k, and let C be a nonempty closed convex subset of H

OF SEMIGROUPS OF NONEXPANSIVE MAPPINGS AND SOLVING SYSTEMS OF VARIATIONAL INEQUALITIES

HOSSEIN PIRI

Communicated by the former editorial board

In this paper, we introduce a new general iterative process for approximating a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the set of solutions a system of variational inequality for finite family of inverse strongly monotone mappings in a real Hilbert space. Our result extends and improves the result announced by H. Piri and A.H. Badali [13] and some others.

AMS 2010 Subject Classification: 47H09, 47H10, 43A07, 47J25.

Key words: projection, common fixed point, amenable semigroup, iterative pro- cess, strong convergence, system of variational inequality.

1. INTRODUCTION

Let H be a real Hilbert space with inner product h., .i and norm are denoted by k . k, and let C be a nonempty closed convex subset of H. A mapping T:C → C is called nonexpansive if k T x−T y k≤k x−y k, for all x, y ∈ C. By F ix(T), we denote the set of fixed points of T (i.e., F ix(T) = {x∈H:T x=x}). It is well known thatF ix(T) is closed and convex. Recall that a self-mappingf:C→C is a contraction onC if there exists a constant α∈[0,1) such thatkf(x)−f(y)k≤αkx−yk for allx, y∈C.

Let B:C → H be a mapping. The variational inequality problem, de- noted byV I(C, B), is to findx∈C such that

hBx, y−xi ≥0, (1.1)

for ally ∈C. The variational inequality problem has been extensively studied in literature. See, for example, [19, 20] and the references therein.

Definition 1.1.Let B:C→H be a mapping. Then B is called

(i) strongly positive with constantγ > 0, if there is a constantγ >0 such that

hBx, xi ≥γ kxk², ∀x∈C.

MATH. REPORTS16(66),2(2014), 295–317

(ii) η−strongly monotone if there exists a positive constantη such that hBx−By, x−yi ≥ηkx−yk², ∀x, y∈C,

(iii) k−Lipschitzian if there exists a positive constant ksuch that kBx−Byk≤kkx−y k, ∀x, y∈C,

(iv) β−inverse strongly monotone if there exists a positive real numberβ >0 such that

hBx−By, x−yi ≥β kBx−Byk², ∀x, y∈C.

It is obvious that anyβ−inverse strongly monotone mappingBis ¹_β−Lip- schitzian.

In 2001 Yamada [18] introduced the following hybrid iterative method for solving the variational inequality

xn+1=T xn−µαnF(T xn), n∈N, (1.2)

whereF isk−Lipschitzian andη−strongly monotone operator withk >0,η >

0, 0< µ < ^2η_k2, then he proved that if{α_n} satisfying appropriate conditions, then {x_n}generated by (1.2) converges strongly to the unique solution of the variational inequality:

hF x^∗, x−x^∗i ≥0, ∀x∈F ix(T).

In 2006, Marino and Xu [12] consider the following iterative method:

xn+1 = (I −αnA)T xn+αnγf(xn), n≥0.

(1.3)

It is proved that if the sequence{α_n}of parameters satisfies the following conditions:

(C₁) α_n→0, (C₂)

∞

P

n=0

α_n=∞, (C₃) either

∞

P

n=0

|α_n+1−α_n|<∞or lim

n→∞

αn+1

αn = 1.

Then, the sequence{x_n}generated by (1.3) converges strongly, asn→ ∞, to the unique solution of the variational inequality:

h(γf−A)x^∗, x−x^∗i ≤0, ∀x∈F ix(T), which is the optimality condition for minimization problem

x∈F ix(Tmin )

1

2hAx, xi −h(x),

whereh is a potential function for γf i.e.,h⁰(x) =γf(x), for all x∈H. Some people also study the application of the iterative method (1.3) [11, 14].

In 2010, Tian [16] combined the iterative (1.3) with the iterative method (1.2) and considered the iterative methods:

xn+1 = (I−µαnF)T xn+αnγf(xn), n≥0, (1.4)

and he proved that if the sequence{α_n}of parameters satisfies the conditions (C1),(C2) and (C3), then the sequences {x_n} generated by (1.4) converges strongly to the unique solution x^∗ ∈F ix(T) of the variational inequality

h(µF −γf)x^∗, x−x^∗i ≥0, ∀x∈F ix(T).

Motivated and inspired by Atsushiba and Takahashi [2], Ceng and Yao [4], Kim [8], Lau et al. [9], Lauet al. [10], Marino and Xu [12], Tian [16], Xu [17] and Yamada [18], Piri and Badali [13] introduced the following iterative algorithm: Letx0∈C and

y_n=β_nx_n+ (1−β_n)P_C(x_n−δ_nBx_n),

xn+1 =αnγf(xn) + (I−αnµF)Tµnyn, n≥0.

(1.5)

whereP_Cis a metric projection of H onto C, B isβ−inverse strongly monotone, ϕ = {T_t : t ∈ S} is a nonexpansive semigroup on H such that the set F = F ix(ϕ)∩V I(C, B) 6= ∅, X is a subspace of B(S) such that 1 ∈ X and the mapping t → hT_tx, yi is an element of X for each x, y ∈ H, and {µ_n} is a sequence of means on X. They prove that if {µ_n} is left regular and the sequences {α_n}, {β_n} and {δ_n} of parameters satisfy appropriate conditions, then the sequences{x_n} and {y_n} generated by (1.5) converge strongly to the unique solutionx^∗∈ F of the variational inequalities:

h(µF −γf)x^∗, x−x^∗i ≥0, ∀x∈ F, hBx^∗, y−x^∗i ≥0 ∀y∈C,

In this paper, motivated and inspired by Atsushiba and Takahashi [2], Ceng and Yao [4], Kim [8], Lau et al. [9], Lauet al. [10], Marino and Xu [12], Katchang Kumam [7], Tian [16], Xu [17], Yamada [18] and Piri and Badali [13], we introduce the following general iterative algorithm: Let x₀ ∈C and

y_m+1,n=x_n,

yi,n=γi,nPC(I−δi,nAi)yi+1,n

+ (1−γi,n)PC(I−δi+1,nAi+1)yi+1,n, i= 1,2,· · ·, m, x_n+1=α_nγf(T_µnP_C(I−δ_1,nA₁)y_1,n) +β_nx_n

+ ((1−β_n)I−α_nµF)T_µnP_C(I−δ_2,nA₂)y_1,n, (1.6)

where P_C is a metric projection of H onto C, for i = 1,2,· · ·, m+ 1, A_i be inverse strongly monotone, ϕ ={T_t :t ∈ S} is a nonexpansive semigroup on

C such that the set F = T_m+1

i=1 V I(C, A_i)∩F ix(ϕ) 6= ∅, X is a subspace of B(S) such that 1 ∈ X and the mapping t → hT_tx, yi is an element of X for each x, y ∈ H, and {µ_n} is a sequence of means on X. Our purpose in this paper is to introduce this general iterative algorithm for approximating a com- mon element of the set of common fixed point of a semigroup of nonexpansive mappings and the set of solutions of system of variational inequalities for finite family of inverse strongly monotone mapping. We will prove that if {µ_n} is left regular sequence of means and the sequences{α_n},{β_n},{γ_i,n}^m,_i=1,n=1^∞ and {δ_i,n}^m,_i=1,n=1^∞ of parameters satisfy appropriate conditions, then the sequences {x_n} and {y_i,n}^m,_i=1,n=1^∞ generated by (1.6) converge strongly to the unique so- lutionx^∗∈ F of the system of the variational inequalities:

h(µF −γf)x^∗, x−x^∗i ≥0, ∀x∈ F,

hA_ix^∗, y−x^∗i ≥0, ∀y∈C, i= 1,2,· · ·, m+ 1.

2.PRELIMINARIES

This section collects some lemmas which will be used in the proofs of the main results in the next section.

Lemma 2.1 ([6]). For a givenx∈H, y∈C,

y=P_Cx⇔ hy−x, z−yi ≥0, ∀z∈C.

It is well known that P_C is a firmly nonexpansive mapping of H onto C and satisfies

kPCx−PCyk²≤ hP_Cx−PCy, x−yi, ∀x, y∈H (2.1)

Moreover, PC is characterized by the following properties: PCx∈C and for all x∈H, y∈C,

hx−P_Cx, y−P_Cxi ≤0.

(2.2)

It is easy to see that (2.2) is equivalent to the following inequality kx−yk²≥kx−P_Cxk² +ky−P_Cxk².

(2.3)

Using Lemma 2.1, one can see that the variational inequality (1.1) is equivalent to a fixed point problem.

It is easy to see that the following is true:

u∈V I(C, A)⇔u=PC(u−λAu), λ >0.

(2.4)

A set-valued mapping U:H → 2^H is called monotone if for all x, y ∈ H, f ∈ U x and g ∈ U y imply hx −y, f −gi ≥ 0. A monotone mapping U:H → 2^H is maximal if the graph of G(U) of U is not properly contained

in the graph of any other monotone mapping. It is known that a monotone mappingU is maximal if and only if for (x, f)∈H×H,hx−y, f−gi ≥0 for every (y, g)∈G(U) implies that f ∈U x. LetA be a monotone mapping ofC into H and letN_Cx be the normal cone to C atx ∈C, that is, N_Cx ={y ∈ H :hz−x, yi ≤0,∀z∈C}and define

U x=

Ax+NCx, x∈C,

∅ x /∈C.

(2.5)

ThenU is the maximal monotone and 0∈U xif and only ifx∈V I(C, A);

see [15].

Let C be a nonempty subset of a Hilbert space H and T: C → H a mapping. Then T is said to be demiclosed at v∈H if, for any sequence {x_n} inC, the following implication holds:

xn* u∈C and T xn→v imply T u=v, where→ (resp. * ) denotes strong (resp. weak) convergence.

Lemma 2.2 ([1]). Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T:C → H is nonexpansive. Then, the mapping I−T is demiclosed at zero.

Lemma 2.3 ([1]). Let H be a real Hilbert space. Then, for allx, y∈H (i) kx−yk²≤kxk² +2hy, x+yi,

(ii) kx−yk²≥kxk² +2hy, xi.

Lemma 2.4 ([18]). Let {x_n} and {y_n} be bounded sequences in a Banach spaceE and let{α_n}be a sequence in[0,1]with0<lim inf

n→∞ α_n≤lim sup

n→∞

α_n<1.

Suppose xn+1=αnxn+ (1−αn)yn for all integers n≥0 and lim sup

n→∞

(kyn+1−ynk − kxn+1−xnk)≤0.

Then, lim

n→∞kyn−xnk= 0.

Lemma 2.5 ([17]). Let {a_n} be a sequence of nonnegative real numbers such that

an+1≤(1−bn)an+bncn, n≥0,

where {b_n} and {c_n} are sequences of real numbers satisfying the following conditions:

(i) {b_n} ⊂(0,1),

∞

P

n=0

bn=∞, (ii) either lim sup

n→∞ cn≤0 or

∞

P

n=0

|b_ncn|<∞.

Then, lim

n→∞an= 0.

Lemma2.6 ([13]). LetHbe a real Hilbert space andF be ak−Lipschitzian and η−strongly monotone operator with k > 0, η > 0. Let 0 < µ < ^2η_k2 and τ = µ(η − ^µk₂²). Then for t ∈ (0,min{1,¹_τ}), I −tµF is contraction with constant 1−tτ.

LetS be a semigroup and letB(S) be the space of all bounded real valued functions defined on S with supremum norm. For s ∈ S and f ∈ B(S), we define elementsl_sf and r_sf inB(S) by

(l_sf)(t) =f(st), (r_sf)(t) =f(ts), ∀t∈S.

Let X be a subspace ofB(S) containing 1 and let X^∗ be its topological dual. An element µ of X^∗ is said to be a mean on X ifk µk=µ(1) = 1. We often write µt(f(t)) instead of µ(f) for µ ∈ X^∗ and f ∈ X. Let X be left invariant (resp. right invariant), i.e., ls(X) ⊂X (resp. rs(X) ⊂ X) for each s ∈ S. A mean µ on X is said to be left invariant (resp. right invariant) if µ(lsf) = µ(f) (resp. µ(rsf) = µ(f)) for eachs∈ S and f ∈X. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup, see [10]. A net {µ_α} of means on X is said to be strongly left regular if

limα kls∗

µα−µα k= 0, for each s∈S, wherel^∗_s is the adjoint operator ofls.

LetS be a semigroup and letC be a nonempty closed and convex subset of a reflexive Banach spaceE. A familyϕ={T_t :t∈S} of mapping fromC into itself is said to be a nonexpansive semigroup on C if Tt is nonexpansive andT_ts=T_tT_sfor eacht, s∈S. ByF ix(ϕ) we denote the set of common fixed points of ϕ,i.e.

F ix(ϕ) = \

t∈S

{x∈C :Ttx=x}.

Lemma 2.7 ([10]). Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let ϕ = {T_t : t ∈ S} be a nonexpansive semigroup onH such that {T_tx:t∈S}is bounded for somex∈ C, letX be a subspace ofB(S) such that1∈X and the mappingt→ hT_tx, y^∗i is an element of X for eachx∈C and y^∗ ∈E^∗, andµis a mean on X. If we write Tµx instead of R

Ttxdµ(t), then the following hold.

(i) T_µ is nonexpansive mapping from C into C.

(ii) T_µx=x for each x∈F ix(ϕ).

(iii) Tµx∈co{T_tx:t∈S} for each x∈C.

Notation. The open ball of radiusr centered at 0 is denoted byB_r and for a subsetDofH, bycoD, we denote the closed convex hull ofD. For >0 and a mappingT:D→H, we letF(T;D) be the set of−approximate fixed points sets of T,i.e. F(T;D) ={x∈D:kx−T xk≤}.

3. MAIN RESULTS

Theorem 3.1. Let C be a nonempty closed convex subset of real Hilbert space H, F:C → C be a k−Lipschitzian and η−strongly monotone operator withk >0,η >0,f:C→Cbe a contraction with coefficient0< α <1,γ be a number in(0,_α^τ), whereτ =µ(η−^µk₂²)and0< µ < ^2η_k2. Fori= 1,2,· · ·, m+1, letA_i:C→Hbeδ_i−inverse strongly monotone mapping. LetSbe a semigroup and ϕ= {T_t : t ∈S} be a nonexpansive semigroup of C into itself such that F =Tm+1

i=1 V I(C, Ai)∩F ix(ϕ)6=∅. LetX be a left invariant subspace ofB(S) such that 1 ∈ X, and the function t → hT_tx, yi is an element of X for each x∈C andy∈H. Let{µ_n}be a left regular sequence of means onX such that P∞

n=1 kµn+1−µnk<∞. Let {α_n}, {γ_i,n}^m,_i=1,n=1^∞ be sequences in (0,1), {β_n} and {δ_i,n}^m+1,_i=1,n=1^∞ be a sequence in [0,1) satisfy the following conditions:

(B₁) 0 < lim infn→∞γ_i,n ≤ lim sup_n→∞γ_i,n < 1 and {δ_i,n} ⊂ (0,2δ_i) i = 1,2,· · · , m+ 1,

(B₂) limn→∞α_n= 0, P∞

n=1α_n=∞ and limn→∞β_n= 0.

(B3) P∞

n=1 |αn+1−αn|<∞, P∞

n=1 |βn+1−βn|<∞ P∞

n=1 |δi,n+1−δi,n|<∞,P∞

n=1 |γi,n+1−γi,n|<∞ i= 1,2,· · · , m+ 1.

If x0 ∈C and {x_n} and {y_i,n}^m,_i=1,n=1^∞ be generated by the iteration algo- rithm:

ym+1,n=xn,

yi,n=γi,nPC(I−δi,nAi)yi+1,n

+ (1−γi,n)PC(I−δi+1,nAi+1)yi+1,n, i= 1,2,· · · , m, x_n+1=α_nγf(T_µnP_C(I−δ_1,nA₁)y_1,n) +β_nx_n

+ ((1−β_n)I−α_nµF)T_µnP_C(I−δ_2,nA₂)y_1,n,

then, {x_n} and {y_i,n}^m,_i=1,n=1^∞ converge strongly, as n → ∞, to x^∗ ∈ F, which is a unique solution of system of the variational inequalities.

h(µF −γf)x^∗, x−x^∗i ≥0, ∀x∈ F,

hA_ix^∗, y−x^∗i ≥0, ∀y∈C, i= 1,2,· · ·, m+ 1.

Proof. Since {δ_i,n}^m+1,_i=1,n=1^∞ satisfy in condition (B₁) and A_i is δ_i−inverse strongly monotone mapping, for any x, y∈C, we have

k(I−δ_i,nA_i)x−(I−δ_i,nA_i)yk²

=k(x−y)−δi,n(Aix−Aiy)k²

=kx−yk² −2δ_i,nhx−y, Aix−Aiyi+δ_i,n² kAix−Aiyk²

≤kx−yk² −2δ_i,nδ_ikA_ix−A_iyk² +δ_i,n² kA_ix−A_iyk²

=kx−yk² +δ_i,n(δ_i,n−2δ_i)kA_ix−A_iy k²

≤kx−yk² It follows that

k(I−δ_i,nA_i)x−(I −δ_i,nA_i)yk≤kx−y k, i= 1,2,· · · , m+ 1.

(3.1)

Let p ∈ F, in the context of the variational inequality problem the characterization of projection (2.4) implies that p = PC(I −δi,nAi)p, i = 1,2,· · · , m+ 1.Using (3.1), we get

kyi,n−pk²

=kγi,nPC(I −δi,nAi)yi+1,n+ (1−γi,n)PC(I−δi+1,nAi+1)yi+1,n−pk²

≤γ_i,nkP_C(I−δ_i,nA_i)y_i+1,n−pk²+(1−γ_i,n)kP_C(I−δ_i+1,nA_i+1)y_i+1,n−pk²

≤γ_i,nkyi+1,n−pk²+(1−γi,n)kyi+1,n−pk²=kyi+1,n−pk² Which implies that

ky1,n−pk≤ky2,n−pk≤ · · · ≤kym,n−pk≤kym+1,n−pk=kxn−pk (3.2)

We claim that {x_n}is bounded. Let p∈ F, using Lemma 2.6, we have kx_n+1−pk=kα_nγf(T_µnP_C(I−δ_1,nA₁)y_1,n) +β_nx_n

+ ((1−β_n)I−α_nµF)T_µnP_C(I−δ_2,nA₂)y_1,n−pk

=k((1−βn)I−αnµF)TµnPC(I−δ2,nA2)y1,n

−((1−βn)I−αnµF)TµnPC(I−δ2,nA2)pk

+β_nkx_n−pk+α_nkγf(T_µnP_C(I −δ_1,nA₁)y_1,n)−µF pk

= (1−β_n−α_nτ)kT_µnP_C(I−δ_2,nA₂)y_1,n−T_µnP_C(I−δ_2,nA₂)pk +β_nkx_n−pk+α_nkγf(T_µnP_C(I −δ_1,nA₁)y_1,n)

−γf(p)k+kγf(p)−µF pk

≤(1−βn−αnτ)ky1,n−pk+βnkxn−pk+αnγαky1,n−pk +αnkγf(p)−µF pk

≤(1−α_n(τ −γα))kx_n−pk+α_nkγf(p)−µF pk

By induction, kxn−pk≤max

(τ−γα)⁻¹ kγf(p)−µF pk,kx0−pk =M0. Hence,{x_n}is bounded and also{y_i,n}^m,_i=1,n=1^∞ and{f(P_C(I−δ_1,nA₁)y_1,n)}

are bounded. Set D = {y ∈ H :k y −p k≤ M0}. We remark that D is ϕ-invariant bounded closed convex set and {x_n}^∞_n=1,{y_i,n}^m,_i=1,n=1^∞ ,{P_C(I− δ_i,nA_i)y_i+1,n}^m,_i=1,n=1^∞ ⊂D. Now, we claim that

(3.3) lim sup

n→∞ sup

y∈D

kTµny−TtTµnyk= 0, ∀t∈S.

This assertion is proved in [13]. We now show that

n→∞lim kx_n+1−x_nk= 0 (3.4)

From definition of {x_n}, we have kxn+1−xnk

=kα_nγf(T_µnP_C(I−δ_1,nA₁)y_1,n) +β_nx_n + ((1−β_n)I −α_nµF)T_µnP_C(I−δ_2,nA₂)y_1,n

−αn−1γf(T_µn−1P_C(I−δ1,n−1A₁)y1,n−1) +βn−1xn−1

+ ((1−βn−1)I−αn−1µF)Tµn−1PC(I−δ2,n−1A2)y1,n−1 k

=kαnγ[f(TµnPC(I−δ1,nA1)y1,n)−f(Tµn−1PC(I−δ1,n−1A1)y1,n−1)]

+ (α_n−αn−1)γf(T_µn−1P_C(I−δ1,n−1A₁)y1,n−1) +β_n[x_n−xn−1] + (β_n−βn−1)xn−1+ [((1−β_n)I−α_nµF)T_µnP_C(I−δ_2,nA₂)y_1,n

−((1−β_n)I −α_nµF)T_µn−1P_C(I−δ2,n−1A₂)y1,n−1] + [((1−βn)I−αnµF)Tµn−1PC(I−δ2,n−1A2)y1,n−1

−((1−βn−1)I−αn−1µF)Tµn−1PC(I−δ2,n−1A2)y1,n−1]k

≤αnγαkTµnPC(I−δ1,nA1)y1,n−Tµn−1PC(I−δ1,n−1A1)y1,n−1 k +|α_n−αn−1 |kγf(T_µn−1P_C(I−δ1,n−1A₁)y1,n−1)k

+β_nkx_n−xn−1 k+|β_n−βn−1 |kxn−1 k

+ (1−β_n−α_nτ)kTµnP_C(I−δ_2,nA2)y1,n−Tµn−1P_C(I−δ_2,n−1A2)y1,n−1k +|βn−βn−1 |kTµn−1PC(I−δ2,n−1A2)y1,n−1k

+|αn−αn−1 |kµF Tµn−1PC(I−δ2,n−1A2)y1,n−1k From (2.4) for p∈ F, we have

kTµn−1PC(I−δi,n−1Ai)y1,n−1−TµnPC(I−δi,nAi)y1,nk

≤kTµn−1PC(I−δi,n−1Ai)y1,n+1−Tµn−1PC(I−δi,n+1Ai)y1,n k +kT_µn+1P_C(I−δi,n−1A_i)y_1,n−T_µn−1P_C(I−δ_i,nA_i)y_1,n k

+kT_µn−1P_C(I−δ_i,nA_i)y_1,n−T_µnP_C(I−δ_i,nA_i)y_1,nk

≤ky1,n−1−y1,nk+|δi,n−1−δi,n|kAiy1,n k +kµn−1−µnk[ky1,n−pk+kpk]

By taking

u_i,n=P_C(I−δ_i,nA_i)y_i+1,n,

v_i,n=P_C(I−δ_i+1,nA_i+1)y_i+1,n, i= 1,2,· · · , m.

From definition of {y_i,n}^m+1,_i=1,n=1^∞, we have ky_i,n+1−y_i,nk

=kγi,n+1ui,n+1+ (1−γi,n+1)vi,n+1−γi,nui,n−(1−γi,n)vi,nk

=kγi,n+1(ui,n+1−ui,n) + (γi,n+1−γi,n)ui,n+ (1−γi,n+1)(vi,n+1−vi,n) + (γi,n−γi,n+1)vi,nk

=γ_i,n+1 ku_i,n+1−u_i,nk+(1−γ_i,n+1)kv_i,n+1−v_i,nk +|γ_i,n+1−γ_i,n|[ku_i,nk+kv_i,nk]

=γi,n+1 kP_C(I−δi,n+1Ai)yi+1,n+1−P_C(I−δi,n+1Ai)yi+1,n

+PC(I −δi,n+1Ai)yi+1,n−PC(I−δi,nAi)yi+1,nk

+ (1−γ_i,n+1)kPC(I−δ_i+1,n+1Ai+1)yi+1,n+1−P_C(I−δi+1,n+1Ai+1)yi+1,n

+P_C(I −δ_i+1,n+1A_i+1)y_i+1,n−P_C(I−δ_i+1,nA_i+1)y_i+1,nk +|γ_i,n+1−γ_i,n|[ku_i,nk+kv_i,nk]

≤γ_i,n+1 ky_i+1,n+1−y_i+1,n k+γ_i,n+1 |δ_i,n+1−δ_i,n|kA_iy_i+1,nk + (1−γi,n+1)kyi+1,n+1−yi+1,nk

+ (1−γi,n+1)|δi+1,n+1−δi+1,n |kAi+1yi+1,n k +|γ_i,n+1−γ_i,n|[ku_i,nk+kv_i,nk]

=ky_i+1,n+1−y_i+1,nk+γ_i,n+1|δ_i,n+1−δ_i,n+1 |kA_iy_i+1,nk + (1−γ_i,n+1)|δ_i+1,n+1−δ_i+1,n kA_i+1y_i+1,nk

+|γi,n+1−γi,n|[kui,nk+kvi,nk]

which implies that for some approximate constant M_i >0, ky_i,n+1−y_i,n≤ky_i+1,n+1−y_i+1,n k+[|δ_i,n+1−δ_i,n|

+|δ_i+1,n+1−δ_i+1,n |+|γ_i,n+1−γ_i,n|]M_i. From this and (3.2), we get

kyi,n+1−yi,nk

≤kx_n+1−x_nk+

m

X

j=i

[|δ_j,n+1−δ_j,n |

+|δ_j+1,n+1−δ_j+1,n |+|γ_j,n+1−γ_j,n|]M_j. (3.5)

Therefore, kxn+1−xnk

≤(1−βn−αn(τ−γα))[kxn−xn−1 k+

m

X

j=1

[|δj,n−δj,n−1 |

+|δj+1,n−δj+1,n−1|+|γj,n−γj,n−1 |]M_j+|δ2,n−1−δ2,n |kA2y1,nk +|δ1,n−1−δ_1,n |kA₁y_1,n k+kµn−1−2µ_nk(ky_1,n−pk+kpk)]

+|α_n−αn−1 |kγf(T_µn−1P_C(I−δ1,n−1A₁)y1,n−1)k +β_nkx_n−xn−1 k+|β_n−βn−1|kxn−1 k

+|βn−βn−1|kTµn−1PC(I−δ2,n−1A2)y1,n−1 k +|αn−αn−1 |kµF Tµn−1PC(I−δ2,n−1A2)y1,n−1 k

≤(1−αn(τ −γα))kxn−xn−1k+[

m

X

j=1

[|δj,n−δj,n−1 |

+|δj+1,n−δj+1,n−1|+|γj,n−γj,n−1 |]M_j+|δ1,n−1−δ2,n |kA2y1,nk +|δ2,n−1−δ2,n |kA2y1,n k+2kµn−1−µnk(ky1,n−pk+kpk)]

+|α_n−αn−1 |[kγf(T_µn−1P_C(I−δ1,n−1A₁)y1,n−1)k +kµF T_µn−1P_C(I−δ2,n−1A₂)y1,n−1 k]

+|βn−βn−1|[kxn−1 k+kTµn−1PC(I−δ2,n−1A2)y1,n−1k]

(3.6)

Thus, it follows from (3.6), condition (B₃) and Lemma 2.4 that

n→∞lim (kx_n+1−x_nk) = 0.

In this stage, we will show

n→∞lim kxn−Ttxnk= 0, ∀t∈S.

(3.7)

Let t∈S and >0. By [3, Theorem 1.2], there existsδ >0 such that coF_δ(T_t;D) +B_δ⊂F(T_t;D), ∀t∈S.

(3.8)

From condition B₁ and (3.3), there exists N₁∈Nsuch that

α_n≤ δ

2(τ +µk)M₀, β_n≤ δ

4M₀, T_µnP_C(I−δ_2,nA₂)y_1,n∈F_δ(T_t;D), n≥N₁. By Lemma 2.6 and (2.4), forp∈ F we have

α_nkγf(P_C(I−δ_1,nA₁)y_1,n)−µF T_µnP_C(I−δ_2,nA₂)y_1,n k

≤α_n[γ kf(P_C(I −δ_1,nA₁)y_1,n)−f(p)k+kγf(p)−µF pk +kµF p−µF TµnPC(I−δ2,nA2)y1,n k]

≤αn(γαky1,n−pk+kγf(p)−µF pk+µkky1,n−pk)

≤α_n(γαM₀+ (τ −γα)M₀+µkM₀)

≤α_n(τ +µk)M₀ ≤ δ 2, and

β_nkx_n−T_µnP_C(I−δ_2,nA₂)y_1,nk

≤β_n[kx_n−pk+kT_µnP_C(I−δ_2,nA₂)y_1,n−pk]

≤βn[kxn−pk+ky1,n−pk]

≤2βnkxn−pk

≤2β_nM₀ ≤ δ 2. Therefore, we have

xn+1 =TµnPC(I−δ2,nA2)y1,n+αn[γf(PC(I−δ1,nA1)y1,n)

−µF T_µnP_C(I −δ_2,nA₂)y_1,n] +β_n[x_n−T_µnP_C(I−δ_2,nA₂)y_1,n]

∈F_δ(T_t;D) +Bδ 2 +Bδ

2

⊂F(T_t;D), n≥N₁. This shows that

kx_n−T_tx_nk≤, ∀n≥N₁. Since >0 is arbitrary, we get (3.7).

Let Q=PF. Then Q(I−µF +γf) is a contraction of H into itself. In fact, we see that

kQ(I−µF +γf)x−Q(I−µF +γf)yk

≤k(I−µF +γf)x−(I−µF +γf)y k

≤k(I−µF)x−(I−µF)yk+γ kf(x)−f(y)k

≤(1−τ)kx−y k+γαkx−y k

and hence, Q(I−µF +γf) is a contraction due to (1−(τ −γα))∈(0,1).

Therefore, by Banach’s contraction principle, PF(I −µF +γf) has a unique fixed point x^∗. Then using (2.4), x^∗ is the unique solution of the variational inequality:

h(γf −µF)x^∗, x−x^∗i ≤0, ∀x∈ F. (3.9)

We show that

lim sup

n→∞

hγf(x^∗)−µF x^∗, x_n−x^∗i ≤0.

(3.10)

We can choose a subsequence {x_nj} of {x_n}such that lim sup

n→∞

hγf(x^∗)−µF x^∗, xn−x^∗i= lim

j→∞hγf(x^∗)−µF x^∗, xnj−x^∗i.

(3.11)

Because {x_nj} is bounded, therefore {x_nj} has subsequence{x_njk} such that x_njk * z. With no loss of generality, we may assume that x_nj * z. It follows from (3.7) and Lemma 2.2 that z∈F ix(T).

In this stage, we will show that

n→∞lim kui,n−yi+1,n k= 0

n→∞lim kv_i,n−y_i+1,nk= 0, i= 1,2,· · ·, m.

(3.12)

Let p∈ F, from (3.2), Lemma 2.3 and definition of {x_n}, we have kx_n+1−pk²

=kαnγf(T PC(I−δ1,nA1)y1,n) +βnxn

+ ((1−β_n)I−α_nµF)T P_C(I−δ_2,nA₂)y_1,n−pk²

=k[((1−β_n)I−α_nF)T P_C(I−δ_2,nA₂)y_1,n−((1−β_n)I−α_nF)p]

+βn[xn−p] +αn[γf(T PC(I−δ1,nA1)y1,n)−F p]k²

≤k((1−β_n)I −α_nF)T P_C(I−δ_2,nA₂)y_1,n−((1−β_n)I−α_nF)pk² + 2β_nhx_n−p, x_n+1−pi

+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi

≤(1−β_n−α_nτ )² ky_1,n−pk² +2β_nhx_n−p, x_n+1−pi + 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤ky1,n−pk²+2βnhx_n−p, xn+1−pi

+ 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤kyi,n−pk²+2βnhx_n−p, xn+1−pi

+ 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi.

(3.13)

From (2.4), (3.2), (3.13) and Lemma 2.3 , we have (for i= 1,2,· · · , m) kx_n+1−pk²

≤kγi,nPC(I−δi,nAi)yi+1,n+ (1−γi,n)PC(I−δi+1,nAi+1)yi+1,n−pk² + 2βnhx_n−p, xn+1−pi+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi

≤γ_i,nkP_C(I−δ_i,nA_i)y_i+1,n−pk² +(1−γ_i,n)kP_C(I−δ_i+1,nA_i+1)y_i+1,n−pk² + 2βnhx_n−p, xn+1−pi+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi

≤γ_i,nk(y_i+1,n−p)−δ_i,n(A_iy_i+1,n−A_ip)k² +(1−γ_i,n)ky_i+1,n−pk²

+ 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤γi,n[kyi+1,n−pk²+δ_i,n² kAiyi+1,n−A_ipk²−2δ_i,nhA_iyi+1,n−A_ip, yi+1,n−pi]

+ (1−γ_i,n)ky_i+1,n−pk² +2β_nhx_n−p, x_n+1−pi + 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤kyi+1,n−pk² +δ²_i,nkAiyi+1,n−Aipk² −2δ_i,nδikAiyi+1,n−Aipk² + 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤kxn−pk² +δi,n(δi,n−2δi)kAiyi+1,n−Aipk²

+ 2βnhx_n−p, xn+1−pi+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi which implies that

−δi,n(δi,n−2δi)kAiyi+1,n−Aipk²

≤kx_n−pk²− kx_n+1−pk² +2β_nhx_n−p, x_n+1−pi + 2α_nhγf(T P_C(I −δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤[kx_n−pk+kx_n+1−pk]kx_n+1−x_nk + 2βnhx_n−p, xn+1−pi

+ 2αnhγf(T PC(I −δ1,nA1)y1,n)−F p, xn+1−pi (3.14)

Using (3.4), (3.14) and condition B₂, we get

limn→∞ kA_iy_i+1,n−A_ipk= 0, i= 1,2,· · · , m.

(3.15)

Repeating the same argument as above, we conclude that limn→∞ kA_i+1y_i+1,n−A_i+1pk= 0, i= 1,2,· · ·, m.

(3.16)

From (2.1), we have kui,n−pk²

=kP_C(I−δ_i,nA_i)y_i+1,n−P_C(I−δ_i,nA_ip)k²

≤ h(I−δ_i,nA_i)y_i+1,n−(I−δ_i,nA_i)p, u_i,n−pi

= 1

2[k(I−δi,nAi)yi+1,n−(I−δi,nAi)pk² +kui,n−pk²

− k(I−δ_i,nA_i)y_i+1,n−(I−δ_i,nA_i)p−(u_i,n−p)k²]

≤ 1

2[ky_i+1,n−pk²+ku_i,n−pk²

− k(I−βi,nAi)yi+1,n−(I−δi,nAi)p−(ui,n−p)k²]

= 1

2[ky_i+1,n−pk²+ku_i,n−pk² − ky_i+1,n−u_i,nk²

+ 2δ_i,nhy_i+1,n−u_i,n, A_iy_i+1,n−A_ipi −δ_i,n² kA_iy_i+1,n−A_ipk²].

So, we obtain

ku_i,n−pk² ≤ky_i+1,n−pk²− ky_i+1,n−u_i,nk² + 2δ_i,nhy_i+1,n−u_i,n, A_iy_i+1,n−A_ipi

−δ²_i,nkAiyi+1,n−Aipk², i= 1,2,· · · , m.

(3.17)

By using same method as (3.17), we have kv_i,n−pk² ≤ky_i+1,n−pk²− ky_i+1,n−v_i,nk²

+ 2δ_i+1,nhy_i+1,n−v_i,n, A_i+1y_i+1,n−A_i+1pi

−δ²_i+1,n kAi+1yi+1,n−Ai+1pk², i= 1,2,· · ·, m.

(3.18)

From (3.13), (3.17), (3.18) and Lemma 2.3, we have kxn+1−pk²

≤ky_i,n−pk²+2β_nhx_n−p, x_n+1−pi

+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi

≤kγ_i,nP_C(I−δ_i,nA_i)y_i+1,n+ (1−γ_i,n)P_C(I−δ_i+1,nA_i+1)y_i+1,n−pk² + 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I −δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤γi,nkui,n−pk² +(1−γi,n)kvi,n−pk²

+ 2βnhx_n−p, xn+1−pi+ 2αnhγf(T PC(I −δ1,nA1)y1,n)−F p, xn+1−pi

≤γ_i,n[ky_i+1,n−pk²− ky_i+1,n−u_i,nk²+2δ_i,nhy_i+1,n−u_i,n, A_iy_i+1,n−A_ipi

−δ_i,n² kAiyi+1,n−Aipk²] + (1−γi,n)[kyi+1,n−pk² − kyi+1,n−vi,nk² + 2δ_i+1,nhy_i+1,n−v_i,n, A_i+1y_i+1,n−A_i+1pi−δ²_i+1,nkA_i+1y_i+1,n−A_i+1pk²] + 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I −δ_1,nA₁)y_1,n)−F p, x_n+1−pi

=kyi+1,n−pk²+γi,n[− kyi+1,n−u_i,nk²+2δi,nhy_i+1,n−u_i,n, Aiyi+1,n−A_ipi

−δ_i,n² kA_iy_i+1,n−A_ipk²] + (1−γ_i,n)[− ky_i+1,n−v_i,nk²

+ 2δi+1,nhy_i+1,n−v_i,n, Ai+1yi+1,n−A_i+1pi−δ²_i+1,nkAi+1yi+1,n−Ai+1pk²] + 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I −δ_1,nA₁)y_1,n)−F p, x_n+1−pi

≤kx_n−pk² +γ_i,n[− ky_i+1,n−u_i,nk² +2δ_i,nhy_i+1,n−u_i,n, A_iy_i+1,n−A_ipi

−δ_i,n² kAiyi+1,n−Aipk²] + (1−γi,n)[− kyi+1,n−vi,nk²

+ 2δ_i+1,nhy_i+1,n−v_i,n, A_i+1y_i+1,n−A_i+1pi−δ²_i+1,n kA_i+1y_i+1,n−A_i+1pk²] + 2βnhx_n−p, xn+1−pi+ 2αnhγf(T P_C(I −δ1,nA1)y1,n)−F p, xn+1−pi Which implies that

γi,nkyi+1,n−ui,nk²

≤[kx_n−pk+kx_n+1−pk]kx_n+1−x_nk

+γ_i,n[2δ_i,nhy_i+1,n−u_i,n, A_iy_i+1,n−A_ipi

−δ_i,n² kAiyi+1,n−Aipk²] + (1−γi,n)[+2δi+1,nhy_i+1,n−vi,n, A_i+1y_i+1,n−A_i+1pi −δ_i+1,n² kA_i+1y_i+1,n−A_i+1pk²]

+ 2β_nhx_n−p, x_n+1−pi+ 2α_nhγf(T P_C(I−δ_1,nA₁)y_1,n)−F p, x_n+1−pi and

(1−γ_i,n)ky_i+1,n−v_i,nk²

≤[kxn−pk+kxn+1−pk]kxn+1−xnk+γi,n[2δi,nhy_i+1,n−ui,n, A_iy_i+1,n−A_ipi+δ_i,n² kA_iy_i+1,n−A_ipk²] + (1−γ_i,n)[2δ_i+1,nhy_i+1,n

−v_i,n, A_i+1y_i+1,n−A_i+1pi −δ_i+1,n² kA_i+1y_i+1,n−A_i+1pk²]

+ 2βnhx_n−p, xn+1−pi+ 2αnhγf(T PC(I−δ1,nA1)y1,n)−F p, xn+1−pi Therefore, using condition B₂, (3.4), (3.15) and (3.16) we get (3.12).

Simple calculation shows that, kyi,n−xnk≤

m

X

j=i

kyj,n−yj+1,nk, , i= 1,2,· · ·, m.

(3.19)

From (3.12) and (3.19), we get

n→∞lim kyi,n−xnk= 0, i= 1,2,· · · , m.

(3.20)

Now, let us show that fori= 1,2,· · ·, m+1,z∈V I(C, A_i). LetU_i:H → 2^H be a set-valued mapping is defined by

Uix=

Aix+NCx, x∈C,

∅, x /∈C.

where N_Cx is the normal cone to C at x ∈ C. Since A_i is monotone. Thus, U_i is maximal monotone see [15]. Let (x, y) ∈ G(U_i), hence, y−A_ix ∈ N_Cx and sinceui,n=PC(I−δi,nAi)yi+1,n therefore,hx−ui,n, y−Aixi ≥0. On the other hand from (2.3), we have

hx−ui,n, ui,n−(yi+1,n−δi,nAiyi+1,n)i ≥0, that is

hx−ui,n,ui,n−yi+1,n

δ_i,n +Aiyi+1,ni ≥0.

Therefore, we have hx−ui,nj, yi

≥ hx−u_i,nj, A_ixi

≥ hx−u_i,nj, A_ixi − hx−u_i,nj,u_i,nj−y_i+1,nj δi,nj

+A_iy_i+1,nji

=hx−ui,nj, Aix−u_i,nj−y_i+1,nj δi,nj

−Aiyi+1,nji

=hx−u_i,nj, A_ix−A_iu_i,nji+hx−u_i,nj, A_iu_i,nj−A_iy_i+1,nji

− hx−u_i,nj,ui,nj−yi+1,nj

δ_i,nj i

≥ hx−u_i,nj, A_iu_i,nj−A_iy_i+1,nji − hx−u_i,nj,ui,nj−yi+1,nj

δ_i,nj i

≥ hx−u_i,nj, A_iu_i,nj−A_iy_i+1,nji− kx−u_nj kk u_i,nj−y_i+1,nj δi,nj

k. (3.21)

Since xnj * z Ai is _δ¹

i−Lipschitzian, from (3.12), (3.20) and (3.21) we obtain

hx−z, yi ≥0.

Since U_i is maximal monotone, we havez∈U_i⁻¹0, and hence, z∈V I(C, A_i), i= 1,2,· · ·, m.

Repeating the same argument as above for vi,n, we conclude that z∈V I(C, Ai), i= 2,3,· · ·, m+ 1.

Therefore z∈ F and applying (3.9) and (3.11), we have lim sup

n→∞

h(γf −F)x^∗, x_n−x^∗i ≤0.

Finally, we prove that xn → x^∗ as n → ∞. Using (2.4), (3.2), and Lemma 2.3, we have

kxn+1−x^∗k²

=kαnγf(T PC(I−δ1,nA1)y1,n) +βnxn

+ ((1−β_n)I−α_nF)T P_C(I−δ_2,nA₂)y_1,n−x^∗ k²

=k[((1−β_n)I−α_nF)T P_C(I−δ_2,nA₂)y_1,n

−((1−βn)I−αnF)T PC(I−δ3,nA3)x^∗]

+β_n[x_n−x^∗] +α_n[γf(T P_C(I−δ_1,nA₁)y_1,n)−F x^∗]k²

≤k[((1−β_n)I−α_nF)T P_C(I−δ_2,nA₂)y_1,n

−((1−β_n)I−α_nF)T P_C(I−δ_2,nA₂)x^∗]

+βn[xn−x^∗]k²+2αnhγf(T PC(I−δ1,nA1)y1,n)−F x^∗, xn+1−x^∗i

≤[(1−β_n−α_nτ)ky_1,n−x^∗ k+β_nkx_n−x^∗k]² + 2αnhγf(PC(I−δ1,nA1)T yn)−F x^∗, xn+1−x^∗i

≤k[(1−β_n−α_nτ)ky_1,n−x^∗k+β_nkx_n−x^∗k]²