An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems

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Journal of Global OptimizationAn International Journal Dealing withTheoretical and Computational Aspectsof Seeking Global Optima and TheirApplications in Science, Managementand Engineering ISSN 0925-5001 J Glob OptimDOI 10.1007/s10898-012-9927-y

An iterative algorithm for common fixedpoints for nonexpansive semigroups andstrictly pseudo-contractive mappings withoptimization problems

Phayap Katchang & Poom Kumam

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J Glob OptimDOI 10.1007/s10898-012-9927-y

An iterative algorithm for common fixed points fornonexpansive semigroups and strictly pseudo-contractivemappings with optimization problems

Phayap Katchang · Poom Kumam

Received: 29 January 2012 / Accepted: 17 May 2012© Springer Science+Business Media, LLC. 2012

Abstract In this paper, we introduce an iterative algorithm for finding a common elementof the set of solutions of a system of mixed equilibrium problems, the set of solutions of avariational inclusion problems for inverse strongly monotone mappings, the set of commonfixed points for nonexpansive semigroups and the set of common fixed points for an infinitefamily of strictly pseudo-contractive mappings in Hilbert spaces. Furthermore, we prove astrong convergence theorem of the iterative sequence generated by the proposed iterativealgorithm under some suitable conditions which solves some optimization problems. Ourresults extend and improve the recent results of Chang et al. (Appl Math Comput 216:51–60,2010), Hao (Appl Math Comput 217(7):3000–3010, 2010), Jaiboon and Kumam (NonlinearAnal 73:1180–1202, 2010) and many others.

Keywords System of mixed equilibrium problem · Variational inclusion · Optimizationproblems · Nonexpansive mapping · Nonexpansive semigroups · Infinite family of strictlypseudo-contractive mappings · η-strongly convex functions · Metric projection

Mathematics Subject Classification 47H09 · 47H10 · 47H17 · 47J20 · 49J30 · 49J40 ·49M05 · 90C99

The first author was supported by The KMUTT Post-doctoral scholarship at King Mongkut’s University ofTechnology Thonburi (KMUTT). Furthermore, The second author was supported by the Higher EducationResearch Promotion and National Research University Project of Thailand, Office of the Higher EducationCommission (NRU-CSEC No. 55000613)..

P. KatchangDepartment of Mathematics and Statistics, Faculty of Science and Agricultural Technology,Rajamangala University of Technology Lanna Tak, Tak 63000, Thailande-mail: [email protected]

P. Katchang · P. Kumam (B)Department of Mathematics, Faculty of Science, King Mongkut’s University of TechnologyThonburi (KMUTT) Bangmod, Bangkok 10140, Thailande-mail: [email protected]

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1 Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ‖ · ‖. Let C be a nonemptyclosed convex subset of H . Recall that, a mapping T : C → C is nonexpansive if

‖T x − T y‖ ≤ ‖x − y‖, ∀x, y ∈ C.

We denote the set of fixed points of T by F(T ), that is F(T ) = {x ∈ C : x = T x}.A mapping f : C → C is said to be an α-contraction if there exists a coefficient α ∈ (0, 1)

such that

‖ f (x) − f (y)‖ ≤ α‖x − y‖, ∀x, y ∈ C.

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

(1) monotone if

〈Bx − By, x − y〉 ≥ 0, ∀x, y ∈ H ;(2) σ -strongly monotone if there exists a positive real number σ such that

〈Bx − By, x − y〉 ≥ σ‖x − y‖2, ∀x, y ∈ H.

For constant σ > 0, this implies that

‖Bx − By‖ ≥ σ‖x − y‖,that is, B is σ -expansive and when σ = 1, it is expansive;

(3) σ -inverse-strongly monotone if there exists a positive real number σ such that

〈Bx − By, x − y〉 ≥ σ‖Bx − By‖2, ∀x, y ∈ H ;(4) k-strictly pseudo-contractive, if there exists a constant k ∈ [0, 1) such that

‖Bx − By‖2 ≤ ‖x − y‖2 + k‖(I − B)x − (I − B)y‖2, ∀x, y ∈ H.

Let A be a strongly positive linear bounded operator on H if there is a constant γ̄ > 0with the property

〈Ax, x〉 ≥ γ̄ ‖x‖2, ∀x ∈ H. (1.1)

Optimization problem (for short, OP) as the following

minx∈F

μ

2〈Ax, x〉 + 1

2‖x − u‖2 − h(x), (1.2)

where F = ∩∞n=1Cn, C1, C2, . . . are infinitely closed convex subsets of H such that

∩∞n=1Cn �= ∅, u ∈ H, μ ≥ 0 is a real number, A is a strongly positive linear bounded

operator on H and h is a potential function for γ f (i.e., h′(x) = γ f (x) for x ∈ H ). This kindof optimization problem has been studied extensively by many authors, see, for example,[4,9,27,30] when F = ∩∞

n=1Cn and h(x) = 〈x, b〉, where b is a given point in H .A family S = {S(s) : 0 ≤ s < ∞} of mappings of C into itself is called a nonexpansive

semigroup on C if it satisfies the following conditions:

(i) S(0)x = x for all x ∈ C ;(ii) S(s + t) = S(s)S(t) for all s, t ≥ 0;

(iii) ‖S(s)x − S(s)y‖ ≤ ‖x − y‖ for all x, y ∈ C and s ≥ 0;(iv) for all x ∈ C, s �→ S(s)x is continuous.

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We denote by F(S) the set of all common fixed points of S = {S(s) : s ≥ 0}, i.e., F(S) =∩s≥0 F(S(s)). It is known that F(S) is closed and convex.

Let φ : C → R be a real-valued function and let {�k : C × C → R, k = 1, 2, . . . , N }be a finite family of equilibrium functions, i.e., �k(u, u) = 0 for each u ∈ C. The system ofmixed equilibrium problems (for short, SMEP) for function (�1,�2, . . . , �N , φ) which isto find z ∈ C such that

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

�1(z, y) + φ(y) − φ(z) ≥ 0, ∀y ∈ C,

�2(z, y) + φ(y) − φ(z) ≥ 0, ∀y ∈ C,

...

�N (z, y) + φ(y) − φ(z) ≥ 0, ∀y ∈ C.

(1.3)

The set of solutions of (1.3) is denoted by ∩Nk=1 M E P(�k, φ), where M E P(�k, φ) is the

set of solutions of the mixed equilibrium problem (for short, MEP), which is to find z ∈ Csuch that

�k(z, y) + φ(y) − φ

(z) ≥ 0, ∀y ∈ C. (1.4)

In particular, if φ ≡ 0, and N = 1, then the problem (1.3) reduces to the equilibrium problem(for short, EP), which is to find z ∈ C such that

�(z, y) ≥ 0, ∀y ∈ C. (1.5)

It is well-known that the SMEP includes fixed point problem, optimization problem, varia-tional inequality problem, and Nash equilibrium problem as its special cases (see [1,10,17,23,25] for more details).

For solving the solutions of a nonexpansive semigroup and the solutions the system ofmixed equilibrium problems, Chang et al. [7] studied the following approximation method;⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

�1(u(1)

n , x) + φ(x) − φ

(u(1)

n) + 1

r1

⟨K ′(u(1)

n) − K ′(xn), η

(x, u(1)

n)⟩ ≥ 0, ∀x ∈ C,

�2(u(2)

n , x) + φ(x) − φ

(u(2)

n) + 1

r2

⟨K ′(u(2)

n) − K ′(xn), η

(x, u(2)

n)⟩ ≥ 0, ∀x ∈ C,

...

�N(u(N )

n , x) + φ(x) − φ

(u(N )

n) + 1

rN

⟨K ′(u(N )

n) − K ′(xn), η

(x, u(N )

n)⟩ ≥ 0, ∀x ∈ C,

xn+1 = αn f (Wn xn) + βn xn + γn1tn

∫ tn0 S(s)Wnu(N )

n ds,

(1.6)

where⎧⎪⎨

⎪⎩

u(1)n = J�1

r1 xn,

u(k)n = J�k

rk u(k−1)n = J�k

rk J�k−1rk−1 u(k−2)

n = J�krk · · · J�2

r2 u(1)n ,

= J�krk · · · J�2

r2 J�1r1 xn, k = 2, 3, . . . , N ,

(1.7)

J�krk : C → C, k = 1, 2, . . . , N is the mapping defined by (2.13) below, Wn is the mapping

defined by (2.11) and S = {S(s) : 0 ≤ s < ∞} is a nonexpansive semigroup. They provedthat, {xn} converges strongly to a fixed point of F(S)∩ F(W )∩ (∩N

k=1 M E P(�k, φ))

undercontrol conditions on the parameters.

Let B : H → H be a single-valued nonlinear mapping and M : H → 2H be a set-valuedmapping. We consider the following variational inclusion problem, which is to find a pointu ∈ H such that

θ ∈ B(u) + M(u), (1.8)

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where θ is the zero vector in H. The set of solutions of problem (1.8) is denoted by I (B, M).If M = ∂δC , where C is a nonempty closed convex subset of H and δC : H → [0,∞] is theindicator function of C , i.e.,

δC (x) ={

0, x ∈ C,

+ ∞, x /∈ C.

Then the variational inclusion problem (1.8) is equivalent to find u ∈ C such that

〈Bu, v − u〉 ≥ 0, ∀v ∈ H. (1.9)

This problem is called Hartman-Stampacchia variational problem ([3,13,15]).A set-valued mapping M : H → 2H is called monotone if for all x, y ∈ H, f ∈ Mxand g ∈ My imply 〈x − y, f − g〉 ≥ 0. A monotone mapping M : H → 2H is maximalif the graph of G(M) of M is not properly contained in the graph of any other monotonemapping. It is known that a monotone mapping M is maximal if and only if for (x, f ) ∈H × H, 〈x − y, f − g〉 ≥ 0 for every (y, g) ∈ G(M) implies f ∈ Mx .Let the set-valued mapping M : H → 2H be a maximal monotone. We define the resolventoperator JM,λ associate with M and λ as follows:

JM,λ(u) = (I + λM)−1(u), u ∈ H, (1.10)

where λ is a positive number. It is worth mentioning that the resolvent operator JM,λ issingle-valued, nonexpansive and 1-inverse strongly monotone ([5,19,20,31]).

Inspired and motivated by Chang et al. [7], Hao [12] and Jaiboon and Kumam [14], thepurpose of this paper is to introduce an iterative algorithm for finding a common element ofthe set of solutions of (1.3), the set of solutions of (1.8) for inverse strongly monotone map-pings, the set of common fixed points for nonexpansive semigroups and the set of commonfixed points for an infinite family of strictly pseudo-contractive mappings. Consequently, weprove the strong convergence theorem in a real Hilbert space under control conditions onthe parameters. Furthermore, we can apply our results for solving some optimization prob-lems. Our results extend and improve the corresponding results in Chang et al. [7], Hao [12],Jaiboon and Kumam [14] and many others.

2 Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H . We denotestrong convergence (weak convergence) by notation → (⇀). In a real Hilbert space H , it iswell known that

‖x − y‖2 = ‖x‖2 − ‖y‖2 − 2〈x − y, y〉, (2.1)

‖x + y‖2 ≤ ‖x‖2 + 2〈y, x + y〉, (2.2)

‖x + y‖2 ≥ ‖x‖2 + 2〈y, x〉, (2.3)

and

‖λx + (1 − λ)y‖2 = λ‖x‖2 + (1 − λ)‖y‖2 − λ(1 − λ)‖x − y‖2 (2.4)

for all x, y ∈ H and λ ∈ R.

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Recall that for every point x ∈ H , there exists a unique nearest point in C , denoted byPC x , such that

‖x − PC x‖ ≤ ‖x − y‖, ∀y ∈ C.

PC is called the metric projection of H onto C. It is well known that PC is a nonexpansivemapping of H onto C and satisfies

〈x − y, PC x − PC y〉 ≥ ‖PC x − PC y‖2, ∀x, y ∈ H. (2.5)

Obviously, this immediately implies that

‖(x − y) − (PC x − PC y)‖2 ≤ ‖x − y‖2 − ‖PC x − PC y‖2, ∀x, y ∈ H. (2.6)

Moreover, PC x is characterized by the following properties: PC x ∈ C and

〈x − PC x, y − PC x〉 ≤ 0, (2.7)

‖x − y‖2 ≥ ‖x − PC x‖2 + ‖y − PC x‖2 (2.8)

for all x ∈ H, y ∈ C .In order to prove our main results, we need the following Lemmas.

Lemma 2.1 [32].Let V : C → H be a k-strict pseudo-contraction, then

(1) the fixed point set F(V ) of V is closed convex so that the projection PF(V ) is welldefined;

(2) define a mapping T : C → H by

T x = t x + (1 − t)V x, ∀x ∈ C. (2.9)

If t ∈ [k, 1), then T is a nonexpansive mapping such that F(V ) = F(T ).

A family of mappings {Vi : C → H}∞i=1 is called a family of uniformly k-strict pseudo-contractions, if there exists a constant k ∈ [0, 1) such that

‖Vi x − Vi y‖2 ≤ ‖x − y‖2 + k‖(I − Vi )x − (I − Vi )y‖2, ∀x, y ∈ C, ∀i ≥ 1.

Let {Vi : H → H}∞i=1 be a countable family of uniformly k-strict pseudo-contractions.Let {Ti : H → H}∞i=1 be the sequence of nonexpansive mappings defined by (2.9), i.e.,

Ti x = t x + (1 − t)Vi x, ∀x ∈ H, ∀i ≥ 1, t ∈ [k, 1). (2.10)

Let {Ti } be a sequence of nonexpansive mappings of H into itself defined by (2.10) andlet {μi } be a sequence of nonnegative numbers in [0,1]. For each n ≥ 1, define a mappingWn of H into itself as follows:

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Un,n+1 = I,

Un,n = μnTnUn,n+1 + (1 − μn)I,

Un,n−1 = μn−1Tn−1Un,n + (1 − μn−1)I,

... (2.11)

Un,k = μk TkUn,k+1 + (1 − μk)I,

Un,k−1 = μk−1Tk−1Un,k + (1 − μk−1)I,

...

Un,2 = μ2T2Un,3 + (1 − μ2)I,

Wn = Un,1 = μ1T1Un,2 + (1 − μ1)I.

Such a mapping Wn is nonexpansive from H into itself and it is called the W -mappinggenerated by T1, T2, . . . , Tn and μ1, μ2, . . . , μn .

For each n, k ∈ N, let the mapping Un,k be defined by (2.11). Then we can have thefollowing crucial conclusions concerning Wn . You can find them in [22]. Now we only needthe following similar version in Hilbert spaces.

Lemma 2.2 [22]. Let C be a nonempty closed convex subset of a real Hilbert space H.Let T1, T2, . . . be nonexpansive mappings of C into itself such that ∩∞

n=1 F(Tn) is nonempty,let μ1, μ2, . . . be real numbers such that 0 ≤ μn ≤ b < 1 for every n ≥ 1. Then,

(1) Wn is nonexpansive and F(Wn) = ∩ni=1 F(Ti ), ∀n ≥ 1;

(2) for every x ∈ C and k ∈ N, the limit limn→∞ Un,k x exists;(3) a mapping W : C → C defined by

W x := limn→∞ Wn x = lim

n→∞ Un,1x, ∀x ∈ C (2.12)

is a nonexpansive mapping satisfying F(W ) = ∩∞i=1 F(Ti ) and it is called the W -map-

ping generated by T1, T2, . . . and μ1, μ2, . . ..

Lemma 2.3 [6]. Let C be a nonempty closed convex subset of a Hilbert space H, {Ti : C →C}be a countable family of nonexpansive mappings with ∩∞

i=1 F(Ti ) �= ∅, {μi } be a real se-quence such that 0 < μi ≤ b < 1, ∀i ≥ 1. If D is any bounded subset of C, then

limn→∞ sup

x∈D‖W x − Wn x‖ = 0.

Lemma 2.4 [18]. Each Hilbert space H satisfies Opial’s condition, i.e., for any sequence{xn} ⊂ H with xn ⇀ x, the inequality

lim infn→∞ ‖xn − x‖ < lim inf

n→∞ ‖xn − y‖,hold for each y ∈ H with y �= x.

Lemma 2.5 [16]. Assume A be a strongly positive linear bounded operator on H with coef-ficient γ̄ > 0 and 0 < ρ ≤ ‖A‖−1. Then ‖I − ρ A‖ ≤ 1 − ργ̄ .

For solving the system of mixed equilibrium problems (1.3), let us assume that function�k : C × C → R, k = 1, 2, . . . , N satisfies the following conditions:

(H1) �k is monotone, i.e., �k(x, y) + �k(y, x) ≤ 0, ∀x, y ∈ C ;

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(H2) for each fixed y ∈ C, x �→ �k(x, y) is convex and upper semicontinuous;(H3) for each x ∈ C, y �→ �k(x, y) is convex.

Let η : C × C → H and B : C → H be two mappings. B is said to be:

(1) monotone if

〈Bx − By, η(x, y)〉 ≥ 0, ∀x, y ∈ C;(2) σ -strongly monotone if there exists a positive real number σ such that

〈Bx − By, η(x, y)〉 ≥ σ‖x − y‖2, ∀x, y ∈ C;(3) L-Lipschitz continuous if there exists a constant L > 0 such that

‖η(x, y)‖ ≤ L‖x − y‖, ∀x, y ∈ C;Let K : C → R be a differentiable functional on a convex set C , which is called:

(1) η-convex [11] if

K (y) − K (x) ≥⟨K ′(x), η(y, x)

⟩, ∀x, y ∈ C,

where K ′(x) is the Fréchet derivative of K at x ;(2) η-strongly convex [2] if there exists a constant ξ > 0 such that

K (y) − K (x) −⟨K ′(x), η(y, x)

⟩≥ ξ

2‖x − y‖2, ∀x, y ∈ C.

In particular, if η(x, y) = x − y for all x, y ∈ C , then K is said to be strongly convex.

Lemma 2.6 [8]. Let C be a nonempty closed convex subset of a real Hilbert space H andlet φ be a lower semicontinuous and convex functional from C to R. Let � be a bifunctionfrom C × C to R satisfying (H1)–(H3). Assume that

(i) η : C × C → H is λ-Lipschitz continuous with constant λ > 0 such that;

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ C,(b) η(·, ·) is affine in the first variable,(c) for each fixed x ∈ C, y �→ η(x, y) is sequentially continuous from the weak

topology to the weak topology;

(ii) K : C → R is η-strongly convex with constant σ > 0 and its derivativeK ′ is sequentially continuous from the weak topology to the strong topology;

(iii) for each x ∈ C, there exist a bounded subset Ex ⊂ C and zx ∈ C such that for anyy ∈ C\Ex ,

�(y, zx ) + φ(zx ) − φ(y) + 1

r

⟨K ′(y) − K ′(x), η(zx , y)

⟩< 0.

For given r > 0, let J�r : C → C be the mapping defined by

J�r (x) =

{

y ∈ C : �(y, z) + φ(z) − φ(y) + 1

r

⟨K ′(y) − K ′(x), η(z, y)

⟩≥ 0, ∀z ∈ C

}

(2.13)

forall x ∈ C. Then

(1) J�r is single-valued,

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(2) F(J�r ) = M E P(�, φ), where M E P(�, φ) is the set of soultion of the mixed

equilibrium problem,

�(x, y) + φ(y) − φ(x) ≥ 0, ∀y ∈ C.

(3) M E P(�, φ) is closed and convex.

Lemma 2.7 [24]. Let {xn} and {vn} be bounded sequences in a Banach space X and let {βn}be a sequence in [0, 1] with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn+1 =(1−βn)vn +βn xn for all integers n ≥ 0 and lim supn→∞(‖vn+1 −vn‖−‖xn+1 − xn‖) ≤ 0.

Then, limn→∞ ‖vn − xn‖ = 0.

Lemma 2.8 [28]. Assume {xn} is a sequence of nonnegative real numbers such that

xn+1 ≤ (1 − an)xn + bn, ∀n ≥ 0,

where {an} is a sequence in (0, 1) and {bn} is a sequence in R such that

(1)∑∞

n=1 an = ∞(2) lim supn→∞ bn

an≤ 0 or

∑∞n=1 |bn | < ∞.

Then limn→∞ xn = 0.

Lemma 2.9 [29]. Let C be a nonempty closed convex subset of a real Hilbert space H, andg : C → R ∪ {∞} be a proper lower-semicontinuous differentiable convex function. If z isa solution to the minimization problem

g(z) = infx∈C

g(x),

then⟨g′(x), x − z

⟩≥ 0, x ∈ C.

In particular, if z solves problem O P, then⟨u + [

γ f − (I + μA)]z, x − z

⟩≤ 0.

Lemma 2.10 [21]. Let C be a nonempty bounded closed convex subset of a Hilbert spaceH and let S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C, then for any h ≥ 0,

limt−→∞ sup

x∈C

∥∥∥∥

1

t

t∫

0

T (s)xds − T (h)(1

t

t∫

0

T (s)xds)

∥∥∥∥ = 0.

Lemma 2.11 [26]. Let C be a nonempty bounded closed convex subset of H, {xn} be asequence in C and S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C. If thefollowing conditions are satisfied:

(i) xn ⇀ z;(ii) lim sups−→∞ lim supn−→∞ ‖S(s)xn − xn‖ = 0, then z ∈ S.

Lemma 2.12 [5]. M : H → 2H be a maximal monotone mapping and B : H → H bea Lipschitz continuous mapping. Then the mapping S = M + B : H → 2H is a maximalmonotone mapping.

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Remark 2.13 From Lemma 2.12 implies that I (B, M) is closed and convex if M : H → 2H

is a maximal monotone mapping and B : H → H is a Lipschitz continuous mapping.

Lemma 2.14 [31]. u ∈ H is a solution of variational inclusion (1.8) if and only ifu = JM,λ(u − λBu), ∀λ > 0, i.e.,

I (B, M) = F(JM,λ(I − λB)), ∀λ > 0.

3 Main results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) tofind the solutions of finding a common element of the set of solutions of (1.3), the set ofsolutions of (1.8) for inverse strongly monotone mappings, the set of common fixed pointsfor nonexpansive semigroups and the set of common fixed points for an infinite family ofstrictly pseudo-contractive mappings in a real Hilbert space.

Theorem 3.1 Let H be a real Hilbert space, f be a contraction of H into itself withα ∈ (0, 1)and A be a strongly positive linear bounded operator on H with coefficient γ̄ > 0 and0 < γ <

(1+μ)γ̄α

. Let φ be a lower semicontinuous and convex functional from H to R

and let {�k : H × H → R, k = 1, 2, . . . , N } be a finite family of equilibrium functionssatisfying conditions (H1)–(H3). Let S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semi-group on H and let {tn} be a positive real divergent sequence such that limn→∞ tn

tn+1= 1.

Let {Vi : H → H}∞i=1 be a countable family of uniformly k-strict pseudo-contractions,{Ti : H → H}∞i=1 be the countable family of nonexpansive mappings defined by Ti x =t x+(1−t)Vi x,∀x ∈ H,∀i ≥ 1, t ∈ [k, 1), Wn be the W -mapping defined by (2.11) and W bea mapping defined by (2.12) with F(W ) �= ∅. Let M1, M2 : H → 2H be maximal monotonemappings and B1, B2 : H → H beσ1, σ2-inverse-strongly monotone mappings, respectively.Suppose that � := F(S) ∩ F(W ) ∩ ( ∩N

k=1 M E P(�k, φ)) ∩ I (B1, M1) ∩ I (B2, M2) �= ∅.

Let μ > 0, γ > 0 and rk > 0, k = 1, 2, . . . , N, which are constants. For given x1 ∈ Harbitrarily and fixed u ∈ H, suppose the {xn}, {yn}, {zn} and

{u(k)

n}, k = 1, 2, . . . , N be the

sequences generated iteratively by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�1(u(1)

n , x) + φ(x) − φ

(u(1)

n) + 1

r1

⟨K ′(u(1)

n) − K ′(xn), η

(x, u(1)

n)⟩ ≥ 0, ∀x ∈ H,

�2(u(2)

n , x) + φ(x) − φ

(u(2)

n) + 1

r2

⟨K ′(u(2)

n) − K ′(xn), η

(x, u(2)

n)⟩ ≥ 0, ∀x ∈ H,

.

.

.

�N(u(N )

n , x) + φ(x) − φ

(u(N )

n) + 1

rN

⟨K ′(u(N )

n) − K ′(xn), η

(x, u(N )

n)⟩ ≥ 0, ∀x ∈ H,

zn = JM2,δ

(u(N )

n − δB2u(N )n

),

yn = JM1,τ (zn − τ B1zn),

xn+1 = αn[u + γ f (Wn xn)

]+βn xn + [(1 − βn)I − αn(I + μA)

] 1tn

∫ tn0 S(s)Wn ynds,

(3.1)

where⎧⎪⎨

⎪⎩

u(1)n = J�1

r1 xn,

u(k)n = J�k

rk u(k−1)n = J�k

rk J�k−1rk−1 u(k−2)

n = J�krk · · · J�2

r2 u(1)n ,

= J�krk · · · J�2

r2 J�1r1 xn, k = 2, 3, . . . , N ,

(3.2)

J�krk : H → H, k = 1, 2, . . . , N is the mapping defined by (2.13), {αn} and {βn} are two

sequences in (0, 1) for all n ∈ N, τ ∈ (0, 2σ1) and δ ∈ (0, 2σ2). Assume the followingconditions are satisfied:

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(C1) η : H × H → H is λ-Lipschitz continuous with constant λ > 0 such that

(a) η(x, y) + η(y, x) = 0, ∀x, y ∈ H,

(b) x �→ η(x, y) is affine,(c) for each fixed y ∈ H, y �→ η(x, y) is sequentially continuous from the weak

topology to the weak topology;

(C2) K : H → R is η-strongly convex with constant ξ > 0 and its derivative K ′ is notonly sequentially continuous from the weak topology to the strong topology but alsoLipschitz continuous with a Lipschitz constant ν > 0 such that ξ > λν;

(C3) For each k ∈ {1, 2, . . . , N } and for all x ∈ C, there exist a bounded subset Ex ⊂ Hand zx ∈ H such that for any y ∈ H\Ex ,

�k(y, zx ) + φ(zx ) − φ(y) + 1

rk

⟨K ′(y) − K ′(x), η(zx , y)

⟩< 0;

(C4) limn→∞ αn = 0 and∑∞

n=1 αn = ∞;(C5) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1.

Then, {xn} converges strongly to x∗ ∈ �, which solves the following optimization problem:

minx∗∈�

μ

2〈Ax∗, x∗〉 + 1

2‖x∗ − u‖2 − h(x∗). (3.3)

Proof By the condition (C4) and (C5), we may assume, without loss of generality, thatαn ≤ (1 − βn)(1 + μ‖A‖)−1 for all n ∈ N. First, we show that I − τ B1 and I − δB2 arenonexpansive. Indeed, for all x, y ∈ H and τ ∈ (0, 2σ1), we note that

‖(I − τ B1)u − (I − τ B1)v‖2 = ‖(u − v) − τ(B1u − B1v)‖2

= ‖u − v‖2 − 2τ 〈u − v, B1u − B1v〉 + τ 2‖B1u − B1v‖2

≤ ‖u − v‖2 + τ(τ − 2σ1)‖B1u − B1v‖2

≤ ‖u − v‖2, (3.4)

which implies that the mapping I − τ B1 is nonexpansive. So is I − δB2.Since A is a strongly positive bounded linear operator on H, we have

‖A‖ = sup{|〈Ax, x〉| : x ∈ H, ‖x‖ = 1}.Observe that

⟨((1 − βn)I − αn(I + μA)

)x, x

⟩= 1 − βn − αn − αnμ〈Ax, x〉≥ 1 − βn − αn − αnμ‖A‖≥ 0,

so this shows that (1 − βn)I − αn(I + μA) is positive. It follows that

‖(1 − βn)I − αn(I + μA)‖ = sup

{∣∣∣

⟨((1 − βn)I − αn(I + μA)

)x, x

⟩∣∣∣ : x ∈ H, ‖x‖ = 1

}

= sup

{

1 − βn − αn − αnμ〈Ax, x〉 : x ∈ H, ‖x‖ = 1

}

≤ 1 − βn − αn − αnμγ̄ .

We shall divide the proofs into several steps.

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Step 1 We show that {xn} is bounded.

Let x∗ ∈ � := F(S) ∩ F(W ) ∩ ( ∩Nk=1 M E P(�k, φ)

) ∩ I (B1, M1) ∩ I (B2, M2).

In fact, by the assumption that for each k ∈ {1, 2, . . . , N }, J�krk is nonexpansive. Let AN :=

J�NrN · · · J�2

r2 J�1r1 and A0 = I . Then, we have x∗ = AN x∗ and u(N )

n = AN xn . Sincex∗ ∈ I (B1, M1) and x∗ ∈ I (B2, M2), then x∗ = JM1,τ (x∗ − τ B1x∗) = JM2,δ(x∗ − δB2x∗).Since x∗ = S(s)x∗,∀s ≥ 0 and x∗ = Wn x∗,∀n ≥ 1. Therefore, we have

x∗ = AN x∗ = JM1,τ (x∗ − τ B1x∗)= Wn JM1,τ (x∗ − τ B1x∗)= S(s)Wn JM1,τ (x∗ − τ B1x∗).

Because I − τ B1, I − δB2, JM1,τ , JM2,δ and AN are nonexpansive mappings, we have

‖yn − x∗‖ = ‖JM1,τ (zn − τ B1zn) − JM1,τ (x∗ − τ B1x∗)‖≤ ‖(I − τ B1)zn − (I − τ B1)x∗‖≤ ‖zn − x∗‖= ‖JM2,δ

(u(N )

n − δB2u(N )n

)− JM2,δ(x∗ − δB2x∗)‖

≤ ‖(I − δB2)u(N )n − (I − δB2)x∗‖

≤ ‖u(N )n − x∗‖

= ‖AN xn − AN x∗‖≤ ‖xn − x∗‖ (3.5)

which yields that

‖xn+1 − x∗‖ = ‖αnu + αn(γ f (Wn xn) − (I + μA)x∗) + βn(xn − x∗)

+ ((1 − βn)I − αn(I + μA)

)( 1

tn

tn∫

0

S(s)Wn ynds − x∗)‖

≤ αn‖u‖ + αn‖γ f (Wn xn) − (I + μA)x∗‖ + βn‖xn − x∗‖+ (

1 − βn − αn(1 + μγ̄ ))‖yn − x∗‖

≤ αn‖u‖ + αn‖γ f (Wn xn) − γ f (x∗)‖ + αn‖γ f (x∗) − (I + μA)x∗‖ + βn‖xn − x∗‖+ (

1 − βn − αn(1 + μγ̄ ))‖xn − x∗‖

≤ αn‖u‖ + αnγα‖xn − x∗‖ + αn‖γ f (x∗) − (I + μA)x∗‖ + βn‖xn − x∗‖+ (

1 − βn − αn(1 + μγ̄ ))‖xn − x∗‖

= αn(‖u‖ + ‖γ f (x∗) − (I + μA)x∗‖) +

(1 − αn(1 + μγ̄ ) + αnγα

)‖xn − x∗‖

=(

1 − αn((1 + μγ̄ ) − γα

))‖xn − x∗‖

+ αn((1 + μγ̄ ) − γα

)‖u‖ + ‖γ f (x∗) − (I + μA)x∗‖(1 + μγ̄ ) − γα

. (3.6)

It follows from (3.6) and induction that

‖xn − x∗‖ ≤ max

{

‖x1 − p‖, ‖u‖ + ‖γ f (x∗) − (I + μA)x∗‖(1 + μγ̄ ) − γα

}

, n ≥ 1. (3.7)

Hence, {xn} is bounded, so are {yn}, {zn}, {Wn xn}, { f (Wn xn)}, {u(k)n } for all k = 1, 2, . . . , N

and {Kn Wn yn}, where Kn = 1tn

∫ tn0 S(s)ds.

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Step 2 We prove that limn→∞ ‖xn+1 − xn‖ = 0 and limn→∞∥∥∥u(N )

n+1 − u(N )n

∥∥∥ = 0.

Again, since I − τ B1 and I − δB2 are nonexpansive mappings, we have the followingestimates:

‖yn+1 − yn‖ = ‖JM1,τ (zn+1 − τ B1zn+1) − JM1,τ (zn − τ B1zn)‖≤ ‖(zn+1 − τ B1zn+1) − (zn − τ B1zn)‖≤ ‖zn+1 − zn‖=

∥∥∥JM2,δ

(u(N )

n+1 − δB2u(N )n+1

)− JM2,δ

(u(N )

n − δB2u(N )n

)∥∥∥

≤∥∥∥

(u(N )

n+1 − δB2u(N )n+1

)−(

u(N )n − δB2u(N )

n

)∥∥∥

≤∥∥∥u(N )

n+1 − u(N )n

∥∥∥

= ‖AN xn+1 − AN xn‖≤ ‖xn+1 − xn‖. (3.8)

On the other hand, since Ti and Un,i are nonexpansive, we have

‖Wn+1 yn − Wn yn‖ = ‖μ1T1Un+1,2 yn − μ1T1Un,2 yn‖≤ μ1‖Un+1,2 yn − Un,2 yn‖= μ1‖μ2T2Un+1,3 yn − μ2T2Un,3 yn‖≤ μ1μ2‖Un+1,3 yn − Un,3 yn‖...

≤ μ1μ2 · · · μn‖Un+1,n+1 yn − Un,n+1 yn‖

≤ M1

n∏

i=1

μi , (3.9)

where M1 ≥ 0 is a constant such that ‖Un+1,n+1 yn − Un,n+1 yn‖ ≤ M1 for all n ≥ 0.It follows from (3.8) and (3.9), we have

‖Wn+1 yn+1 − Wn yn‖ ≤ ‖Wn+1 yn+1 − Wn+1 yn‖ + ‖Wn+1 yn − Wn yn‖

≤ ‖yn+1 − yn‖ + M1

n∏

i=1

μi

≤ ‖xn+1 − xn‖ + M1

n∏

i=1

μi . (3.10)

It follows that

‖Kn+1Wn+1 yn+1 − Kn Wn yn‖ =∥∥∥∥

1

tn+1

tn+1∫

0

S(s)Wn+1 yn+1ds − 1

tn

tn∫

0

S(s)Wn ynds

∥∥∥∥

≤ 1

tn+1

tn+1∫

0

‖S(s)Wn+1 yn+1 − S(s)Wn yn‖ds

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+∥∥∥∥

1

tn+1

tn+1∫

0

S(s)Wn ynds − 1

tn

tn∫

0

S(s)Wn ynds

∥∥∥∥

≤ ‖Wn+1 yn+1 − Wn yn‖ +∥∥∥∥

1

tn+1

tn+1∫

tn

S(s)Wn ynds

+ 1

tn+1

tn∫

0

S(s)Wn ynds − 1

tn

tn∫

0

S(s)Wn ynds

∥∥∥∥

≤ ‖Wn+1 yn+1 − Wn yn‖ + 1

tn+1

tn+1∫

tn

‖S(s)Wn yn‖ds

+∣∣∣∣

1

tn+1− 1

tn

∣∣∣∣

tn∫

0

‖S(s)Wn yn‖ds

≤ ‖Wn+1 yn+1 − Wn yn‖ + 2

(

1 − tntn+1

)

M2

≤ ‖xn+1 − xn‖ + M1

n∏

i=1

μi + 2

(

1 − tntn+1

)

M2, (3.11)

where M2 = max{‖S(s)Wn yn‖}.Setting xn+1 = (1 − βn)vn + βn xn,∀n ≥ 1, we have

vn = xn+1 − βn xn

1 − βn= αn

(u + γ f (Wn xn)

) + ((1 − βn)I − αn(I + μA)

)Kn Wn yn

1 − βn.

Then, we obtain

vn+1 − vn = αn+1(u + γ f (Wn+1xn+1)

)+((1 − βn+1)I − αn+1(I + μA)

)Kn+1Wn+1 yn+1

1 − βn+1

− αn(u + γ f (Wn xn)

) + ((1 − βn)I − αn(I + μA)

)Kn Wn yn

1 − βn

= αn+1

1 − βn+1

(u + γ f (Wn+1xn+1)

) − αn

1 − βn

(u + γ f (Wn xn)

)+Kn+1Wn+1 yn+1 − Kn Wn yn

+ αn

1 − βn(I + μA)Kn Wn yn − αn+1

1 − βn+1(I + μA)Kn+1Wn+1 yn+1

= αn+1

1 − βn+1

((u + γ f (Wn+1xn+1)

)−(I + μA)Kn+1Wn+1 yn+1

)

+ αn

1 − βn

((I + μA)Kn Wn yn − u − γ f (Wn xn)

) + Kn+1Wn+1 yn+1 − Kn Wn yn . (3.12)

It follows from (3.11) and (3.12) that

‖vn+1 − vn‖ − ‖xn+1 − xn‖ ≤ αn+1

1 − βn+1

(‖u‖ + ‖γ f (Wn+1xn+1)‖ + ‖(I + μA)Kn+1Wn+1 yn+1‖

)

+ αn

1 − βn

(‖(I + μA)Kn Wn yn‖ + ‖u‖ + ‖γ f (Wn xn)‖

)

+ M1

n∏

i=1

μi + 2

(

1 − tntn+1

)

M2. (3.13)

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By the conditions (C4), (C5) and from tn ∈ (0,∞), tn → ∞ and 0 < μi ≤ b < 1,∀i ≥ 1,we have

lim supn→∞

(‖vn+1 − vn‖ − ‖xn+1 − xn‖) ≤ 0.

Hence, by Lemma 2.7, we obtain

limn→∞ ‖vn − xn‖ = 0.

It follows that

limn→∞ ‖xn+1 − xn‖ = lim

n→∞(1 − βn)‖vn − xn‖ = 0. (3.14)

Applying (3.14) into (3.8), we obtain that

limn→∞ ‖yn+1 − yn‖ = lim

n→∞ ‖zn+1 − zn‖ = limn→∞

∥∥∥u(N )

n+1 − u(N )n

∥∥∥ = 0.

Step 3 We show that limn→∞ ‖Kn Wn yn − yn‖ = 0, limn→∞ ‖yn − S(s)yn‖ = 0 andlimn→∞ ‖u(k+1)

n − u(k)n ‖ = 0, where Kn = 1

tn

∫ tn0 S(s)ds.

Since xn+1 = αn(u +γ f (Wn xn)

)+βn xn + ((1−βn)I −αn(I +μA)

)Kn Wn yn , we have

‖xn − Kn Wn yn‖ ≤ ‖xn − xn+1‖ + ‖xn+1 − Kn Wn yn‖= ‖xn − xn+1‖

+‖αn(u + γ f (Wn xn)

) + βn xn + ((1 − βn)I − αn(I + μA)

)Kn Wn yn − Kn Wn yn‖

= ‖xn − xn+1‖ +∥∥∥∥αn

((u + γ f (Wn xn)

)−(I + μA)Kn Wn yn

)+ βn(xn − Kn Wn yn)

∥∥∥∥

≤ ‖xn − xn+1‖+αn

(

‖u‖+‖γ f (Wn xn)‖+‖(I+μA)Kn Wn yn‖)

+βn‖xn − Kn Wn yn‖,

that is

‖xn − Kn Wn yn‖ ≤ 1

1 − βn‖xn − xn+1‖ + αn

1 − βn

(

‖u‖ + ‖γ f (Wn xn)‖ + ‖(I + μA)Kn Wn yn‖)

.

By (C4), (C5) and (3.14) it follows that

limn→∞ ‖Kn Wn yn − xn‖ = 0. (3.15)

Since J�NrN : H → H is firmly nonexpansive, u(N )

n = AN xn , where AN :=J�N

rN · · · J�2r2 J�1

r1 and x∗ ∈ �, we have

‖u(N )n − x∗‖2 = ‖AN xn − AN x∗‖2

≤ 〈AN xn − AN x∗, xn − x∗〉= 〈u(N )

n − x∗, xn − x∗〉= 1

2(‖u(N )

n − x∗‖2 + ‖xn − x∗‖2 − ‖xn − u(N )n ‖2),

and hence

‖u(N )n − x∗‖2 ≤ ‖xn − x∗‖2 − ‖xn − u(N )

n ‖2. (3.16)

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Observe that

‖xn+1 − x∗‖2 = ‖((1 − βn)I − αn(I + μA))(Kn Wn yn − x∗) + βn(xn − x∗)

+αn(u + γ f (Wn xn) − (I + μA)x∗)‖2

= ‖((1 − βn)I − αn(I + μA))(Kn Wn yn − x∗) + βn(xn − x∗)‖2

+α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2

+ 2βnαn

⟨xn − x∗, u + γ f (Wn xn) − (I + μA)x∗⟩

+2αn

⟨((1 − βn)I − αn(I + μA)

)(Kn Wn yn − x∗), u + γ f (Wn xn) − (I + μA)x∗⟩

≤[(1 − βn − αn − αnμγ̄ )‖Kn Wn yn − x∗‖ + βn‖xn − x∗‖

]2

+α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2

+ 2βnαn

⟨xn − x∗, u + γ f (Wn xn) − (I + μA)x∗⟩

+ 2αn

⟨((1 − βn)I − αn(I + μA)

)(Kn Wn yn − x∗), u + γ f (Wn xn) − (I + μA)x∗⟩

=[(1 − βn − αn − αnμγ̄ )‖Kn Wn yn − x∗‖ + βn‖xn − x∗‖

]2 + cn

≤ (1 − βn − αn − αnμγ̄ )2‖Kn Wn yn − x∗‖2 + β2n ‖xn − x∗‖2

+ 2(1 − βn − αn − αnμγ̄ )βn‖Kn Wn yn − x∗‖‖xn − x∗‖| + cn

≤ (1 − βn − αn − αnμγ̄ )2‖Kn Wn yn − x∗‖2 + β2n ‖xn − x∗‖2

+ (1 − βn − αn − αnμγ̄ )βn

[

‖Kn Wn yn − x∗‖2 + ‖xn − x∗‖2]

+ cn

=[(1 − αn − αnμγ̄ )2 − 2(1 − αn − αnμγ̄ )βn + β2

n

]‖Kn Wn yn − x∗‖2 + β2

n ‖xn − x∗‖2

+[

(1 − αn − αnμγ̄ )βn − β2n

][

‖Kn Wn yn − x∗‖2 + ‖xn − x∗‖2]

+ cn

=[(1 − αn − αnμγ̄ )2 − (1 − αn − αnμγ̄ )βn

]‖Kn Wn yn − x∗‖2

+(1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn, (3.17)

where

cn = α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2 + 2βnαn

⟨xn − x∗, u + γ f (Wn xn) − (I + μA)x∗⟩

+ 2αn

⟨((1 − βn)I − αn(I + μA)

)(Kn Wn yn − x∗), u + γ f (Wn xn) − (I + μA)x∗⟩.

It follows from condition (C4) that

limn→∞ cn = 0. (3.18)

Put (3.16) into (3.17), we have

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖u(N )n − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

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≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ ){‖xn − x∗‖2 − ‖xn − u(N )

n ‖2}

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn − αnμγ̄ )2‖xn − x∗‖2

− (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖xn − u(N )n ‖2 + cn

≤ ‖xn − x∗‖2 − (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖xn − u(N )n ‖2 + cn .(3.19)

It follows that

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖xn − u(N )n ‖2 ≤ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2 + cn

≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖) + cn .

Therefore, by (3.14) and (3.18), we get

limn→∞ ‖xn − u(N )

n ‖ = 0. (3.20)

Since

‖u(N )n − Kn Wn yn‖ ≤ ‖u(N )

n − xn‖ + ‖xn − Kn Wn yn‖,and by (3.15) and (3.20), we have

limn→∞ ‖u(N )

n − Kn Wn yn‖ = 0. (3.21)

Since B1 is a σ1-inverse-strongly monotone mapping with τ ∈ (0, 2σ1) and for anyx∗ ∈ �, we have

‖yn − x∗‖2 = ‖JM1,τ (zn − τ B1zn) − JM1,τ (x∗ − τ B1x∗)‖2

≤ ‖(zn − x∗) − τ(B1zn − B1x∗)‖2

= ‖zn − x∗‖2 − 2τ 〈zn − x∗, B1zn − B1x∗〉 + τ 2‖B1zn − B1x∗‖2

≤ ‖xn − x∗‖2 + τ(τ − 2σ1)‖B1zn − B1x∗‖2. (3.22)

Similarly, since B2 is a σ2-inverse-strongly monotone mapping with δ ∈ (0, 2σ2), we alsohave

‖zn − x∗‖2 ≤ ‖xn − x∗‖2 + δ(δ − 2σ2)‖B2u(N )n − B2x∗‖2. (3.23)

Substituting (3.22) into (3.17), we have

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ ){‖xn − x∗‖2 + τ(τ − 2σ1)‖B1zn − B1x∗‖2

}

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn − αnμγ̄ )2‖xn − x∗‖2

+(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )τ(τ − 2σ1)‖B1zn − B1x∗‖2 + cn

≤ ‖xn − x∗‖2

+(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )τ(τ − 2σ1)‖B1zn − B1x∗‖2 + cn . (3.24)

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Again, substituting (3.23) into (3.17), we get

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖zn − y∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ ){‖xn − x∗‖2 + δ(δ − 2σ2)‖B2u(N )

n − B2x∗‖2}

+(1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn − αnμγ̄ )2‖xn − x∗‖2

+ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )δ(δ − 2σ2)‖B2u(N )n − B2x∗‖2 + cn

≤ ‖xn − x∗‖2

+ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )δ(δ − 2σ2)‖B2u(N )n − B2x∗‖2 + cn . (3.25)

Therefore, by (3.24) and (3.25), we have

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )τ(2σ1 − τ)‖B1zn − B1x∗‖2

≤ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2 + cn

≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖) + cn

and

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )δ(2σ2 − δ)‖B2u(N )n − B2x∗‖2

≤ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2 + cn

≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖) + cn .

It follows from (3.14) and (3.18), we obtain

limn→∞ ‖B1zn − B1x∗‖ = 0 (3.26)

and

limn→∞ ‖B2u(N )

n − B2x∗‖ = 0. (3.27)

On the other hand, since JM1,τ is firmly nonexpansive, we have

‖yn − x∗‖2 = ‖JM1,τ (zn − τ B1zn) − JM1,τ (x∗ − τ B1x∗)‖2

≤ 〈(zn − τ B1zn) − (x∗ − τ B1x∗), yn − x∗〉= 1

2{‖(zn − τ B1zn) − (x∗ − τ B1x∗)‖2 + ‖yn − x∗‖2

−‖(zn − τ B1zn) − (x∗ − τ B1x∗) − (yn − x∗)‖2}≤ 1

2{‖zn − x∗‖2 + ‖yn − x∗‖2 − ‖(zn − yn) − τ(B1zn − B1x∗)‖2}

= 1

2{‖zn − x∗‖2 + ‖yn − x∗‖2

−‖zn − yn‖2 + 2τ 〈zn − yn, B1zn − B1x∗〉 − τ 2‖B1zn − B1x∗‖2}≤ 1

2{‖zn − x∗‖2 + ‖yn − x∗‖2

−‖zn − yn‖2 + 2τ‖zn − yn‖‖B1zn − B1x∗‖ − τ 2 |B1zn − B1x∗‖2}

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≤ 1

2{‖xn − x∗‖2 + ‖yn − x∗‖2 − ‖zn − yn‖2 + 2τ |zn − yn‖‖B1zn − B1x∗‖,

which yields that

‖yn − x∗‖2 ≤ ‖xn − x∗‖2 − ‖zn − yn‖2 + 2τ‖zn − yn‖‖B1zn − B1x∗‖. (3.28)

Similarly, since JM2,δ is firmly nonexpansive, we also have

‖zn − x∗‖2 ≤ ‖xn − x∗‖2 − ‖u(N )n − zn‖2 + 2δ‖u(N )

n − zn‖‖B2u(N )n − B2x∗‖. (3.29)

Substituting (3.28) into (3.17), we have

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ ){‖xn − x∗‖2 − ‖zn − yn‖2

+2τ‖zn − yn‖‖B1zn − B1x∗‖}

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn −αnμγ̄ )2‖xn − x∗‖2 − (1 − αn −αnμγ̄ )(1 − βn −αn −αnμγ̄ )‖zn − yn‖2

+2τ(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖zn − yn‖‖B1zn − B1x∗‖ + cn

≤ ‖xn − x∗‖2 − (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖zn − yn‖2

+2τ‖zn − yn‖‖B1zn − B1x∗‖ + cn . (3.30)

Again, substituting (3.29) into (3.17), we get

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖zn − y∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ ){‖xn − x∗‖2 − ‖u(N )

n − zn‖2

+2δ‖u(N )n − zn‖‖B2u(N )

n − B2x∗‖}

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn − αnμγ̄ )2‖xn − x∗‖2 − (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖u(N )n − zn‖2

+ 2δ(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖u(N )n − zn‖‖B2u(N )

n − B2x∗‖ + cn

≤ ‖xn − x∗‖2 − (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖u(N )n − zn‖2

+ 2δ‖u(N )n − zn‖‖B2u(N )

n − B2x∗‖ + cn . (3.31)

Therefore, by (3.30) and (3.31), we have

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖zn − yn‖2 ≤ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2

+2τ‖zn − yn‖‖B1zn − B1x∗‖ + cn

≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖)+2τ‖zn − yn‖‖B1zn − B1x∗‖ + cn

and

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖u(N )n − zn‖2 ≤ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2

+ 2δ‖u(N )n − zn‖‖B2u(N )

n − B2x∗‖ + cn

≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖)+ 2δ‖u(N )

n − zn‖‖B2u(N )n − B2x∗‖ + cn .

It follows from (3.14), (3.18), (3.26) and (3.27), we obtain

limn→∞ ‖zn − yn‖ = 0 (3.32)

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and

limn→∞ ‖u(N )

n − zn‖ = 0. (3.33)

Therefore, it follows from (3.21), (3.32) and (3.33), we get

limn→∞ ‖Kn Wn yn − yn‖ = 0, (3.34)

and from (3.15) and (3.34), we have

limn→∞ ‖xn − yn‖ = 0. (3.35)

Since {Wn yn} is a bounded sequence in H , from Lemma 2.10 for all s ≥ 0, we have

limn→∞ ‖Kn Wn yn − S(s)Kn Wn yn‖ = lim

n→∞

∥∥∥∥

1

tn

tn∫

0

S(s)Wn ynds − S(s)

(1

tn

tn∫

0

S(s)Wn ynds

)∥∥∥∥ = 0,

(3.36)

and since

‖yn − S(s)yn‖ ≤ ‖yn − Kn Wn yn‖ + ‖Kn Wn yn − S(s)Kn Wn yn‖+‖S(s)Kn Wn yn − S(s)yn‖ ≤ 2‖yn − Kn Wn yn‖ + ‖Kn Wn yn − S(s)Kn Wn yn‖

It follows from (3.34) and (3.36), we get

limn→∞ ‖yn − S(s)yn‖ = 0. (3.37)

On the other hand, since J�krk : H → H is firmly nonexpansive, Ak := J�k

rk · · · J�2r2 J�1

r1 ,

k = 1, 2, . . . , N and x∗ ∈ �, we have

‖Ak+1xn − x∗‖2 = ‖J�k+1rk+1 Ak xn − J�k+1

rk+1 x∗‖2

≤ 〈J�k+1rk+1 Ak xn − x∗, Ak xn − x∗〉

= 1

2

(

‖J�k+1rk+1 Ak xn − x∗‖2 + ‖Ak xn − x∗‖2 − ‖J�k+1

rk+1 Ak xn − Ak xn‖2)

,

and hence

‖Ak+1xn − x∗‖2 ≤ ‖xn − x∗‖2 − ‖Ak+1xn − Ak xn‖2. (3.38)

From (3.5) and (3.17), for each k = 1, 2, . . . , N − 1, we have

‖xn+1 − p‖2 ≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖yn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖AN xn − x∗‖2

+ (1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖Ak+1xn − x∗‖2

+ (1 − ł phan − αnμγ̄ )βn‖xn − x∗‖2 + cn

≤ (1−αn−αnμγ̄ )(1−βn−αn−αnμγ̄ ){‖xn−x∗‖2−‖Ak+1xn−Ak xn‖2

}

+(1 − αn − αnμγ̄ )βn‖xn − x∗‖2 + cn

= (1 − αn − αnμγ̄ )2‖xn − x∗‖2

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−(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖Ak+1xn − Ak xn‖2 + cn

≤ ‖xn − x∗‖2 − (1 − αn − αnμγ̄ )

×(1 − βn − αn − αnμγ̄ )‖Ak+1xn − Ak xn‖2 + cn . (3.39)

It follows that

(1 − αn − αnμγ̄ )(1 − βn − αn − αnμγ̄ )‖Ak+1xn − Ak xn‖2 ≤ ‖xn − x∗‖2

−‖xn+1 − x∗‖2 + cn ≤ ‖xn − xn+1‖(‖xn − x∗‖ + ‖xn+1 − x∗‖) + cn .

Therefore, by (3.14) and (3.18), we get

limn→∞ ‖Ak+1xn − Ak xn‖ = 0 that is lim

n→∞ ‖u(k+1)n − u(k)

n ‖ = 0. (3.40)

Step 4 We prove that

lim supn→∞

⟨u + [

γ f − (I + μA)]x∗, xn − x∗⟩ ≤ 0,

where x∗ is a solution of the optimization problem:

minx∈�

μ

2〈Ax∗, x∗〉 + 1

2‖x∗ − u‖2 − h(x∗).

To show this inequality, we can choose a subsequence {yni } of {yn} such that

limi→∞

⟨u + [

γ f − (I + μA)]x∗, yni − x∗⟩= lim sup

n→∞

⟨u + [

γ f − (I + μA)]x∗, yn − x∗⟩.

(3.41)

Since {yni } is bounded, there exists a subsequence {yni j} of {yni } which converges weakly

to z ∈ H . Without loss of generality, we can assume that yni ⇀ z. From (3.35), we getxni ⇀ z. Next, we show that z ∈ � := F(S)∩ F(W )∩(∩N

k=1 M E P(�k, φ))∩ I (B1, M1)∩

I (B2, M2).

(1) First, we prove that z ∈ F(S). Indeed, from Lemma 2.11 and (3.37), we get z ∈ F(S),i.e., z = S(s)z,∀s ≥ 0.

(2) Next, we show that z ∈ F(W ) = ∩∞n=1 F(Wn), where F(Wn) = ∩n

i=1 F(Ti ),∀n ≥ 1and F(Wn+1) ⊂ F(Wn). Assume that z /∈ F(W ), then there exists a positive integerm such that z /∈ F(Tm) and so z /∈ ∩m

i=1 F(Ti ). Hence for any n ≥ m, z /∈ ∩ni=1

F(Ti ) = F(Wn), i.e., z �= Wnz. This together with z = S(s)z,∀s ≥ 0 showsz = S(s)z �= S(s)Wnz,∀s ≥ 0, therefore we have z �= Kn Wnz,∀n ≥ m. It followsfrom the Opial’s condition and (3.34) that

lim infi→∞ ‖yni − z‖ < lim inf

i→∞ ‖yni − Kni Wni z‖≤ lim inf

i→∞ (‖yni − Kni Wni yni ‖ + ‖Kni Wni yni − Kni Wni z‖)≤ lim inf

i→∞ ‖yni − z‖,which is a contradiction. Thus, we get z ∈ F(W ).

(3) Now, we prove that z ∈ ∩Nk=1 M E P(�k, φ). Since Ak+1 = J�k+1

rk+1 Ak,

k = 1, 2, . . . , N − 1 and u(k+1)n = Ak+1xn , we have

�(Ak+1xn, x) + φ(x) − φ(Ak+1xn) + 1

rk+1⟨K ′(Ak+1xn) − K ′(Ak xn), η(x, Ak+1xn)

⟩≥ 0, ∀x ∈ H.

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It follows that

1

rk+1

⟨K ′(Ak+1xni ) − K ′(Ak xni ), η(x, Ak+1xni )

⟩

≥ −�(Ak+1xni , x) − φ(x) + φ(Ak+1xni ) (3.42)

for all x ∈ C . From (3.40) and by conditions (C1)(c) and (C2), we get

limni →∞

1

rk+1

⟨K ′(Ak+1xni ) − K ′(Ak xni ), η(x, Ak+1xni )

⟩= 0.

By the assumption that φ is lower semicontinuous, then it is weakly lower semicontin-uous and by the condition (H2) that x �−→ (−�i (x, y)) is lower semicontinuous, thenit is weakly lower semicontinuous. Since yni ⇀ z, it follows from (3.21), (3.34) and

(3.40) that u(k)ni ⇀ z for each k = 1, 2, . . . , N − 1. Taking the lower limit ni → ∞ in

(3.42), we have

�k+1(z, x) + φ(x) − φ(z) ≥ 0, ∀x ∈ H, ∀k = 0, 1, 2, . . . , N − 1.

(3.43)

Therefore z ∈ ∩Nk=1 M E P(�k, φ).

(4) Next, we show that z ∈ I (B1, M1) and z ∈ I (B2, M2). In fact, since B1 is a σ1-inverse-strongly monotone mapping, implies that B1 is a 1

σ1-Lipschitz continuous monotone

mapping and domain of B1 equal to H . It follows from Lemma 2.12 that M1 + B1

is a maximal monotone. Let (y, g) ∈ G(M1 + B1), that is, g − B1 y ∈ M1(y). Sinceyni = JM1,τ (zni − τ B1zni ), we have zni − τ B1zni ∈ (I + τ M1)(yni ), that is,

1

τ(zni − yni − τ B1zni ) ∈ M1(yni ). (3.44)

By M1 + B1 is a maximal monotone, we have

〈y − yni , g − B1 y − 1

λ(zni − yni − τ B1zni )〉 ≥ 0, (3.45)

and so

〈y − yni , g〉 ≥ 〈y − yni , B1 y + 1

τ(zni − yni − τ B1zni )〉

= 〈y − yni , B1 y − B1 yni + B1 yni − B1zni + 1

τ(zni − yni )〉

≥ 0 + 〈y − yni , B1 yni − B1zni 〉 + 〈y − yni ,1

τ(zni − yni )〉 (3.46)

It follows from (3.32) and yni ⇀ z that

limi→∞〈y − yni , g〉 = 〈y − z, g〉 ≥ 0. (3.47)

It follows from the maximal monotonicity of M1 + B1 that θ ∈ (M1 + B1)(z), thatis, z ∈ I (B1, M1). By the same way, from (3.33) and zni ⇀ z, we can obtainz ∈ I (B2, M2). Hence z ∈ � is proved.

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Now, from Lemma 2.9, (3.35) and (3.41), we have

lim supn→∞

⟨u + [

γ f − (I + μA)]x∗, xn − x∗⟩ = lim sup

n→∞⟨u + [

γ f − (I + μA)]x∗, yn − x∗⟩

= limi→∞

⟨u + [

γ f − (I + μA)]x∗, yni − x∗⟩

=⟨u + [

γ f − (I + μA)]x∗, z − x∗⟩ ≤ 0.

(3.48)

By (3.34), (3.35) and (3.48), we obtain

lim supn→∞

⟨u + [

γ f − (I + μA)]x∗, Kn Wn yn − x∗⟩ ≤ 0. (3.49)

Step 5 Finally, we show that xn → x∗. From (3.1), we obtain

‖xn+1 − x∗‖2

= ‖αn(u + γ f (Wn xn)

) + βn xn + ((1 − βn)I − αn(I + μA)

)Kn Wn yn − x∗‖2

= ‖((1 − βn)I − αn(I + μA))(Kn Wn yn − x∗)

+βn(xn − x∗) + αn(u + γ f (Wn xn) − (I + μA)x∗)‖2

= ‖((1 − βn)I − αn(I + μA))(Kn Wn yn − x∗) + βn(xn − x∗)‖2

+ 2αn

⟨((1 − βn)I − αn(I + μA)

)(Kn Wn yn − x∗), u

+γ f (Wn xn) − (I + μA)x∗⟩

+ 2αnβn

⟨xn − x∗, u + γ f (Wn xn) − (I + μA)x∗⟩ + α2

n‖u

+γ f (Wn xn) − (I + μA)x∗‖2

≤[(

1 − βn − αn(1 + μγ̄ ))‖Kn Wn yn − x∗‖ + βn‖xn − x∗‖

]2

+ 2αn(1 − βn)γ⟨Kn Wn yn − x∗, f (Wn xn) − f (x∗)

⟩

+2αn(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

− 2α2nγ

⟨(I + μA)(Kn Wn yn − x∗), f (Wn xn) − f (x∗)

⟩

− 2α2n

⟨(I + μA)(Kn Wn yn − x∗), u + γ f (x∗) − (I + μA)x∗⟩

+ 2αnβnγ⟨xn − x∗, f (Wn xn) − f (x∗)

⟩

+2αnβn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2

≤[(

1 − βn − αn(1 + μγ̄ ))‖Kn Wn yn − x∗‖ + βn‖xn − x∗‖

]2

+ 2αn(1 − βn)γ ‖Kn Wn yn − x∗‖‖ f (Wn xn) − f (x∗)‖+2αn(1 − βn)

⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

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− 2α2nγ ‖(I + μA)(Kn Wn yn − x∗)‖‖ f (Wn xn) − f (x∗)‖

− 2α2n‖(I + μA)(Kn Wn yn − x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖

+ 2αnβnγ ‖xn − x∗‖‖ f (Wn xn) − f (x∗)‖+2αnβn ×

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2

≤[(

1 − βn − αn(1 + μγ̄ ))‖xn − x∗‖ + βn‖xn − x∗‖

]2

+ 2αn(1 − βn)γ α‖xn − x∗‖2

+2αn(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

− 2α2nγα‖(I + μA)(Kn Wn yn − x∗)‖‖xn − x∗‖

− 2α2n‖(I + μA)(Kn Wn yn − x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖

+ 2αnβnγα‖xn − x∗‖2 + 2αnβn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+α2n‖u + γ f (Wn xn) − (I + μA)x∗‖2

=[(

1 − αn(1 + μγ̄ ))2 + 2αnγα

]‖xn − x∗‖2

+αn

{

2(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

−2αnγα‖(I + μA)(Kn Wn yn − x∗)‖‖xn − x∗‖−2αn‖(I + μA)(Kn Wn yn − x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖+2βn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+αn‖u + γ f (Wn xn) − (I + μA)x∗‖2}

=[1 − 2αn(1 + μγ̄ ) + α2

n(1 + μγ̄ )2 + 2αnγα]‖xn − x∗‖2

+αn

{

2(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

−2αnγα‖(I + μA)(Kn Wn yn − x∗)‖‖xn − x∗‖−2αn‖(I + μA)(Kn Wn yn − x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖+2βn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+αn‖u + γ f (Wn xn) − (I + μA)x∗‖2}

.

=[1 − 2αn

(1 + μγ̄ − γα

)]‖xn − x∗‖2

+αn

{

2(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+2βn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

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+αn

[(1 + μγ̄ )2‖xn − x∗‖2

−2γα‖(I + μA)(Kn Wn yn − x∗)‖‖xn − x∗‖−2‖(I + μA)(Kn Wn yn − x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖

+‖u + γ f (Wn xn) − (I + μA)x∗‖2]}

. (3.50)

Since {xn}, { f (Wn xn)}

and {Kn Wn yn} are bounded, there exist M > 0 such that(1+μγ̄ )2‖xn −x∗‖2 −2γα‖(I +μA)(Kn Wn yn −x∗)‖‖xn −x∗‖−2‖(I +μA)(Kn Wn yn −x∗)‖‖u + γ f (x∗) − (I + μA)x∗‖ + ‖u + γ f (Wn xn) − (I + μA)x∗‖2 ≤ M for all n ≥ 0.It follows that

‖xn+1 − x∗‖2 ≤ (1 − αnan)‖xn − x∗‖2 + αnbn, (3.51)

where

an = 2(1 + μγ̄ − γα

),

bn = 2(1 − βn)⟨Kn Wn yn − x∗, u + γ f (x∗) − (I + μA)x∗⟩

+2βn

⟨xn − x∗, u + γ f (x∗) − (I + μA)x∗⟩ + αn M.

Applying Lemma 2.8 to (3.51), we conclude that xn → x∗. This completes the proof. ��Corollary 3.2 Let H be a real Hilbert space, f be a contraction of H into itself withα ∈ (0, 1) and A be a strongly positive linear bounded operator on H with coefficientγ̄ > 0. Let φ be a lower semicontinuous and convex functional from H to R and let � :H × H → R be a finite family of equilibrium functions satisfying conditions (H1)–(H3).Let S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on H and let {tn} be a positivereal divergent sequence such that limn→∞ tn

tn+1= 1. Let {Vi : H → H}∞i=1 be a countable

family of uniformly k-strict pseudo-contractions, {Ti : H → H}∞i=1 be the countable familyof nonexpansive mappings defined by Ti x = t x +(1− t)Vi x,∀x ∈ H,∀i ≥ 1, t ∈ [k, 1), Wn

be the W -mapping defined by (2.11) and W be a mapping defined by (2.12) with F(W ) �= ∅.Let M1, M2 : H → 2H be maximal monotone mappings and B1, B2 : H → H be σ1, σ2-inverse-strongly monotone mappings, respectively. Suppose that � := F(S) ∩ F(W ) ∩M E P(�, φ) ∩ I (B1, M1) ∩ I (B2, M2) �= ∅. Let μ > 0, γ > 0 and r > 0, which areconstants. For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose the {xn}, {yn}, {zn} and{un} be the sequences generated iteratively by⎧⎪⎪⎪⎨

⎪⎪⎪⎩

�(un, x) + φ(x) − φ(un) + 1r

⟨K ′(un) − K ′(xn), η(x, un)

⟩ ≥ 0, ∀x ∈ H,

zn = JM2,δ(un − δB2un),

yn = JM1,τ (zn − τ B1zn),

xn+1 = αn[u + γ f (Wn xn)

]+βn xn + [(1 − βn)I − αn(I + μA)

] 1tn

∫ tn0 S(s)Wn ynds,

(3.52)

where un = J�r xn such that J�

r : H → H is the mapping defined by (2.13) and {αn}and {βn} are two sequences in (0, 1) for all n ∈ N. If the function η : H × H → H andK : H → R satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then {xn} convergesstrongly to x∗ ∈ �, which solves the following optimization problem:

minx∗∈�

μ

2〈Ax∗, x∗〉 + 1

2‖x∗ − u‖2 − h(x∗).

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Proof Taking N = 1 in Theorem 3.1. Hence, the conclusion follows. This completes theproof. ��Corollary 3.3 Let H be a real Hilbert space, f be a contraction of H into itself withα ∈ (0, 1) and A be a strongly positive linear bounded operator on H with coefficientγ̄ > 0. Let S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on H and let {tn}be a positive real divergent sequence such that limn→∞ tn

tn+1= 1. Let {Vi : H → H}∞i=1

be a countable family of uniformly k-strict pseudo-contractions, {Ti : H → H}∞i=1 bethe countable family of nonexpansive mappings defined by Ti x = t x + (1 − t)Vi x,∀x ∈H,∀i ≥ 1, t ∈ [k, 1), Wn be the W -mapping defined by (2.11) and W be a mapping definedby (2.12) with F(W ) �= ∅. Let M1, M2 : H → 2H be maximal monotone mappings andB1, B2 : H → H be σ1, σ2-inverse-strongly monotone mappings, respectively. Suppose that� := F(S) ∩ F(W ) ∩ ( ∩N

k=1 M E P(�k, φ)) ∩ I (B1, M1) ∩ I (B2, M2) �= ∅. Let μ > 0

and γ > 0, which are constants. For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose the{xn}, {yn} and {zn} be the sequences generated iteratively by⎧⎪⎨

⎪⎩

zn = JM2,δ(xn − δB2xn),

yn = JM1,τ (zn − τ B1zn),

xn+1 = αn[u + γ f (Wn xn)

]+βn xn + [(1 − βn)I − αn(I + μA)

] 1tn

∫ tn0 S(s)Wn ynds,

(3.53)

where {αn} and {βn} are two sequences in (0, 1) for all n ∈ N. If the sequence {xn} satisfythe conditions (C1)–(C6) as given in Theorem 3.1, then {xn} converges strongly to x∗ ∈ �,which solves the following optimization problem:

minx∗∈�

μ

2〈Ax∗, x∗〉 + 1

2‖x∗ − u‖2 − h(x∗).

Proof Put �(x, y) ≡ φ(x) ≡ 0 for all x, y ∈ H and r = 1. Take K (x) = ‖x‖2

2 andη(y, x) = y − x , for all x, y ∈ H . Then, we get un = PC xn = xn in Corollary 3.2. Hence,the conclusion follows. This completes the proof. ��Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H, f be acontraction of C into itself with α ∈ (0, 1) and A be a strongly positive linear bounded oper-ator on H with coefficient γ̄ > 0. Let S = {S(s) : 0 ≤ s < ∞} be a nonexpansive semigroupon C and let {tn} be a positive real divergent sequence. Let {Vi : C → C}∞i=1 be a countablefamily of uniformly k-strict pseudo-contractions, {Ti : C → C}∞i=1 be the countable familyof nonexpansive mappings defined by Ti x = t x +(1− t)Vi x,∀x ∈ C,∀i ≥ 1, t ∈ [k, 1), Wn

be the W -mapping defined by (2.11) and W be a mapping defined by (2.12) with F(W ) �= ∅.Let M1, M2 : H → 2H be a maximal monotone mappings and B1, B2 : H → H be a σ1, σ2-inverse-strongly monotone mappings, respectively. Suppose that � := F(S)∩F(W )∩(∩N

k=1M E P(�k, φ)

) ∩ I (B1, M1) ∩ I (B2, M2) �= ∅. Let μ > 0 and γ > 0, which are constants.For given x1 ∈ H arbitrarily and fixed u ∈ H, suppose the {xn}, {yn} and {zn} be thesequences generated iteratively by⎧⎪⎨

⎪⎩

zn = PC (xn − δB2xn),

yn = PC (zn − τ B1zn),

xn+1 = αn[u + γ f (Wn xn)

]+βn xn + [(1 − βn)I − αn(I + μA)

] 1tn

∫ tn0 S(s)Wn ynds,

(3.54)

where {αn} and {βn} are two sequences in (0, 1) for all n ∈ N. If the sequence {xn} satisfythe conditions (C1)–(C6) as given in Theorem 3.1, then {xn} converges strongly to x∗ ∈ �,which solves the following optimization problem:

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minx∗∈�

μ

2〈Ax∗, x∗〉 + 1

2‖x∗ − u‖2 − h(x∗).

Proof If M1 = M2 = ∂δC in Corollary 3.3, then the variational inclusion prob-lem (1.8) is equivalent to the variational inequality problem (1.9). On the other hand, sinceM1 = M2 = ∂δC , we see that JM1,τ = JM2,δ = PC . Hence, the conclusion follows. Thiscompletes the proof. ��Acknowledgments P. Katchang gratefully acknowledges support provided by King Mongkut’s Universityof Technology Thonburi (KMUTT) during the first author’s stay at King Mongkut’s University of TechnologyThonburi (KMUTT) as a post doctoral fellow. We also would like to thank the Higher Education ResearchPromotion and National Research University Project of Thailand, Office of the Higher Education Commission(NRU-CSEC project No.55000613) for financial support.

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