# A New Computational Technique for Common Solutions between Systems of Generalized Mixed Equilibrium and Fixed Point Problems

Authors: Pongsakorn Sunthrayuth; Poom Kumam
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general
system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.


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## Body

## 1. Introduction

Throughout this paper, we denoted byℝ the set of all real numbers. We always assume that H is a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥ and C is a nonempty, closed, and convex subset of H. PC denotes the metric projection of H onto C. A mapping T:C→C is said to be L-Lipschitzian if there exists a constant L>0 such that
(1)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈C.
If 0<L<1, then T is a contraction, and if L=1, then T is a nonexpansive mapping. We denote by Fix(T) the set of all fixed points set of the mapping T; that is, Fix(T)={x∈C:Tx=x}.A mappingF:C→H is said to be monotone if
(2)〈Fx-Fy,x-y〉≥0,∀x,y∈C.
A mapping F:C→H is said to be strongly monotone if there exists η>0 such that
(3)〈Fx-Fy,x-y〉≥η∥x-y∥2,∀x,y∈C.
Let φ:C→ℝ be a real-valued function, Θ:C×C→ℝ an equilibrium bifunction, and Ψ:C→H a nonlinear mapping. The generalized mixed equilibrium problem is to find x*∈C such that
(4)Θ(x*,y)+φ(y)-φ(x*)+〈Ψx*,y-x*〉≥0,∀y∈C,
which was introduced and studied by Peng and Yao [1]. The set of solutions of problem (4) is denoted by GMEP(Θ,φ,Ψ). As special cases of problem (4), we have the following results.  (1)
IfΨ=0, then problem (4) reduces to mixed equilibrium problem. Find x*∈C such that
(5)Θ(x*,y)+φ(y)-φ(x*)≥0,∀y∈C,
which was considered by Ceng and Yao [2]. The set of solutions of problem (5) is denoted by MEP(Θ).(2)
Ifφ=0, then problem (4) reduces to generalized equilibrium problem. Find x*∈C such that
(6)Θ(x*,y)+〈Ψx*,y-x*〉≥0,∀y∈C,
which was considered by S. Takahashi and W. Takahashi [3]. The set of solutions of problem (6) is denoted by GEP(Θ,Ψ).(3)
IfΨ=φ=0, then problem (4) reduces to equilibrium problem. Find x*∈C such that
(7)Θ(x*,y)≥0,∀y∈C.
The set of solutions of problem (7) is denoted by EP(Θ).(4)
IfΘ=φ=0, then problem (4) reduces to classical variational inequality problem. Find x*∈C such that
(8)〈Ψx*,y-x*〉≥0,∀y∈C.
The set of solutions of problem (8) is denoted by VI(C,Ψ). It is known that x*∈C is a solution of the problem (8) if and only if x* is a fixed point of the mapping PC(I-λΨ), where λ>0 is a constant and I is the identity mapping. The problem (4) is very general in the sense that it includes several problems, namely, fixed point problems, optimization problems, saddle point problems, complementarity problems, variational inequality problems, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of problem (4) (see, e.g., [4–9]). Several iterative methods to solve the fixed point problems, variational inequality problems, and equilibrium problems are proposed in the literature (see, e.g., [1–3, 10–18]) and the references therein.LetA1,A2:C→H be two mappings. Ceng and Yao [12] considered the following problem of finding (x*,y*)∈C×C such that
(9)G2(x*,x)+〈A2y*,x-x*〉+1λ2〈x*-y*,x-x*〉≥0,∀x∈C,G1(y*,y)+〈A1x*,y-y*〉+1λ1〈y*-x*,y-y*〉≥0,∀y∈C,
which is called a general system of generalized equilibria, where λ1>0 and λ2>0 are two constants. In particular, if G1=G2=G and A1=A2=A, then problem (9) reduces to the following problem of finding (x*,y*)∈C×C such that
(10)G(x*,x)+〈Ay*,x-x*〉+1λ2〈x*-y*,x-x*〉≥0,∀x∈C,G(y*,y)+〈Ax*,y-y*〉+1λ1〈y*-x*,y-y*〉≥0,∀y∈C,
which is called a new system of generalized equilibria, where λ1>0 and λ2>0 are two constants.IfG1=G2=Θ, A1=A2=A, and x*=y*, then problem (9) reduces to problem (7).IfG1=G2=0, then problem (9) reduces to a general system of variational inequalities. Find (x*,y*)∈C×C such that
(11)〈λ2A2y*+x*-y*,x-x*〉≥0,∀x∈C,〈λ1A1x*+y*-x*,y-y*〉≥0,∀y∈C,
where λ1>0 and λ2>0 are two constants, which is introduced and considered by Ceng et al. [19].In 2010, Ceng and Yao [12] proposed the following relaxed extragradient-like method for finding a common solution of generalized mixed equilibrium problems, a system of generalized equilibria (9), and a fixed point problem of a k-strictly pseudocontractive self-mapping S on C as follows:
(12)zn=Srn(Θ,φ)(xn-rnΨxn),yn=Sλ1G1[Sλ2G2(zn-λ2A2zn)-λ1A1Sλ2G2(zn-λ2A2zn)],xn+1=αnu+βnxn+γnyn+δnSyn,∀n≥0,
where Ψ,A1,A2:C→H are α-inverse strongly monotone, α~1-inverse strongly monotone, and α~2-inverse strongly monotone, respectively. They proved strong convergence of the related extragradient-like algorithm (12) under some appropriate conditions {αn}, {βn}, {γn}, and {δn}⊂[0,1] satisfyingαn+βn+γn+δn=1, for all n≥0, to x^=PΩx^, where Ω=Fix(S)∩GMEP(Θ,φ,Ψ)∩Fix(K), with the mapping K:C→C defined by
(13)Kx=Sλ1G1[Sλ2G2(x-λ2A2x)-λ1A1Sλ2G2(x-λ2A2x)],∀x∈C.
Very recently, Ceng et al. [11] introduced an iterative method for finding fixed points of a nonexpansive mapping T on a nonempty, closed, and convex subset C in a real Hilbert space H as follows:
(14)xn+1=PC[αnγVxn+(I-αnμF)Txn],∀n≥0,
where PC is a metric projection from H onto C, V is an L-Lipschitzian mapping with a constant L≥0, and F is a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0 and 0<μ<2η/κ2. Then, they proved that the sequences generated by (14)  converge strongly to a unique solution of variational inequality as follows:
(15)〈(μF-γV)x*,x*-x〉≥0,∀x∈Fix(T).In this paper, motivated and inspired by the previous facts, we first introduce the following problem of finding(x1*,x2*,…,xM*)∈C×C×⋯×C such that
(16)GM(x1*,x1)+〈AMxM*,x1-x1*〉+1λM〈x1*-xM*,x1-x1*〉≥0,∀x1∈C,GM-1(xM*,xM)+〈AM-1xM-1*,xM-xM*〉+1λM-1〈xM*-xM-1*,xM-xM*〉≥0,∀xM∈C,⋮G2(x3*,x3)+〈A2x2*,x3-x3*〉+1λ2〈x3*-x2*,x3-x3*〉≥0,∀x3∈C,G1(x2*,x2)+〈A1x1*,x2-x2*〉+1λ1〈x2*-x1*,x2-x2*〉≥0,∀x2∈C,
which is called a more general system of generalized equilibria in Hilbert spaces, where λi>0 for all i∈{1,2,…,M}. In particular, if M=2,x1*=x*,x2*=y*,x1=x, and x2=y, then problem (16) reduces to problem (9). Finally, by combining the relaxed extragradient-like algorithm (12) with the general iterative algorithm (14), we introduce a new iterative method for finding a common element of a fixed point problem of a nonexpansive semigroup, the set solutions of a general system of generalized equilibria in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to a common element of the previous three sets under some suitable conditions. Furthermore, we apply our results to finding the minimum-norm solutions which solve some quadratic minimization problem. The main result extends various results existing in the current literature.


## 2. Preliminaries

LetS be a semigroup. We denote by ℓ∞ the Banach space of all bounded real-valued functionals on S with supremum norm. For each s∈S, we define the left and right translation operators l(s) and r(s) on ℓ∞(S) by
(17)(l(s)f)(t)=f(st),(r(s)f)(t)=f(ts),
for each t∈S and f∈ℓ∞(S), respectively. Let X be a subspace of ℓ∞(S) containing 1. An element μ in the dual space X* of X is said to be a mean on X if ∥μ∥=μ(1)=1. It is well known that μ is a mean on X if and only if
(18)infs∈Sf(s)≤μ(f)≤sups∈Sf(s),
for each f∈X. We often write μt(f(t)) instead of μ(f) for μ∈X* and f∈X.LetX be a translation invariant subspace of ℓ∞(S) (i.e., l(s)X⊂X and r(s)X⊂X for each s∈S) containing 1. Then, a mean μ on X is said to be left invariant (resp., right invariant) if μ(l(s)f)=μ(f) (resp., μ(r(s)f)=μ(f)) for each s∈S and f∈X. A mean μ on X is said to be invariant if μ is both left and right invariant [20–22]. S is said to be left (resp., right) amenable if X has a left (resp., right) invariant mean. S is a amenable if S is left and right amenable. In this case, ℓ∞(S) also has an invariant mean. As is well known, ℓ∞(S) is amenable when S is commutative semigroup; see [23]. A net {μα} of means on X is said to be left regular if
(19)limα∥ls*μα-μα∥=0,
for each s∈S, where ls* is the adjoint operator of ls.LetC be a nonempty, closed, and convex subset of H. A family 𝒮={T(s):s∈S} is called a nonexpansive semigroup on C if for each s∈S, the mapping T(s):C→C is nonexpansive and T(st)=T(ts) for each s,t∈S. We denote by Fix(𝒮) the set of common fixed point of 𝒮; that is,
(20)Fix(𝒮)=⋂s∈S‍Fix(T(s))=⋂s∈S‍{x∈C:T(s)x=x}.
Throughout this paper, the open ball of radius r centered at 0 is denoted by Br, and for a subset D of H by co¯D, we denote the closed convex hull of D. For ϵ>0 and a mapping T:D→H, the set of ϵ-approximate fixed point of T is denoted by Fϵ(T,D); that is, Fϵ(T,D)={x∈D:∥x-Tx∥≤ϵ}.In order to prove our main results, we need the following lemmas.Lemma 1 (see [23–25]).
Letf be a function of a semigroup S into a Banach space E such that the weak closure of {f(t):t∈S} is weakly compact, and let X be a subspace of ℓ∞(S) containing all the functions t↦〈f(t),x*〉 with x*∈E*. Then, for any μ∈X*, there exists a unique element fμ in E such that
(21)〈fμ,x*〉=μt〈f(t),x*〉,
for all x*∈E*. Moreover, if μ is a mean on X, then
(22)∫f(t)dμ(t)∈co¯{f(t):t∈S}.
One can write fμ by ∫f(t)dμ(t).Lemma 2 (see [23–25]).
LetC be a closed and convex subset of a Hilbert space H, 𝒮={T(s):s∈S} a nonexpansive semigroup from C into C such that Fix(𝒮)≠∅, and X a subspace of ℓ∞ containing 1, the mapping t↦〈T(t)x,y〉 an element of X for each x∈C and y∈H, and μ a mean on X.
If one writesT(μ)x instead of ∫‍Ttxdμ(t), then the following hold: (i)
T(μ) is nonexpansive mapping from C into C;(ii)
T(μ)x=x for each x∈Fix(𝒮);(iii)
T(μ)x∈co¯{Ttx:t∈S} for each x∈C;(iv)
ifμ is left invariant, then T(μ) is a nonexpansive retraction from C onto Fix(𝒮).LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, and let C be a nonempty, closed, and convex subset of H. We denote the strong convergence and the weak convergence of {xn} to x∈H by xn→x and xn⇀x, respectively. Also, a mapping I:C→C denotes the identity mapping. For every point x∈H, there exists a unique nearest point of C, denoted by PCx, such that
(23)∥x-PCx∥≤∥x-y∥,∀y∈C.
Such a projection PC is called the metric projection of H onto C. We know that PC is a firmly nonexpansive mapping of H onto C; that is,
(24)〈x-y,PCx-PCy〉≥∥PCx-PCy∥2,∀x,y∈H.
It is known that
(25)z=PCx⟺〈x-z,y-z〉≤0,∀x∈H,y∈C.
In a real Hilbert space H, it is well known that
(26)∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,(27)∥λx+(1-λ)y∥2=λ∥x∥2+(1-λ)∥y∥2-λ(1-λ)∥x-y∥2,
for all x,y∈H and λ∈[0,1].IfA:C→H is α-inverse strongly monotone, then it is obvious that A is 1/α-Lipschitz continuous. We also have that, for all x,y∈C and λ>0,
(28)∥(I-λA)x-(I-λ)y∥2=∥(x-y)-λ(Ax-Ay)∥2=∥x-y∥2-2λ〈Ax-Ay,x-y〉+λ2∥Ax-Ay∥2≤∥x-y∥2+λ(λ-2α)∥Ax-Ay∥2.
In particular, if λ<2α, then I-λA is a nonexpansive mapping from C to H.For solving the equilibrium problem, let us assume that the bifunctionΘ:C×C→ℝ satisfies the following conditions:  (A1)
Θ(x,x)=0 for all x∈C;(A2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0 for each x,y∈C;(A3)
Θ is upper semicontinuous, that is, for each x,y,z∈C,
(29)limsupt→0+Θ(tz+(1-t)x,y)≤Θ(x,y);(A4)
Θ(x,·) is convex and weakly lower semicontinuous for each x∈C;(B1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for all z∈C∖Dx,
(30)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Lemma 3 (see [1]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Θ:C×C→ℝ be a bifunction satisfying conditions (A1)-(A4), and let φ:C→ℝ be a lower semicontinuous and convex function. For r>0 and x∈H, define a mapping Sr(Θ,φ):H→C as follows:
(31)Sr(Θ,φ)(x)={1ry∈C:Θ(y,z)+φ(z)-φ(y)+1r〈y-z,z-x〉≥0,∀z∈C}.
Assume that either (B1) or (B2) holds. Then, the following hold:  (i)
Sr(Θ,φ)≠∅ for all x∈H and Sr(Θ,φ) is single valued;(ii)
Sr(Θ,φ) is firmly nonexpansive, that is, for all x,y∈H,
(32)∥Sr(Θ,φ)x-Sr(Θ,φ)y∥2≤〈Sr(Θ,φ)x-Sr(Θ,φ)y,x-y〉;(iii)
Fix(Sr(Θ,φ))=MEP(Θ,φ);(iv)
MEP(Θ,φ) is closed and convex.Remark 4.
Ifφ=0, then Sr(Θ,φ) is rewritten as SrΘ (see [12, Lemma 2.1] for more details).Lemma 5 (see [26]).
Let{xn} and {ln} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)ln+βnxn  for all integers n≥0 and limsupn→∞(∥ln+1-ln∥-∥xn+1-xn∥)≤0. Then, limn→∞∥ln-xn∥=0.Lemma 6 (Demiclosedness Principle [27]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:C→C be a nonexpansive mapping with Fix(T)≠∅. If {xn} is a sequence in C that converges weakly to x and if {(I-T)xn} converges strongly to y, then (I-T)x=y; in particular, if y=0, then x∈Fix(T).Lemma 7 (see [28]).
Assume that{an} is a sequence of nonnegative real numbers such that
(33)an+1≤(1-σn)an+δn,
where {σn} is a sequence in (0,1) and {δn} is a sequence in ℝ such that  (i)
∑n=0∞‍σn=∞;(ii)
limsupn→∞(δn/σn)≤0 or ∑n=0∞‍|δn|<∞. Then, limn→∞an=0.The following lemma can be found in [29, 30]. For the sake of the completeness, we include its proof in a real Hilbert space version.Lemma 8.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let F:C→X be a κ-Lipschitzian and η-strongly monotone operator. Let 0<μ<2η/κ2 and τ=μ(η-μκ2/2). Then, for each t∈(0,min{1,1/2τ}), the mapping S:C→H defined by S∶=I-tμF is a contraction with constant 1-tτ.Proof.
Since0<μ<2η/κ2 and t∈(0,min{1,1/2τ}), this implies that 1-tτ∈(0,1). For all x,y∈C, we have
(34)∥Sx-Sy∥2=∥(I-tμF)x-(I-tμF)y∥2=∥(x-y)x-tμ(Fx-Fy)∥2=∥x-y∥2-2tμ〈Fx-Fy,x-y〉+t2μ2∥Fx-Fy∥2≤∥x-y∥2-2tμη∥x-y∥2+t2μ2κ2∥x-y∥2≤[1-tμ(2η-μκ2)]∥x-y∥2=[1-2tμ(η-μκ22)]∥x-y∥2≤[1-tμ(η-μκ22)]2∥x-y∥2=(1-tτ)2∥x-y∥2.
It follows that
(35)∥Sx-Sy∥≤(1-tτ)∥x-y∥.
Hence, we have that S∶=I-tμF is a contraction with constant 1-tτ. This completes the proof.Lemma 9.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ai:C→H  (i=1,2,…,M) be a finite family of αi-inverse strongly monotone operator. Let K:C→C be a mapping defined by
(36)Kx∶=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)···Sλ1G1(I-λ1A1)x,∀x∈C.
If 0<λi<2αi for all i=1,2,…,M, then K:C→C is nonexpansive.Proof.
PutQi=SλiGi(I-λiAi)Sλi-1Gi-1(I-λi-1Ai-1)⋯Sλ1G1(I-λ1A1) for i=1,2,…,M and Q0=I. Then, K=QM. For all x,y∈C, it follows from (28) that
(37)∥Kx-Ky∥=∥QMx-QMy∥=∥SλMGM(I-λMAM)QM-1x-SλMGM(I-λMAM)QM-1y∥≤∥(I-λMAM)QM-1x-(I-λMAM)QM-1y∥≤∥QM-1x-QM-1y∥⋮≤∥Q0x-Q0y∥=∥x-y∥,
which implies that K is nonexpansive. This completes the proof.Lemma 10.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ai:C→H  (i=1,2,…,M) be a nonlinear mapping. For given (x1*,x2*,…,xM*)∈C×C×···×C, where x*=x1*, xi*=Sλi-1Gi-1(I-λi-1Ai-1)xi-1*, for i=2,3,…,M, and x1*=SλMGM(I-λMAM)xM*. Then, (x1*,x2*,…,xM*) is a solution of the problem (16) if and only if x* is a fixed point of the mapping K defined as in Lemma 9.Proof.
Let(x1*,x2*,…,xM*)∈C×C×⋯×C be a solution of the problem (16). Then, we have
(38)GM(x1*,x1)+〈AMxM*,x1-x1*〉+1λM〈x1*-xM*,x1-x1*〉≥0,∀x1∈C,GM-1(xM*,xM)+〈AM-1xM-1*,xM-xM*〉+1λM-1〈xM*-xM-1*,xM-xM*〉≥0,∀xM∈C,⋮G2(x3*,x3)+〈A2x2*,x3-x3*〉+1λ2〈x3*-x2*,x3-x3*〉≥0,∀x3∈C,G1(x2*,x2)+〈A1x1*,x2-x2*〉+1λ1〈x2*-x1*,x2-x2*〉≥0,∀x2∈C,⇕x1*=SλMGM(I-λMAM)xM*x2*=Sλ1G1(I-λ1A1)x1*,⋮xM-1*=SλM-2GM-2(I-λM-2AM-2)xM-2*,xM*=SλM-1GM-1(I-λM-1AM-1)xM-1*,⇕x*=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)⋯Sλ1G1(I-λ1A1)x*=Kx*.
This completes the proof.


## 3. Main Results

Theorem 11.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Θk:C×C→ℝ  (k=1,2,…,N) a finite family of bifunctions which satisfy (A1)–(A4), φk:C→ℝ  (k=1,2,…,N) a finite family of lower semicontinuous and convex functions, and Ψk:C→H  (k=1,2,…,N) a finite family of a μk-inverse strongly monotone mapping and Ak:C→H  (k=1,2,…,M) a finite family of an αk-inverse strongly monotone mapping. Let S be a semigroup, and let 𝒮={T(t):t∈S} be a nonexpansive semigroup on C such that Fix(𝒮)≠∅. Let X be a left invariant subspace of ℓ∞(S) such that 1∈X and the function t→〈T(t)x,y〉 is an element of X for x∈C and y∈H. Let {μn} be a left regular sequence of means on X such that limn→∞∥μn+1-μn∥=0. Let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ,η>0, and let V:C→H be an L-Lipschitzian mapping with a constant L≥0. Let 0<μ<2η/κ2 and 0≤γL<τ, where τ=μ(η-μκ2/2). Assume that ℱ∶=⋂k=1N‍GMEP(Θk,φk,Ψk)∩(K)∩Fix(𝒮)≠∅, where K is defined as in Lemma 9. For given x1∈C, let {xn} be a sequence defined by
(39)un=SrN,n(ΘN,φN)(I-rN,nΨN)SrN-1,n(ΘN-1,φN-1)×(I-rN-1,nΨN-1)⋯Sr1,n(Θ1,φ1)(I-r1,nΨ1)xn,yn=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)⋯Sλ1G1(I-λ1A1)un,xn+1=βnxn+(1-βn)PC[αnγVxn+(I-αnμF)T(μn)yn],∀n≥1,
where {αn}, {βn} are sequences in (0,1), and {rk,n}k=1N is a sequence such that {rk,n}k=1N⊂[ak,bk]⊂(0,2γk)satisfying the following conditions:  (C1)
limn→∞αn=0 and ∑n=1∞‍αn=∞;(C2)
0<liminfn→∞βn≤limsupn→∞βn<1;(C3)
liminfn→∞rk,n>0 and limn→∞(rk,n/rk,n+1)=1 for all k∈{1,2,…,N}. Then, the sequence {xn} defined by (39) converges strongly to x^∈ℱ as n→∞, where x^ solves uniquely the variational inequality
(40)〈(μF-γV)x^,x^-v〉≤0,∀v∈ℱ.
Equivalently, one has x^=Pℱ(I-μF+γV)x^.Proof.
Note that from condition(C1), we may assume, without loss of generality, that αn≤min{1,1/2τ} for all n∈ℕ. First, we show that {xn} is bounded. Set
(41)Gnk∶=Srk,n(Θk,φk)(I-rk,nΨk)Srk-1,n(Θk-1,φk-1)×(I-rk-1,nΨk-1)⋯Sr1,n(Θ1,φ1)(I-r1,nΨ1),∀k∈{1,2,…,N},n∈ℕ,Qi:=SλiGi(I-λiAi)Sλi-1Gi-1(I-λi-1Ai-1)⋯Sλ1G1(I-λ1A1),∀i∈{1,2,…,M},Gn0=Q0=I. Then, we have un=GnNxn and yn=QMun. From Lemmas 3 and 9, we have that GnN and QM are nonexpansive. Take x*∈ℱ; we have
(42)∥un-x*∥=∥GnNxn-GnNx*∥≤∥xn-x*∥.
By Lemma 10, we have x*=QMx*. It follows from (42) that
(43)∥yn-x*∥=∥QMun-QMx*∥≤∥un-x*∥≤∥xn-x*∥.
Set
(44)zn∶=PC[αnγVxn+(I-αnμF)T(μn)yn],∀n∈ℕ.
Then, we can rewrite (39) as xn+1=βnxn+(1-βn)zn. From Lemma 8 and (43), we have
(45)∥zn-x*∥=∥PC[αnγVxn+(I-αμF)T(μn)yn]-PCx*∥≤∥αn(γVxn-μFx*)+(I-αnμF)(T(μn)yn-x*)∥≤αn∥γVxn-μFx*∥+(1-αnτ)∥T(μn)yn-x*∥≤αnγ∥Vxn-Vx*∥+αn∥γVx*-μFx*∥+(1-αnτ)∥yn-x*∥≤(1-αn(τ-γL))∥xn-x*∥+αn∥γVx*-μFx*∥.
It follows from (45) that
(46)∥xn+1-x*∥=∥βn(xn-x*)+(1-βn)(zn-x*)∥≤βn∥xn-x*∥+(1-βn)∥zn-x*∥≤βn∥xn-x*∥+(1-βn)×[(1-αn(τ-γL))∥xn-x*∥+αn∥γVx*-μFx*∥]=(1-αn(1-βn)(τ-γL))∥xn-x*∥+αn(1-βn)(τ-γL)∥γVx*-μFx*∥τ-γL.
By induction, we have
(47)∥xn-x*∥≤max{∥x1-x*∥,∥γVx*-μFx*∥τ-γL},∀n≥1.
Hence, {xn} is bounded, and so are {Vxn} and {FT(μn)yn}.
Next, we show that(48)limn→∞∥xn+1-xn∥=0.
Observe that
(49)limn→∞∥T(μn+1)yn-T(μn)yn∥=0.
Indeed,
(50)∥T(μn+1)yn-T(μn)yn∥=sup∥z∥=1|〈T(μn+1)yn-T(μn)yn,z〉|=sup∥z∥=1|(μn+1)s〈T(s)yn,z〉-(μn)s〈T(s)yn,z〉|≤∥μn+1-μn∥sups∈S∥T(s)yn∥.
Since {yn} is bounded and limn→∞∥μn+1-μn∥=0, then (49) holds. We observe that
(51)∥yn+1-yn∥=∥QMun+1-QMun∥≤∥un+1-un∥.
Let {ωn} be a bounded sequence in C. Now, we show that
(52)limn→∞∥GnNωn-Gn+1Nωn∥=0.
For the previous purpose, put Dnk=Srk,n(Θk,φk)(I-rk,nΨk), and we first show that
(53)limn→∞∥Dn+1kωn-Dnkωn∥=0,∀k∈{1,2,…,N}.
In fact, since Dnkωn∈GMEP(Θk,φk,Ψk) and Dn+1kωn∈GMEP(Θk,φk,Ψk), we have
(54)Θk(Dnkωn,y)+φk(y)-φk(Dnkωn)+〈Ψkωn,y-Dnkωn〉+1rk,n〈y-Dnkωn,Dnkωn-ωn〉≥0,∀y∈C,(55)Θk(Dn+1kωn,y)+φk(y)-φk(Dn+1kωn)+〈Ψkωn,y-Dn+1kωn〉+1rk,n+1〈y-Dn+1kωn,Dn+1kωn-ωn〉≥0,∀y∈C.
Substituting y=Dn+1kωn in (54) and y=Dnkωnin (55), then add these two inequalities, and using (A2), we obtain
(56)〈Dn+1kωn-Dnkωn,1rk,n(Dnkωn-ωn)-1rk,n+1(Dn+1kωn-ωn)〉≥0.
Hence,
(57)〈rk,nrn,k+1Dn+1kωn-Dnkωn,Dnkωn-Dn+1kωn+Dn+1kωn-ωn-rk,nrn,k+1(Dn+1kωn-ωn)〉≥0;
we derive from (57) that
(58)∥Dn+1kωn-Dnkωn∥2≤〈Dn+1kωn-Dnkωn,(1-rk,nrk,n+1)(Dn+1kωn-ωn)〉≤∥Dn+1kωn-Dnkωn∥|1-rk,nrk,n+1|∥Dn+1kωn-ωn∥,
which implies that
(59)∥Dn+1kωn-Dnkωn∥≤|1-rk,nrk,n+1|∥Dn+1kωn-ωn∥.
Noticing that condition (C3) implies that (53) holds, from the definition of GnN and the nonexpansiveness of Dnk, we have
(60)∥GnNωn-Gn+1Nωn∥=∥DnNGnN-1ωn-Dn+1NGn+1N-1ωn∥≤∥DnNGnN-1ωn-Dn+1NGnN-1ωn∥+∥Dn+1NGnN-1ωn-Dn+1NGn+1N-1ωn∥≤∥DnNGnN-1ωn-Dn+1NGnN-1ωn∥+∥GnN-1ωn-Gn+1N-1ωn∥≤∥DnNGnN-1ωn-Dn+1NGnN-1ωn∥+∥DnN-1GnN-2ωn-Dn+1N-1Gn+1N-2ωn∥+∥GnN-2ωn-Gn+1N-2ωn∥≤∥DnNGnN-1ωn-Dn+1NGnN-1ωn∥+∥DnN-1GnN-2ωn-Dn+1N-1Gn+1N-2ωn∥+⋯+∥Dn2ωn-Dn+12ωn∥+∥Dn1ωn-Dn+11ωn∥,
for which (52) follows by (53). Since un=GnNxn and un+1=Gn+1Nxn+1, we have
(61)∥un-un+1∥=∥GnNxn-Gn+1Nxn+1∥≤∥GnNxn-Gn+1Nxn∥+∥Gn+1Nxn-Gn+1Nxn+1∥≤∥GnNxn-Gn+1Nxn∥+∥xn-xn+1∥.
Put a constant M1>0 such that
(62)M1=supn≥1{γ∥Vxn+1∥+μ∥FT(tn+1)yn+1∥,γ∥Vxn∥+μ∥FT(tn)yn∥}.
From definition of {zn}, we note that
(63)∥zn+1-zn∥=∥PC[αn+1γVxn+1+(I-αn+1μF)T(μn+1)yn+1]-PC[αnγVxn+(I-αnμF)T(μn)yn]∥≤αn+1∥γVxn+1-μFT(μn+1)yn+1∥+αn∥γVxn+1-μFT(μn)yn∥+∥T(μn+1)yn+1-T(μn)yn∥≤αn+1∥γVxn+1-μFT(μn+1)yn+1∥+αn∥γVxn+1-μFT(μn)yn∥+∥T(μn+1)yn+1-T(μn+1)yn∥+∥T(μn+1)yn-T(μn)yn∥≤(αn+1+αn)M1+∥yn+1-yn∥+∥T(μn+1)yn-T(μn)yn∥.
It follows from (51), (61), and (63) that
(64)∥zn+1-zn∥≤(αn+1+αn)M1+∥GnNxn-Gn+1Nxn∥+∥xn+1-xn∥+∥T(μn+1)yn-T(μn)yn∥.
From condition (C1), (49), and (52), we have
(65)limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0.
Hence, by Lemma 5, we obtain
(66)limn→∞∥zn-xn∥=0.
Consequently,
(67)limn→∞∥xn+1-xn∥=limn→∞(1-βn)∥zn-xn∥=0.
From condition (C1), we have
(68)∥zn-T(μn)yn∥=∥PC[αnγVxn+(I-αnμF)T(μn)yn]-PCT(μn)yn∥≤αn∥γVxn-μFT(μn)yn∥⟶0asn⟶∞.
From (66) and (68), we have
(69)∥xn-T(μn)yn∥≤∥xn-zn∥+∥zn-T(μn)yn∥⟶0asn⟶∞.
Set zn=PCvn, where vn=αnγVxn+(I-αnμF)T(μn)yn. From (25), we have
(70)∥zn-x*∥=〈vn-x*,zn-x*〉+〈PCvn-vn,PCvn-x*〉≤〈vn-x*,zn-x*〉=αn〈γVxn-μFx*,zn-x*〉+〈(I-αnμF)(T(μn)yn-x*),zn-x*〉≤(1-αnτ)∥yn-x*∥∥zn-x*∥+αn〈γVxn-μFx*,zn-x*〉≤(1-αnτ)2(∥yn-x*∥2+∥zn-x*∥2)+αn〈γVxn-μFx*,zn-x*〉≤(1-αnτ)2∥yn-x*∥2+12∥zn-x*∥2+αn〈γVxn-μFx*,zn-x*〉.
It follows that
(71)∥zn-x*∥2≤∥yn-x*∥2+2αn〈γVxn-μFx*,zn-x*〉≤∥yn-x*∥2+2αn∥γVxn-μFx*∥∥zn-x*∥.
By the convexity of ∥·∥2 and (71), we have
(72)∥xn+1-x*∥2=∥βn(xn-x*)+(1-βn)(zn-x*)∥2≤βn∥xn-x*∥2+(1-βn)∥zn-x*∥2≤βn∥xn-x*∥2+(1-βn)×{∥yn-x*∥2+2αn∥γVxn-μFx*∥∥zn-x*∥}≤βn∥xn-x*∥2+(1-βn)∥yn-x*∥2+2αn(1-βn)M2,
where M2>0 is an appropriate constant such that M2=supn≥1{∥γVxn-μFx*∥∥zn-x*∥}.
Next, we show that(73)limn→∞∥Gnk-1xn-Gnkxn∥=0,∀k∈{1,2,…,N}.
From (28), we have
(74)∥Gnkxn-x*∥2=∥Srk,n(Θk,φk)(I-rk,nΨk)Gnk-1xn-Srk,n(Θk,φk)(I-rk,nΨk)x*∥2≤∥(I-rk,nΨk)Gnk-1-(I-rk,nΨk)x*∥2≤∥Gnk-1xn-x*∥2+rk,n(rk,n-2μk)∥ΨkGnk-1xn-Ψkx*∥2≤∥xn-x*∥2+rk,n(rk,n-2μk)∥ΨkGnk-1xn-Ψkx*∥2.
From (42), for all k∈{1,2,…,N}, we note that
(75)∥yn-x*∥2=∥QMun-x*∥≤∥un-x*∥2=∥GnNxn-x*∥2≤∥Gnkxn-x*∥2.
From (72) and (75), we have
(76)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)∥GnNxn-x*∥2+2αn(1-βn)M2≤βn∥xn-x*∥2+(1-βn)∥Gnkxn-x*∥2+2αn(1-βn)M2.
Substituting (74) into (72), we have
(77)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)×{∥xn-x*∥2+rk,n(rk,n-2μk)∥ΨkGnk-1xn-Ψkx*∥2}+2αn(1-βn)M2=∥xn-x*∥2+(1-βn)rk,n(rk,n-2μk)×∥ΨkGnk-1xn-Ψkx*∥2+2αn(1-βn)M2,
which in turn implies that
(78)(1-βn)rk,n(2μk-rk,n)∥ΨkGnk-1xn-Ψkx*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+2αn(1-βn)M2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2αn(1-βn)M2.
Since liminfn→∞(1-βn)>0, 0<rk,n<2μk, for all k∈{1,2,…,N}, from(C1) and (67), we obtain that
(79)limn→∞∥ΨkGnk-1xn-Ψkx*∥=0,∀k∈{1,2,…,N}.
On the other hand, from Lemma 3 and (26), we have
(80)∥Gnkxn-x*∥2=∥Srk,n(Θk,φk)(I-rk,nΨk)Gnk-1xn-Srk,n(Θk,φk)(I-rk,nΨk)x*∥2≤〈(I-rk,nΨk)Gnk-1xn-(I-rk,nΨk)x*,Gnkxn-x*〉=12(∥(I-rk,nΨk)Gnk-1xn-(I-rk,nΨk)x*∥2+∥Gnkxn-x*∥2-∥(I-rk,nΨk)Gnk-1xn-(I-rk,nΨk)x*-(Gnkxn-x*)∥2)≤12(∥Gnk-1xn-x*∥2+∥Gnk-x*∥2-∥Gnk-1xn-Gnkxn-rk,n(ΨkGnk-1xn-Ψkx*)∥2),
which in turn implies that
(81)∥Gnkxn-x*∥2≤∥Gnk-1xn-x*∥2-∥Gnk-1xn-Gnk-rk,n(ΨkGnk-1-Ψkx*)∥2=∥Gnk-1xn-x*∥2-∥Gnk-1xn-Gnkxn∥-rk,n2∥ΨkGnk-1xn-Ψkx*∥2+2rk,n〈Gnk-1xn-Gnkxn,ΨkGnk-1xn-Ψkx*〉≤∥Gnk-1xn-x*∥2-∥Gnk-1xn-Gnkxn∥2+2rk,n∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥≤∥xn-x*∥2-∥Gnk-1xn-Gnkxn∥2+2rk,n∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥.
Substituting (81) into (76), we have
(82)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)×{∥xn-x*∥2-∥Gnk-1xn-Gnkxn∥2+2rk,n∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥∥Gnk-1xn-Gnkxn∥2}+2αn(1-βn)M2=∥xn-x*∥2-(1-βn)∥Gnk-1xn-Gnkxn∥+2rk,n(1-βn)∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥+2αn(1-βn)M2,
which in turn implies that
(83)(1-βn)∥Gnk-1xn-Gnkxn∥≤∥xn-x*∥2-∥xn+1-x*∥2+2rk,n(1-βn)∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥+2αn(1-βn)M2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2rk,n(1-βn)∥Gnk-1xn-Gnkxn∥∥ΨkGnk-1xn-Ψkx*∥+2αn(1-βn)M2.
Since liminfn→∞(1-βn)>0, from(C1), (67), and (79), we obtain that (73) holds. Consequently,
(84)∥xn-un∥=∥Gn0-GnNxn∥≤∥Gn0xn-Gn1xn∥+∥Gn1xn-Gn2xn∥+⋯+∥GnN-1xn-GnNxn∥⟶0asn⟶∞.
Next, we show that
(85)limn→∞∥AiQi-1un-AiQi-1x*∥=0,∀i∈{1,2,…,M}.
From (28), we have
(86)∥QMun-QMx*∥2=∥SλMGM(I-λMAM)QM-1un-SλMGM(I-λMAM)QM-1x*∥2≤∥(I-λMAM)QM-1un-(I-λMAM)QM-1x*∥2≤∥QM-1un-QM-1x*∥2+λM(λM-2αM)×∥AMQM-1un-AMQM-1x*∥2.
By induction, we have
(87)∥QMun-QMx*∥2≤∥un-x*∥+∑i=1M‍λi(λi-2αi)∥AiQi-1un-AiQi-1x*∥2≤∥xn-x*∥+∑i=1M‍λi(λi-2αi)∥AiQi-1un-AiQi-1x*∥2.
From (72) and (75), we have
(88)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)∥QMun-QMx*∥2+2αn(1-βn)M2.
Substituting (87) into (88), we have
(89)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)×{∑i=1M‍∥xn-x*∥2+∑i=1M‍λi(λi-2αi)∥AiQi-1un-AiQi-1x*∥2}+2αn(1-βn)M2=∥xn-x*∥2+(1-βn)∑i=1M‍λi(λi-2αi)∥AiQi-1un-AiQi-1x*∥2+2αn(1-βn)M2,
which in turn implies that
(90)(1-βn)∑i=1M‍λi(2αi-λi)∥AiQi-1un-AiQi-1x*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+2αn(1-βn)M2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+2αn(1-βn)M2.
Since liminfn→∞(1-βn)>0, from(C1) and (67), we obtain that (85) holds.
On the other hand, from (24) and (26), we have
(91)∥QMun-QMx*∥2=∥SλMGM(I-λMAM)QM-1un-SλMGM(I-λMAM)QM-1x*∥2≤〈(I-λMAM)QM-1un-(I-λMAM)QM-1x*,QMun-QMx*〉=12(∥(I-λMAM)QM-1un-(I-λMAM)QM-1x*∥2+∥QMun-QMx*∥2-∥(I-λMAM)QM-1un-(I-λMAM)x*-(QMun-QMx*)∥2)≤12(∥QM-1un-QM-1x*∥2+∥QMun-QMx*∥2-∥QM-1un-QMun+QMx*-QM-1x*-λM(AMQM-1un-AMQM-1x*)∥2),
which in turn implies that
(92)∥QMun-QMx*∥2≤∥QM-1un-QM-1x*∥2-∥QM-1un-QMun+QMx*-QM-1x*-λM(AMQM-1un-AMQM-1x*)∥2=∥QM-1un-QM-1x*∥2-∥QM-1un-QMun-QMx*-QM-1x*∥2-λM2∥AMQM-1un-AMQM-1x*∥2+2λM〈QM-1un-QMun+QMx*-QM-1x*,AMQM-1un-AMQM-1x*〉≤∥QM-1un-QM-1x*∥2-∥QM-1un-QMun+QMx*-QM-1x*∥2+2λM∥QM-1un-QMun+QMx*-QM-1x*∥×∥AMQM-1un-AMQM-1x*∥.
By induction, we have
(93)∥QMun-QMx*∥2≤∥un-x*∥-∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥2+∑i=1N‍2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥≤∥xn-x*∥-∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥2+∑i=1N‍2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥.
Substituting (93) into (88), we have
(94)∥xn+1-x*∥2≤βn∥xn-x*∥2+(1-βn)×{∥xn-x*∥-∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥2+∑i=1M‍2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥∑i=1M‍}+2αn(1-βn)M2≤∥xn-x*∥2-(1-βn)∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥2+(1-βn)×∑i=1M2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥+2αn(1-βn)M2,
which in turn implies that
(95)(1-βn)∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥2≤∥xn-x*∥2-∥xn+1-x*∥2+(1-βn)∑i=1M‍2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥+2αn(1-βn)M2≤(∥xn-x*∥+∥xn+1-x*∥)∥xn+1-xn∥+(1-βn)∑i=1M‍2λi∥Qi-1un-Qiun+Qix*-Qi-1x*∥×∥AiQi-1un-AiQi-1x*∥+2αn(1-βn)M2.
Since liminfn→∞(1-βn)>0, from(C1), (67), and (85), we obtain that
(96)limn→∞∥Qi-1un-Qiun+Qix*-Qi-1x*∥=0,∀i∈{1,2,…,M}.
Consequently,
(97)∥un-yn∥=∥Q0un-QMun∥≤∑i=1M‍∥Qi-1un-Qiun+Qix*-Qi-1x*∥⟶0asn⟶∞.
It follows from (84) and (97) that
(98)∥xn-yn∥≤∥xn-un∥+∥un-yn∥⟶0asn⟶∞.
Next, we show that
(99)limn→∞∥xn-T(t)xn∥=0,∀t∈S.
Put
(100)M*=max{∥x1-x*∥,1τ-γL∥γVx*-μFx*∥}.
Set D={y∈C:∥y-x*∥≤M*}. We remark that D is nonempty, bounded, closed, and convex set, and {xn}, {yn}, and {zn} are in D. We will show that
(101)limsupn→∞supy∈D∥T(μn)y-T(t)T(μn)y∥=0,∀t∈S.
To complete our proof, we follow the proof line as in [31] (see also [23, 32, 33]). Let ϵ>0. By [34, Theorem 1.2], there exists δ>0 such that
(102)co¯Fδ(T(t);D)+Bδ⊂Fϵ(T(t);D),∀t∈S.
Also by [34, Corollary 1.1], there exists a natural number N such that
(103)∥1N+1∑i=0N‍T(tis)y-T(t)(1N+1∑i=0N‍T(ti)y)∥≤δ,
for all t,s∈S and y∈D. Let t∈S. Since {μn} is strongly left regular, there exists n0∈ℕ such that ∥μn-lti*μn∥≤δ/(M*+∥w∥) for all n≥n0 and i=1,2,…,N. Then, we have
(104)supy∈D∥T(μn)y-∫‍1N+1∑i=0N‍T(tis)ydμn(s)∥=supy∈Dsup∥z∥=1|∑i=0N‍〈T(μn)y,z〉-〈∫‍1N+1∑i=0N‍T(tis)ydμn(s),z〉|=supy∈Dsup∥z∥=1|1N+1∑i=0N‍(μn)s〈T(s)y,z〉-1N+1∑i=0N‍(μn)s〈T(ti)y,z〉|≤1N+1×∑i=0N‍supy∈Dsup∥z∥=1|(μn)s〈T(s)y,z〉-(lti*μn)s〈T(s)y,z〉|≤maxi=1,2,3,…,N∥μn-lti*μn∥(M*+∥w∥)≤δ,∀n≥n0.
On the other hand, by Lemma 2, we have
(105)∫‍1N+1∑i=0N‍T(tis)ydμn(s)∈co¯{1N+1∑i=0N‍T(ti)T(s)y:s∈S}.
Combining (103)–(105), we have
(106)T(μn)y=1N+1∑i=1N‍T(tis)ydμn(s)+(T(μn)y-∫‍1N+1∑i=1N‍T(tis)ydμn(s))∈co¯{1N+1∑i=0N‍T(ti)(T(s)y):s∈S}+Bδ⊂co¯Fδ(T(t);D)+Bδ,
for all y∈D and n≥n0. Therefore,
(107)limsupn→∞supy∈D∥T(μn)y-T(t)T(μn)y∥≤ϵ.
Since ϵ>0 is arbitrary, we obtain that (101) holds. Let t∈S and ϵ>0. Then, there exists δ>0 satisfying (102). From (101) and condition (C2), there existsa,b∈(0,1) such that 0<a≤βn≤b<1 and T(μn)y∈Fδ(T(t);D) for all y∈D. From (69), there exists k0∈ℕ such that ∥xn-T(μn)yn∥<δ/b for all n>k0. Then, from (102) and (106), we have
(108)xn+1=βnxn+(1-βn)T(μn)yn=T(μn)yn+βn(xn-T(μn)yn)∈Fδ(T(t);D)+Bδ⊂Fϵ(T(t);D),
for all n>k0. Hence, limsupn→∞∥xn-T(t)xn∥≤ϵ. Since ϵ>0 is arbitrary, we obtain that (99) holds.
Next, we show that(109)limsupn→∞〈γVx^-μFx^,zn-x^〉≤0,
where x^=Pℱ(I-μF+γV)x^. To show this, we choose a subsequence {xni} of {xn} such that
(110)limsupn→∞〈γVx^-μFx^,xn-x^〉=limi→∞〈γVx^-μFx^,xni-x^〉.
Since {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀v. Now, we show that v∈ℱ.(i)
We first show thatv∈Fix(𝒮). From (99), we have ∥xn-T(t)xn∥→0 as n→∞ for all t∈S. Then, from Demiclosedness Principle 2.6, we get v∈Fix(𝒮).(ii)
We show thatv∈Fix(K), where K is defined as in Lemma 9. Then, from (97), we have
(111)∥yn-Kyn∥=∥Kun-Kyn∥≤∥un-yn∥⟶0asn⟶∞,
and from (98), we also have ∥xn-Kxn∥→0. By Demiclosedness Principle 2.6, we get v∈Fix(K).(iii)
We show thatv∈⋂k=1N‍GMEP(Θk,φk,Ψk). Note that Gnkxn=Srk,n(Θk,φk)(I-rk,nΨk)Gnk-1xn, for all k∈{1,2,…,N}. Then, we have
(112)Θk(Gnkxn,y)+φk(y)-φk(Gnk)+〈ΨkGnk-1xn,y-Gnkxn〉+1rk,n〈y-Gnkxn,Gnkxn-Gnk-1xn〉≥0.Replacing n by ni in the last inequality and using (A2), we have
(113)φk(y)-φk(Gnikxni)+〈ΨkGnik-1xni,y-Gnikxni〉+1rk,n〈y-Gnikxni,Gnikxni-Gnik-1xni〉≥Θk(y,Gnikxni).
Let yt=ty+(1-t)v for all t∈(0,1] and y∈C. This implies that yt∈C. Then, we have
(114)〈yt-Gnikxni,Ψkyt〉≥φk(Gnikxni)-φk(yt)+〈yt-Gnikxni,Ψkyt〉-〈yt-Gnikxni,ΨkGnk-1xni〉-〈yt-Gnikxni,Gnikxni-Gnik-1xnirk,ni〉+Θk(yt,Gnikxni)=φk(Gnikxni)-φk(yt)+〈yt-Gnikxni,Ψkyt-ΨkGnikxni〉+〈yt-Gnikxni,ΨkGnikxni-ΨkGnik-1xni〉-〈yt-Gnikxni,Gnikxni-Gnik-1xnirk,ni〉+Θk(yt,Gnikxni).
From (73), we have ∥ΨkGnikxni-ΨkGnik-1xni∥→0 as i→∞. Furthermore, by the monotonicity of Ψk, we obtain 〈yt-Gnikxni,Ψkyt-ΨkGnikxni〉≥0. Then, from (A4), we obtain
(115)〈yt-v,Ψkyt〉≥φk(v)-φk(yt)+Θk(yt,v).
Using (A1), (A4), and (115), we also obtain
(116)0=Θk(yt,yt)+φk(yt)-φk(yt)≤tΘk(yt,y)+(1-t)Θk(yt,v)+(1-t)φk(v)-φk(yt)≤t[Θk(yt,y)+φk(y)-φk(yt)]+(1-t)〈yt-v,Ψkyt〉=t[Θk(yt,y)+φk(y)-φk(yt)]+(1-t)t〈y-v,Ψkyt〉,
and, hence,
(117)0≤Θk(yt,y)+φk(y)-φk(yt)+(1-t)〈y-v,Ψkyt〉.
Letting t→0 and using (A3), we have, for each y∈C,
(118)0≤Θk(v,y)+φk(y)-φk(v)+〈y-v,Ψkv〉.
This implies that v∈GMEP(Θk,φk,Ψk). Hence, v∈⋂k=1N‍GMEP(Θk,φk,Ψk). Therefore,
(119)v∈ℱ∶=⋂k=1N‍GMEP(Θk,φk,Ψk)∩Fix(K)∩Fix(𝒮).
From (66) and (110), we obtain
(120)limsupn→∞〈γVx^-μFx^,zn-x^〉=limsupn→∞〈γVx^-μFx^,xn-x^〉=limi→∞〈γVx^-μFx^,xni-x^〉=〈γVx^-μFx^,v-x^〉≤0.
Finally, we show that xn→x^ as n→∞. Notice that zn=PCvn, where vn=αnγVxn+(I-αnμF)T(μn)yn. Then, from (25), we have
(121)∥zn-x^∥2=〈vn-x^,zn-x^〉+〈PCvn-vn,PCvn-x^〉≤〈vn-x^,zn-x^〉=αnγ〈Vxn-Vx^,zn-x^〉+αn〈γVx^-μFx^,zn-x^〉+〈(I-αnμF)(T(μn)yn-x^),zn-x^〉≤(1-αn(1-βn)(τ-γL))∥xn-x^∥2+αn〈γVx^-μFx^,zn-x^〉.
It follows from (121) that
(122)∥xn+1-x^∥2≤βn∥xn-x^∥2+(1-βn)∥zn-x^∥2≤βn∥xn-x^∥2+(1-βn)×{(1-αn(1-βn)(τ-γL))∥xn-x^∥2+αn〈γVx^-μFx^,zn-x^〉∥xn-x^∥2}≤(1-αn(1-βn)(τ-γL))∥xn-x^∥2+αn(1-βn)〈γVx^-μFx^,zn-x^〉.
Put σn∶=αn(1-βn)(τ-γL) and δn∶=αn(1-βn)〈γVx^-μFx^,zn-x^〉. Then, (122) reduces to formula
(123)∥xn+1-x^∥2≤(1-σn)∥xn-x^∥2+δn.
It is easily seen that ∑n=1∞‍σn=∞, and (using (120))
(124)limsupn→∞δnσn=1τ-γLlimsupn→∞〈γVx^-μFx^,zn-x^〉≤0.
Hence, by Lemma 7, we conclude that xn→x^ as n→∞. This completes the proof.Using the results proved in [35] (see also [32]), we obtain the following results.Corollary 12.
LetC, H, Θk, φk, Ψk, Ak, F, and V be the same as in Theorem 11. Let S and T be nonexpansive mappings on C with ST=TS. Assume that ℱ∶=Fix(S)∩Fix(T)∩⋂n=1∞‍GMEP(Θk,φk,Ψk)∩Fix(K)≠∅, where K is defined as in Lemma 9. Let {αn}, {βn}, and {rk,n}k=1N be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(125)un=SrN,n(ΘN,φN)(I-rN,nΨN)SrN-1,n(ΘN-1,φN-1)×(I-rN-1,nΨN-1)⋯Sr1,n(Θ1,φ1)(I-r1,nΨ1)xn,yn=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)⋯Sλ1G1(I-λ1A1)un,xn+1=βnxn+(1-βn)×PC[αnγVxn+(I-αnμF)1n2∑i=0n-1‍∑j=0n-1‍SiTjyn],∀n≥1,
converges strongly to x^∈ℱ, where x^ solves uniquely the variational inequality (40).Corollary 13.
LetC, H, Θk, φk, Ψk, Ak, F, and V be the same as in Theorem 11. Let 𝒮={T(t):t>0} be a strongly continuous nonexpansive semigroup on C. Assume that Ω∶=Fix(𝒮)∩⋂n=1∞‍GMEP(Θk,φk,Ψk)∩Fix(K)≠∅, where K is defined as in Lemma 9. Let {αn}, {βn}, and {rk,n}k=1N be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(126)un=SrN,n(ΘN,φN)(I-rN,nΨN)SrN-1,n(ΘN-1,φN-1)×(I-rN-1,nΨN-1)⋯Sr1,n(Θ1,φ1)(I-r1,nΨ1)xn,yn=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)⋯Sλ1G1(I-λ1A1)un,xn+1=βnxn+(1-βn)×PC[αnγVxn+(I-αnμF)1tn∫0tn‍T(s)ynds],∀n≥1,
where {tn} is an increasing sequence in (0,∞) with limn→∞(tn/tn+1)=1, converges strongly to x^∈ℱ, where x^ solves uniquely the variational inequality (40).Corollary 14.
LetC, H, Θk, φk, Ψk, Ak, F, and V be the same as in Theorem 11. Let 𝒮={T(t):t>0} be a strongly continuous nonexpansive semigroup on C. Assume that Ω∶=Fix(𝒮)∩⋂n=1∞‍GMEP(Θk,φk,Ψk)∩Fix(K)≠∅, where K is defined as in Lemma 9. Let {αn}, {βn}, and {rk,n}k=1N be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(127)un=SrN,n(ΘN,φN)(I-rN,nΨN)SrN-1,n(ΘN-1,φN-1)×(I-rN-1,nΨN-1)⋯Sr1,n(Θ1,φ1)(I-r1,nΨ1)xn,yn=SλMGM(I-λMAM)SλM-1GM-1×(I-λM-1AM-1)⋯Sλ1G1(I-λ1A1)un,xn+1=βnxn+(1-βn)PC×[(an∫0∞‍exp(-ans)T(s)yn)αnγVxn+(I-αnμF)×(an∫0∞‍exp(-ans)T(s)yn)ds],∀n≥1,
where {an} is a decreasing sequence in (0,∞) with limn→∞an=0, converges strongly to x^∈Ω, where x^ solves uniquely the variational inequality (40).


## 4. Some Applications

In this section, as applications, we will apply Theorem11 to find minimum-norm solutions x^=PΩ(0) of some variational inequalities. Namely, find a point x^ which solves uniquely the following quadratic minimization problem:
(128)∥x^∥2=minx∈Ω∥x∥2.
Minimum-norm solutions have been applied widely in several branches of pure and applied sciences, for example, defining the pseudoinverse of a bounded linear operator, signal processing, and many other problems in a convex polyhedron and a hyperplane (see [36, 37]).Recently, some iterative methods have been studied to find the minimum-norm fixed point of nonexpansive mappings and their generalizations (see, e.g. [38–49] and the references therein).Using Theorem11 and Corollaries 12, 13, and 14, we immediately have the following results, respectively.Theorem 15.
LetC and H be the same as in Theorem 11. Let 𝒮={T(t):t∈S} be a nonexpansive semigroup on C such that ℱ∶=Fix(𝒮)≠∅. Let {αn} and {βn} be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(129)xn+1=βnxn+(1-βn)PC[(1-αn)T(μn)xn],∀n≥1,
converges strongly to x^∈ℱ, where x^=Pℱ(0) is the minimum-norm fixed point of ℱ, where x^ solves uniquely the quadratic minimization problem (128).Theorem 16.
LetC and H be the same as in Corollary 12. Let S and T be nonexpansive mappings on C with ST=TS such that ℱ∶=Fix(S)∩Fix(T)≠∅. Let {αn} and {βn} be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(130)xn+1=βnxn+(1-βn)PC×[(1-αn)1n2∑i=0n-1‍∑j=0n-1‍SiTjxn],∀n≥1,
converges strongly to x^∈ℱ, where x^=Pℱ(0) is the minimum-norm fixed point of ℱ, where x^ solves uniquely the quadratic minimization problem (128).Theorem 17.
LetC and H be the same as in Corollary 13. Let 𝒮={T(t):t>0} be a strongly continuous nonexpansive semigroup on C such that ℱ∶=Fix(𝒮)≠∅. Let {αn} and {βn} be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(131)xn+1=βnxn+(1-βn)PC[(1-αn)1tn∫0tn‍T(s)xnds],∀n≥1,
where {tn} is an increasing sequence in (0,∞) with limn→∞(tn/tn+1)=1, converges strongly to x^∈ℱ, where x^=Pℱ(0) is the minimum-norm fixed point of ℱ, where x^ solves uniquely the quadratic minimization problem (128).Theorem 18.
LetC and H be the same as in Corollary 14. Let 𝒮={T(t):t>0} be a nonexpansive semigroup on C such that ℱ∶=Fix(𝒮)≠∅. Let {αn} and {βn} be sequences satisfying (C1)–(C3). Then, the sequence {xn} defined by
(132)xn+1=βnxn+(1-βn)PC×[(1-αn)(an∫0∞‍exp(-ans)T(s)xnds)],∀n≥1,
where {an} is a decreasing sequence in (0,∞) with limn→∞an=0, converges strongly to x^∈ℱ, where x^=Pℱ(0) is the minimum-norm fixed point of ℱ, where x^ solves uniquely the quadratic minimization problem (128).



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