²+ku_n−pk²− kx_n−u_nk² + 2r_nhx_n−u_n, Bx_n−Bpi −r_n²kBx_n−Bpk²

,

which implies that

ku_n−pk² ≤ kx_n−pk²− kx_n−u_nk²+ 2r_nhx_n−u_n, Bx_n−Bpi −r_n²kBx_n−Bpk²

≤ kx_n−pk²− kx_n−unk²+ 2rnkx_n−unkkBx_n−Bpk

≤ kx_n−pk²− kx_n−u_nk²+MkBx_n−Bpk.

(3.24)

From (3.23) and (3.24), we have

kx_n+1−pk²≤2αnkγf(xn)−Apkkx_n+1−pk+βkx_n−pk²

+ (1−β)(kx_n−pk²− kx_n−u_nk²+MkBx_n−Bpk).

It follows that

(1−β)kx_n−unk² ≤2αnkγf(xn)−Apkkx_n+1−pk+kx_n−pk²

− kx_n+1−pk²+MkBx_n−Bpk

≤2αnkγf(xn)−Apkkx_n+1−pk+kx_n−xn+1k(kx_n−pk +kx_n+1−pk) +MkBx_n−Bpk.

Therefore,

n→∞lim kx_n−u_nk= 0.

Note that{x_n}is a bounded sequence. Let ˜xbe a weak limit of {x_n}. Puttingx^∗=P_{F ix(S)∩Ω}(I−A+γf).

Then there existsR such thatB(x^∗, R) contains{x_n}. Moreover,B(x^∗, R) isT(s)-invariant for everys≥0;

therefore, without loss of generality, we can assume that{T(s)}s≥0is a nonexpansive semigroup onB(x^∗, R).

We notice that, from Theorem 3.1, ˜x∈ωw(yn) =ωs(yn). Then, from Lemma 2.3, it follows that, for every h≥0, limn→∞ky_n−T(h)y_nk= 0 and from the demiclosedness principle, we have ˜x∈F ix(S). By the same argument as that of Theorem 3.1, we can deduce that ˜x∈Ω. Hence, ˜x∈F ix(S)∩Ω. Therefore,

lim sup

n→∞

hγf(x^∗)−Ax^∗, x_n+1−x^∗i= lim

n→∞hγf(x^∗)−Ax^∗,x˜−x^∗i ≤0.

Finally, we prove that xn→x^∗. From (3.22), we have

kx_n+1−x^∗k² ≤α_nγhf(x_n)−f(x^∗), x_n+1−x^∗i+α_nhγf(x^∗)−Ax^∗, x_n+1−x^∗i +βkx_n−x^∗kkx_n+1−x^∗k+ (1−β−γα¯ _n)ky_n−x^∗kkx_n+1−x^∗k

≤α_nργkx_n−x^∗kkx_n+1−x^∗k+α_nhγf(x^∗)−Ax^∗, x_n+1−x^∗i +βkx_n−x^∗kkx_n+1−x^∗k+ (1−β−γα¯ n)kx_n−x^∗kkx_n+1−x^∗k

≤ 1−(¯γ−γρ)α_n

2 (kx_n−x^∗k²+kx_n+1−x^∗k²) +αnhγf(x^∗)−Ax^∗, xn+1−x^∗i,

that is,

kx_n+1−x^∗k² ≤[1−(¯γ−γρ)α_n]kx_n−x^∗k²+ 2α_nhγf(x^∗)−Ax^∗, x_n+1−x^∗i.

Hence, all conditions of Lemma 2.5 are satisfied. Therefore, we immediately deduce thatxn→x^∗. In particular, if we take f = 0 and A=I, then it is clear thatx^∗ =P_{F ix(S)∩Ω}(0) is the unique solution to the minimization problemx^∗ = arg minx∈F ix(S)∩Ωkxk. This completes the proof.

Remark 3.3. We observe that our algorithms presented in this paper include some algorithms in the literature as special cases:

(1) If we take B = 0 and β = 0 and let S = {T(s)}_s≥0 be a nonexpansive semigroup on a real Hilbert space H, then our algorithms (3.1) and (3.14) reduce to the algorithms (1.5) and (1.6) which were considered by Cianciaruso et al. [5].

(2) If we take B = 0, β = 0, γ = 1 and let S = T be a nonexpansive mapping on a real Hilbert space H, then our algorithm (3.14) reduces to the algorithm (1.4) which was considered by Plubtieng and Punpaeng [15].

(3) If we takeA=I, B = 0,β = 0, γ = 1,F = 0, and let S={T(s)}_s≥0 be a nonexpansive semigroup on a real Hilbert spaceH, then our algorithm (3.1) reduces to the following algorithm

x_t=tf(x_t) + (1−t) 1 λt

Z λt

0

T(s)x_tds, which was considered by Plubtieng and Punpaeng [16].