# A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

Authors: Rattanaporn Wangkeeree; Uthai Kamraksa; Rabian Wangkeeree
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a
general composite algorithm for finding a common
element of  the set of solutions of a general equilibrium
problem and  the common fixed point set of a finite family of
asymptotically  nonexpansive mappings in the framework of Hilbert
spaces. Strong convergence of such iterative scheme is obtained
which solving some variational inequalities for a strongly
monotone and strictly pseudocontractive mapping. Our results
extend the corresponding recent results of Yao and Liou (2010).


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## Body

## 1. Introduction

LetC be a nonempty, closed, convex subset of a real Hilbert space H. Recall that a mapping A:C→H is called α-inverse-strongly monotone if there exists a positive real number α such that 〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2, for all x,y∈C.  It is clear that any α-inverse-strongly monotone mapping is monotone and 1/α-Lipschitz continuous. Let f:C→H be a ρ-contraction, that is, there exists a constant ρ∈[0,1) such that ∥f(x)-f(y)∥≤ρ∥x-y∥ for all x,y∈C. A mapping S:C→C is said to be nonexpansive if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C and asymptotically nonexpansive [1] if there exists a sequence {kn}⊂[0,∞) with limn→∞kn=0 such that(1.1)‖Snx-Sny‖≤(1+kn)‖x-y‖,∀x,y∈C.
Denote the set of fixed points of S by Fix(S). For asymptotically nonexpansive self-map S, it is well known that Fix(S) is closed and convex (see, e.g., [1]).The class of asymptotically nonexpansive mappings which is an important generalization of that of nonexpansive mappings was introduced by Goebel and Kirk [1]. They established that if C is a nonempty, closed, convex, bounded subset of a uniformly convex Banach space E and S is an asymptotically nonexpansive self-mapping of C, then S has a fixed point in C.LetA:C→H be a nonlinear mapping and ϕ:C×C→ℝ a bifunction. Consider a general equilibrium problem:(1.2)Findz∈Csuchthatϕ(z,y)+〈Az,y-z〉≥0,∀y∈C.
The set of all solutions of the general equilibrium problem (1.2) is denoted by EP, that is, (1.3)EP⁡={z∈C:ϕ(z,y)+〈Az,y-z〉≥0,∀y∈C}.
If A=0, then (1.2) reduces to the following equilibrium problem of finding z∈C such that(1.4)ϕ(z,y)≥0,∀y∈C.
If ϕ=0, then (1.2) reduces to the variational inequality problem of finding z∈C such that(1.5)〈Az,y-z〉≥0,∀y∈C.
We note that the problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others. See, for example, [2–5].In 2005, Combettes and Hirstoaga [6] introduced an iterative algorithm of finding the best approximation to the initial data and proved a strong convergence theorem. In 2007, by using the viscosity approximation method, S. Takahashi and W. Takahashi [7] introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping. Subsequently, algorithms constructed for solving the equilibrium problems and fixed point problems have further developed by some authors. In particular, Ceng and Yao [8] introduced an iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem (1.2) and the set of common fixed points of finitely many nonexpansive mappings. Maingé and Moudafi [9] introduced an iterative algorithm for equilibrium problems and fixed point problems. Yao et al. [10] considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite nonexpansive mappings. Noor et al. [11] introduced an iterative method for solving fixed point problems and variational inequality problems. Wangkeeree [12] introduced a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. Wangkeeree and Kamraksa [13] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Their results extend and improve many results in the literature. For some works related to the equilibrium problem, fixed point problems, and the variational inequality problem, please see [1–57] and the references therein.However, we note that all constructed algorithms in [7, 9–13, 16, 57] do not work to find the minimum-norm solution of the corresponding fixed point problems and the equilibrium problems. Very recently, Yao and Liou [46] purposed some algorithms for finding the minimum-norm solution of the fixed point problems and the equilibrium problems. They first suggested two new composite algorithms (one implicit and one explicit) for solving the above minimization problem. To be more precisely, let C be a nonempty, closed, convex subset of H, ϕ:C×C→ℝ a bifunction satisfying certain conditions, and S:C→C a nonexpansive mapping such that Ω:=Fix(S)∩EP≠∅. Let f be a contraction on a Hilbert space H. For given x0∈C arbitrarily, let the sequence {xn} be generated iteratively by(1.6)ϕ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,xn+1=μnPC[αnf(xn)+(1-αn)Sxn]+(1-μn)un,n≥0,
where A is an α-inverse strongly monotone mapping. They proved that if {αn} and {μn} are two sequences in [0,1] satisfying the following conditions:(i)limn→∞αn=0, ∑n=0∞αn=∞ and limn→∞(αn+1/αn)=1,(ii)0<liminfn→∞μn≤limsupn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=0,       then, the sequence {xn} generated by (1.6) converges strongly to x*∈Ω which is the unique solution of variational inequality (1.7)〈(I-f)x*,x-x*〉≥0,x∈Ω.
In particular, if we take f=0 in (1.6), then the sequence {xn} generated by(1.8)ϕ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,xn+1=μnPC[(1-αn)Sxn]+(1-μn)un,n≥0,
converges strongly to a solution of the minimization problem which is the problem of finding x* such that(1.9)x*=argmin⁡x∈Ω‖x‖2,
where Ω stands for the intersection set of the solution set of the general equilibrium problem and the fixed points set of a nonexpansive mapping.On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [25, 43, 44] and the references therein. Let B be a strongly positive bounded linear operator on H, that is, there is a constant γ¯>0 with property(1.10)〈Bx,x〉≥γ¯‖x‖2∀x∈H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H(1.11)min⁡x∈Fix⁡(S)12〈Bx,x〉-〈x,b〉,
where b is a given point in H. In 2003, Xu [43] proved that the sequence {xn} defined by the iterative method below, with the initial guess x0∈H chosen arbitrarily:(1.12)xn+1=(I-αnB)Txn+αnu,n≥0,
converges strongly to the unique solution of the minimization problem (1.11) provided the sequence {αn} satisfies certain conditions. Using the viscosity approximation method, Moudafi [29] introduced the following iterative process for nonexpansive mappings (see [43] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x0∈H, define a sequence {xn} recursively by(1.13)xn+1=(1-αn)Txn+αnf(xn),n≥0,
where {αn} is a sequence in (0,1). It is proved [29, 43] that under certain appropriate conditions imposed on {αn}, the sequence {xn} generated by (1.13) strongly converges to the unique solution x* in C of the variational inequality(1.14)〈(I-f)x*,x-x*〉≥0,x∈H.
Recently, Marino and Xu [28] mixed the iterative method (1.12) and the viscosity approximation method (1.13) introduced by Moudafi [29] and considered the following general iterative method:(1.15)xn+1=(I-αnB)Txn+αnγf(xn),n≥0,
where B is a strongly positive bounded linear operator on H. They proved that if the sequence {αn} of parameters satisfies the certain conditions, then the sequence {xn} generated by (1.15) converges strongly to the unique solution x* in H of the variational inequality(1.16)〈(B-γf)x*,x-x*〉≥0,x∈H
which is the optimality condition for the minimization problem: minx∈Fix(S)(1/2)〈Bx,x〉-h(x), where h is a potential function for γf(i.e.,h′(x)=γf(x) for x∈H).Recall that a mappingF:H→H is called   δ-strongly monotone if there exists a positive constant δ such that(1.17)〈Fx-Fy,x-y〉≥δ‖x-y‖2,∀x,y∈H.
Recall also that a mapping F is called  λ-strictly pseudocontractive if there exists a positive constant λ such that(1.18)〈Fx-Fy,x-y〉≤‖x-y‖2-λ‖(x-y)-(Fx-Fy)‖2,∀x,y∈H.
It is easy to see that (1.18) can be rewritten as(1.19)〈(I-F)x-(I-F)y,x-y〉≥λ‖(I-F)x-(I-F)y‖2.Remark 1.1.
IfF is a strongly positive bounded linear operator on H with coefficient γ¯, then F is γ¯-strongly monotone and 1⁄2-strictly pseudocontractive. In fact, since F is a strongly positive, bounded, linear operator with coefficient γ¯, we have
(1.20)〈Fx-Fy,x-y〉=〈F(x-y),x-y〉≥γ¯‖x-y‖2.
Therefore, F is γ¯-strongly monotone. On the other hand,
(1.21)‖(I-F)x-(I-F)y‖2=〈(x-y)-(Fx-Fy),(x-y)-(Fx-Fy)〉=〈x-y,x-y〉-2〈Fx-Fy,x-y〉+〈Fx-Fy,Fx-Fy〉=‖x-y‖2-2〈Fx-Fy,x-y〉+‖Fx-Fy‖2≤‖x-y‖2-2〈Fx-Fy,x-y〉+‖F‖2‖x-y‖2.
Since F is strongly positive if and only if (1/∥F∥)F is strongly positive, we may assume, without loss of generality, that ∥F∥=1. From (1.21), we have
(1.22)〈Fx-Fy,x-y〉≤‖x-y‖2-12‖(I-F)x-(I-F)y‖2=‖x-y‖2-12‖(x-y)-(Fx-Fy)‖2.
Hence, F is 1⁄2-strictly pseudocontractive.In this paper, motivated by the above results, we introduce a general iterative scheme below in a real Hilbert spaceH, with the initial guess x0∈C chosen arbitrary:(1.23)ϕ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,yn=αnγf(xn)+(I-αnF)Si(n+1)p(n+1)xn,xn+1=μnPC[yn]+(1-μn)un,n≥0,
where p(n)=j+1 if jN<n≤(j+1)N, j=1,2,… and n=jN+i(n), i(n)∈{1,2,…,N}, C is a nonempty, closed, convex subset of H, {αn} and {μn} are two sequences in [0,1], ϕ:C×C→ℝ is a bifunction satisfying certain conditions, S1,S2,…,SN:C→C is a finite family of asymptotically nonexpansive mappings with sequences {1+kp(n)i(n)}, respectively, f:C→H is a contraction with coefficient 0<ρ<1, F is δ-strongly monotone and λ-strictly pseudocontractive with δ+λ>1, γ is a positive real number such that γ<(1/ρ)(1-(1-δ)/λ), and A is an α-inverse strongly monotone mapping. We prove that the proposed algorithm converges strongly to x*∈Ω which is the unique solution of the following variational inequality: (1.24)〈(F-γf)x*,x-x*〉≥0,x∈Ω.
In particular,(I)
ifF is a strongly positive bounded linear operator on H, then x* is the unique solution of the variational inequality (1.16),(II)
ifF=I, the identity mapping on H and γ=1, then x* is the unique solution of the variational inequality (1.14),(III)
ifF=I, the identity mapping on H and f=0, then x* is the unique solution of minimization problem (1.9).The results presented in this paper extend and improve the main results in Yao and Liou [46], Marino and Xu [28], and many others.


## 2. Preliminaries

LetC be a nonempty, closed, convex subset of a real Hilbert space H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx such that (2.1)‖x-PCx‖≤‖x-y‖,∀y∈C.PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive mapping of H onto C and satisfies(2.2)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,
for every x,y∈H. Moreover, PCx is characterized by the following properties: PCx∈C and(2.3)〈x-PCx,y-PCx〉≤0,‖x-y‖2≥‖x-PCx‖2+‖y-PCx‖2,
for all x∈H,y∈C. For more details, see [39]. We will make use of the following well-known result.Lemma 2.1.
LetH be a Hilbert space. Then, the following inequality holds:
(2.4)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.Throughout this paper, we assume that a bifunctionϕ:C×C→ℝ satisfies the following conditions:  (A1)
ϕ(x,x)=0 for all x∈C,(A2)
ϕ is monotone, that is, ϕ(x,y)+ϕ(y,x)≤0 for all x,y∈C,(A3)
for eachx,y,z∈C, limt↓0ϕ(tz+(1-t)x,y)≤ϕ(x,y),(A4)
for eachx∈C, the mapping y↦ϕ(x,y) is convex and lower semicontinuous.We need the following lemmas for proving our main results.Lemma 2.2 (see [6]).
LetC be a nonempty, closed, convex subset of a real Hilbert space H. Let ϕ:C×C→ℝ be a bifunction which satisfies conditions (A1)–(A4). Let r>0 and x∈C. Then, there exists z∈C such that
(2.5)ϕ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, if Tr(x)={z∈C:ϕ(z,y)+(1/r)〈y-z,z-x〉≥0,∀y∈C}, then the following hold:  (i)
Tr  is single-valued and Tr is firmly nonexpansive, that is, for any x,y∈H,(2.6)‖Trx-Try‖2≤〈Trx-Try,x-y〉,(ii)
EP is closed and convex and EP=Fix(Tr).Lemma 2.3 (see [30]).
LetC be a nonempty, closed, convex subset of a real Hilbert space H. Let the mapping A:C→H be α-inverse strongly monotone and r>0 a constant. Then, one has
(2.7)‖(I-rA)x-(I-rA)y‖2≤‖x-y‖2+r(r-2α)‖Ax-Ay‖2,∀x,y∈C.
In particular, if 0≤r≤2α, then I-rA is nonexpansive.Lemma 2.4 (see [45]).
LetS be an asymptotically nonexpansive mapping defined on a bounded, closed, convex subset C of a Hilbert space H. If {xn} is a sequence in C such that xn⇀x and ∥Sxn-xn∥→0 as n→∞, then x∈Fix(S).Lemma 2.5 (see [44]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.8)an+1≤(1-αn)an+αnσn+γn,n≥0,
where {αn},{σn}, and {γn} are nonnegative real sequences satisfying the following conditions:  (i)
{αn}⊂[0,1], ∑n=1∞αn=∞,(ii)
limsupn→∞σn≤0,(iii)
∑n=1∞γn<∞.Then, limn→∞an=0.Lemma 2.6 (see [41]).
LetE be a strictly convex Banach space and C a closed, convex subset of E. Let S1,S2,…,SN:C→C be a finite family of nonexpansive mappings of C into itself such that the set of common fixed points of S1,S2,…,SN is nonempty. Let T1,T2,…,TN:C→C be mappings given by
(2.9)Ti=(1-αi)I+αiSi,∀i=1,2,…,N,
where I denotes the identity mapping on C. Then, the finite family {T1,T2,…,TN} satisfies the following:
(2.10)⋂i=1NFix⁡(Ti)=⋂i=1NFix⁡(Si),⋂i=1NFix⁡(Ti)=Fix⁡(TNTN-1TN-2⋯T1)=Fix⁡(T1TN⋯T2)=Fix⁡(TN-1TN-2⋯T1TN).The following lemma can be found in [35, Lemma 2.7]. For the sake of the completeness, we include its proof in a Hilbert space's version.Lemma 2.7.
LetH be a real Hilbert space and F:H→H a mapping.  (i)
IfF is δ-strongly monotone and λ-strictly pseudocontractive with δ+λ>1, then I-F is contractive with constant (1-δ)/λ.(ii)
IfF is δ-strongly monotone and λ-strictly pseudocontractive with δ+λ>1, then for any fixed number τ∈(0,1), I-τF is contractive with constant 1-τ(1-(1-δ)/λ).Proof.
(i) For anyx,y∈H, we have
(2.11)λ‖(I-F)x-(I-F)y‖2≤‖x-y‖2-〈Fx-Fy,x-y〉≤(1-δ)‖x-y‖2,∀x,y∈H.
Thus,
(2.12)‖(I-F)x-(I-F)y‖≤1-δλ‖x-y‖,∀x,y∈H.
Since δ+λ>1, we have (1-δ)/λ∈(0,1). Hence, I-F is contractive with constant (1-δ)/λ.
(ii) SinceI-F is contractive with constant (1-δ)/λ, we have for any τ∈(0,1),
(2.13)‖x-y-τ(Fx-Fy)‖=‖(1-τ)(x-y)+τ[(I-F)x-(I-F)y]‖≤(1-τ)‖x-y‖+τ‖(I-F)x-(I-F)y‖≤(1-τ)‖x-y‖+τ1-δλ‖x-y‖=(1-τ(1-1-δλ))‖x-y‖,∀x,y∈H.
Hence, I-τF is contractive with constant 1-τ(1-(1-δ)/λ).Lemma 2.8.
LetS1,S2,…,SN:C→C be a finite family of asymptotically nonexpansive mappings with sequences {1+kp(n)i(n)}, respectively, such that kp(n)i(n)→0 as n→∞. Then, there exists a sequence {hn}⊂[0,∞) with hn→0 as n→∞ such that
(2.14)‖Si(n)p(n)x-Si(n)p(n)y‖≤(1+hn)‖x-y‖,∀x,y∈C,
where p(n)=j+1 if jN<n≤(j+1)N, j=1,2,… and n=jN+i(n); i(n)∈{1,2,…,N}.Proof.
Define the sequence{hn} by hn:=max{kp(n)i(n):1≤i(n)≤N} and the result follows immediately.In the rest of our discussion in this paper, we will assume thatp(n)=j+1 if jN<n≤(j+1)N, j=1,2,… and n=jN+i(n); i(n)∈{1,2,…,N} and hn:=max{kp(n)i(n):1≤i(n)≤N} for all n≥1, and for each n≥1, n=(p(n)-1)N+i(n).


## 3. Main Results

Now, we are a position to state and prove our main results.Theorem 3.1.
LetC be a nonempty, closed, convex subset of a real Hilbert space H. Let S1,S2,…,SN:C→C be a finite family of asymptotically nonexpansive mappings with sequences {1+kp(n)i(n)}, respectively, such that kp(n)i(n)→0 as n→∞, hn:=max1≤i(n)≤N{kp(n)i(n)} and Γ:=⋂i=1NFix(Si),
(3.1)Γ=Fix⁡(SNSN-1SN-2⋯S1)=Fix⁡(S1SN⋯S2)=⋯=Fix⁡(SN-1SN-2⋯S1SN).
Let A:C→H be an α-inverse strongly monotone mapping. Let ϕ:C×C→ℝ be a bifunction which satisfies conditions (A1)–(A4) such that Ω:=EP∩Γ is nonempty. Let F:C→H be δ-strongly monotone and λ-strictly pseudocontractive with δ+λ>1, f:C→H a ρ-contraction, γ a positive real number such that γ<(1-(1-δ)/λ)/ρ, and r a constant such that r∈(0,2α). For x0∈C arbitrarily, let the sequence {xn} be generated iteratively by (1.23). Suppose that {αn} and {μn} are two sequences in [0,1] satisfying the following conditions:  (C1)
limn→∞αn=0, limn→∞(αn+1/αn)=1, ∑n=0∞αn=∞ and limn→∞(hn/αn)=0,(C2)
0<liminfn→∞μn≤limsupn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=0. Assume that ∑n=1∞supz∈B∥Si(n+1)p(n+1)z-Si(n)p(n)z∥<∞, for each bounded subset B of C. Then, the sequence {xn} converges strongly to x* of the following variational inequality:
(3.2)〈(F-γf)x*,x-x*〉≥0,x∈Ω
or equivalently x̃=PΩ(I-F+γf)x̃, where PΩ is the metric projection of H onto Ω.Proof.
First, we rewrite the sequence{xn} by the following:
(3.3)xn+1=μnPC[yn]+(1-μn)Tr(xn-rAxn),n≥0,
where the mapping Tr is defined in Lemma 2.2. Pick z∈Ω and un=Tr(xn-rAxn). The nonexpansivity of Tr and I-rA implies that
(3.4)‖un-z‖=‖Tr(xn-rAxn)-Tr(z-rAz)‖≤‖xn-z‖,∀z∈Ω.
Setting γ¯:=(1-(1-δ)/λ) and using Lemma 2.7(ii), we have
(3.5)‖yn-z‖=‖αnγ(f(xn)-Fz)+(I-αnF)(Si(n+1)p(n+1)xn-z)‖≤ααnγ‖xn-z‖+αn‖γf(xn)-Fz‖+(1-αnγ¯)(1+hn+1)‖xn-z‖=[1-αn(γ¯-αγ)+(1-αnγ¯)hn+1]‖xn-z‖+αn‖γf(xn)-Fz‖.
By our assumptions, we have (1-αnγ¯)(hn+1/αn)→0 as n→∞. We can assume, without loss of generality, that (1-αnγ¯)(hn+1/αn)<(1/2)(γ¯-αγ). Applying Lemma 2.7, we can calculate the following:
(3.6)‖xn+1-z‖=‖μn(PC[yn]-z)+(1-μn)(un-z)‖≤μn‖PC[yn]-z‖+(1-μn)‖un-z‖≤μn‖yn-z‖+(1-μn)‖xn-z‖≤μn[1-αn(γ¯-αγ)+(1-αnγ¯)hn+1]‖xn-z‖+μnαn‖γf(xn)-Fz‖+(1-μn)‖xn-z‖=[1-μnαn[(γ¯-αγ)-(1-αnγ¯)hn+1αn]]‖xn-z‖+μnαn‖γf(xn)-Fz‖≤[1-12μnαn(γ¯-αγ)]‖xn-z‖+μnαn(1/2)(γ¯-αγ)(1/2)(γ¯-αγ)‖γf(xn)-Fz‖.
By induction, we obtain, for all n≥0,
(3.7)‖xn-z‖≤max⁡{‖x0-z‖,2‖γf(x0)-F(z)‖γ¯-αγ}.
Hence, {xn} is bounded. Consequently, we deduce that {un}, {f(xn)}, and {yn} are all bounded.
Next, we show that(3.8)lim⁡n→∞‖xn+N-xn‖=0.
From (1.23), we have
(3.9)‖yn+N-yn+N-1‖=‖αn+Nγf(xn+N)+(I-αn+NF)Si(n+N+1)p(n+N+1)xn+N-αn+N-1γf(xn+N-1)-(I-αn+N-1F)Si(n+N)p(n+N)xn+N-1‖=‖Si(n+N+1)p(n+N+1)xn+N-1-Si(n+N)p(n+N)xn+N-1αn+Nγ(f(xn+N)-f(xn+N-1))+(αn+N-αn+N-1)γf(xn+N-1)+(I-αn+NF)(Si(n+N+1)p(n+N+1)xn+N-Si(n+N+1)p(n+N+1)xn+N-1)+[(I-αn+NF)-(I-αn+N-1F)]Si(n+N+1)p(n+N+1)xn+N-1+(I-αn+N-1F)(Si(n+N+1)p(n+N+1)xn+N-1-Si(n+N)p(n+N)xn+N-1)‖≤αn+Nγα‖xn+N-xn+N-1‖+|αn+N-αn+N-1|γ‖f(xn+N-1)‖+(1-αn+Nγ¯)(1+hn+N+1)‖xn+N-xn+N-1‖+|αn+N-1-αn+N|‖F‖‖Si(n+N+1)p(n+N+1)xn+N-1‖+(1-αn+N-1γ¯)‖Si(n+N+1)p(n+N+1)xn+N-1-Si(n+N)p(n+N)xn+N-1‖≤αn+Nγα‖xn+N-xn+N-1‖+|αn+N-αn+N-1|γ‖f(xn+N-1)‖+(1-αn+Nγ¯)(1+hn+N+1)‖xn+N-xn+N-1‖+|αn+N-1-αn+N|‖F‖‖Si(n+N+1)p(n+N+1)xn+N-1‖+sup⁡x∈{xn:n∈N}‖Si(n+N+1)p(n+N+1)x-Si(n+N)p(n+N)x‖,
and from (3.3), we have
(3.10)‖xn+N+1-xn+N‖=‖μn+NPC[yn+N]+(1-μn+N)un+N-μn+N-1PC[yn+N-1]-(1-μn+N-1)un+N-1‖=‖μn+N(PC[yn+N]-PC[yn+N-1])+(μn+N-μn+N-1)PC[yn+N-1]+(1-μn+N)(un+N-un+N-1)+(μn+N-1-μn+N)un+N-1‖≤μn+N‖yn+N-yn+N-1‖+(1-μn+N)‖un+N-un+N-1‖+|μn+N-μn+N-1|(‖PC[yn+N-1]‖+‖un+N-1‖),‖un+N-un+N-1‖=‖Tr(xn+N-rAxn+N)-Tr(xn+N-1-rAxn+N-1)‖≤‖(xn+N-rAxn+N)-(xn+N-1-rAxn+N-1)‖≤‖xn+N-xn+N-1‖.
Therefore,
(3.11)‖xn+N+1-xn+N‖≤μn+Nαn+Nγα‖xn+N-xn+N-1‖+μn+N|αn+N-αn+N-1|γ‖f(xn+N-1)‖+μn+N(1-αn+Nγ¯)(1+hn+N+1)‖xn+N-xn+N-1‖+μn+N|αn+N-1-αn+N|‖F‖‖Si(n+N+1)p(n+N+1)xn+N-1‖+μn+Nsup⁡x∈{xn:n∈N}‖Si(n+N+1)p(n+N+1)x-Si(n+N)p(n+N)x‖+(1-μn+N)‖xn+N-xn+N-1‖+|μn+N-μn+N-1|(‖PC[yn+N-1]‖+‖un+N-1‖)≤(1-μn+Nαn+N(γ¯-γα))‖xn+N-xn+N-1‖+μn+Nαn+N×[(hn+N+1αn+N+hn+N+1γ¯)M+|1-αn+N-1αn+N|γ‖f(xn+N-1)‖+|αn+N-1αn+N-1|‖F‖‖Si(n+N+1)p(n+N+1)xn+N-1‖+1μn+N|μn+N-μn+N-1αn+N|(‖PC[yn+N-1]‖+‖un+N-1‖)]+sup⁡x∈{xn:n∈N}‖Si(n+N+1)p(n+N+1)x-Si(n+N)p(n+N)x‖.
By Lemma 2.5, we obtain that
(3.12)lim⁡n→∞∥xn+N+1-xn+N∥=0.
Furthermore,
(3.13)∥xn+N-xn∥≤∥xn+N-xn+N-1∥+∥xn+N-1-xn+N-2∥+⋯+∥xn+1-xn∥⟶0,asn⟶∞.
Hence,
(3.14)lim⁡n→∞∥xn+N-xn∥=0.
Next, we show that
(3.15)lim⁡n→∞∥xn-un∥=0.
By the convexity of the norm ∥·∥, we have
(3.16)‖xn+1-z‖2=‖μn(PC[yn]-z)+(1-μn)(un-z)‖2≤μn‖PC[yn]-z‖2+(1-μn)‖un-z‖2≤μn‖yn-z‖2+(1-μn)‖un-z‖2=μn‖αnγf(xn)+(I-αnF)Si(n+1)p(n+1)xn-z‖2+(1-μn)‖un-z‖2=μn‖αnγf(xn)-αnF(z)+(I-αnF)Si(n+1)p(n+1)xn-(I-αnF)z‖2+(1-μn)‖un-z‖2≤μn‖(I-αnF)Si(n+1)p(n+1)xn-(I-αnF)z‖2+μnαn2‖γf(xn)-F(z)‖2+2μnαn〈(I-αnF)Si(n+1)p(n+1)xn-(I-αnF)z,γf(xn)-F(z)〉+(1-μn)‖un-z‖2≤μn(1-αnγ¯)2(1+hn+1)2‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)‖un-z‖2≤μn(1-αnγ¯)(1+2hn+1+hn+12)‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)‖un-z‖2=μn(1-αnγ¯)(1+hn+1*)‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)‖un-z‖2,
where hn+1*=2hn+1+hn+12. From Lemma 2.3, we get
(3.17)‖un-z‖2=‖Tr(xn-rAxn)-Tr(z-rAz)‖2≤‖(xn-rAxn)-(z-rAz)‖2≤‖xn-z‖2+r(r-2α)‖Axn-Az‖2.
Substituting (3.17) into (3.16), we have
(3.18)‖xn+1-z‖2≤μn(1-αnγ¯)(1+hn+1*)‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)[‖xn-z‖2+r(r-2α)‖Axn-Az‖2]=(1-αnμn(γ¯-hn+1*αn))‖xn-z‖2-αnμnhn+1γ¯‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)r(r-2α)‖Axn-Az‖2.
Therefore,
(3.19)(1-μn)r(2α-r)‖Axn-Az‖2≤(1-αnμn(γ¯-hn+1*αn))‖xn-z‖2-‖xn+1-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)×‖γf(xn)-F(z)‖‖xn-z‖≤‖xn-z‖2-‖xn+1-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖≤(‖xn-z‖+‖xn+1-z‖)‖xn-xn+1‖+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖.
Since liminfn→∞(1-μn)r(2α-r)>0, ∥xn-xn+1∥→0 and αn→0, we derive
(3.20)lim⁡n→∞‖Axn-Az‖=0.
From Lemma 2.2, we obtain
(3.21)‖un-z‖2=‖Tr(xn-rAxn)-Tr(z-rAz)‖2≤〈(xn-rAxn)-(z-rAz),un-z〉=12(‖(xn-rAxn)-(z-rAz)‖2+‖un-z‖2-‖(xn-z)-r(Axn-Az)-(un-z)‖2)≤12(‖xn-z‖2+‖un-z‖2-‖(xn-un)-r(Axn-Az)‖2)=12(‖xn-z‖2+‖un-z‖2-‖xn-un‖2+2r〈xn-un,Axn-Az〉-r2‖Axn-Az‖2).
Thus, we deduce
(3.22)‖un-z‖2≤‖xn-z‖2-‖xn-un‖2+2r‖xn-un‖‖Axn-Az‖.
By (3.16) and (3.22), we have
(3.23)‖xn+1-z‖2≤μn(1-αnγ¯)(1+hn+1*)‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)[‖xn-z‖2-‖xn-un‖2+2r‖xn-un‖‖Axn-Az‖]≤(1-αnμn(γ¯-hn+1*αn))‖xn-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)[-‖xn-un‖2+2r‖xn-un‖‖Axn-Az‖].
Therefore,
(3.24)(1-μn)‖xn-un‖2≤‖xn-z‖2-‖xn+1-z‖2+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+(1-μn)[2r‖xn-un‖‖Axn-Az‖]≤(‖xn-z‖-‖xn+1-z‖)‖xn-xn+1‖+μnαn2‖γf(xn)-F(z)‖2+2αnμn(1-αnγ¯)‖γf(xn)-F(z)‖‖xn-z‖+2r(1-μn)‖xn-un‖‖Axn-Az‖.
Since liminfn→∞(1-μn)>0, αn→0, ∥xn+1-xn∥→0 and ∥Axn-Az∥→0, we derive that
(3.25)lim⁡n→∞‖xn-un‖=0.
Next, we show that
(3.26)lim⁡n→∞‖xn-Si(n+N)Si(n+N-1)Si(n+N-2)⋯Si(n+2)Si(n+1)xn‖=0.
By using (3.14), it suffices to show that
(3.27)lim⁡n→∞‖xn+N-Si(n+N)Si(n+N-1)Si(n+N-2)⋯Si(n+2)Si(n+1)xn‖=0.
Observe that
(3.28)‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖≤‖xn+N-xn+N-1‖+‖xn+N-Si(n+N)p(n+N)xn+N-1‖≤‖xn+N-xn+N-1‖+μn+N-1‖PC[yn+N-1]-Si(n+N)p(n+N)xn+N-1‖+(1-μn+N-1)‖un+N-1-Si(n+N)p(n+N)xn+N-1‖≤‖xn+N-xn+N-1‖+μn+N-1‖yn+N-1-Si(n+N)p(n+N)xn+N-1‖+(1-μn+N-1)‖un+N-1-Si(n+N)p(n+N)xn+N-1‖≤‖xn+N-xn+N-1‖+μn+N-1αn+N-1×(‖γf(xn+N-1)‖+‖FSi(n+N)p(n+N)xn+N-1‖)+(1-μn+N-1)‖un+N-1-xn+N-1‖+(1-μn+N-1)‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖.
Hence,
(3.29)‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖≤1μn+N-1‖xn+N-xn+N-1‖+αn+N-1(‖γf(xn+N-1)‖+‖FSi(n+N)p(n+N)xn+N-1‖)+(1-μn+N-1)μn+N-1‖un+N-1-xn+N-1‖.
From (3.14), (3.25), limn→∞αn=0, and (C2), we have
(3.30)‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖⟶0asn⟶∞.
Since Si(n) is Lipschitz with constant Li(n) for each i(n)∈{1,2,…,N} and for L=max1≤i≤N{Li(n)}, and for any positive number n≥1, n=(p(n)-1)N+i(n), we have
(3.31)‖xn+N-1-Si(n+N)xn+N-1‖≤‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖+‖Si(n+N)p(n+N)xn+N-1-Si(n+N)xn+N-1‖≤‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖+L‖Si(n+N)p(n+N)-1xn+N-1-xn+N-1‖≤‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖+L(‖Si(n+N)p(n+N)-1xn+N-1-Si(n)p(n+N)-1xn-1‖+‖Si(n)p(n+N)-1xn-1-xn-1‖+‖xn-1-xn+N-1‖).
Since for each n>N, n+N=n(modN), and also n=(p(n)-1)N+i(n), so
(3.32)n+N=(p(n)-1+1)N+i(n)=(p(n+N)-1)N+i(n+N),
that is,
(3.33)p(n+N)-1=p(n),i(n+N)=i(n).
Hence,
(3.34)‖Si(n+N)p(n+N)-1xn+N-1-Si(n)p(n+N)-1xn-1‖=‖Si(n)p(n)xn+N-1-Si(n)p(n)xn-1‖≤L‖xn+N-1-xn-1‖.
Also,
(3.35)‖Si(n)p(n+N)-1xn-1-xn-1‖=‖Si(n)p(n)xn-1-xn-1‖.
Therefore, substituting (3.34) and (3.35) into (3.31), we have
(3.36)‖xn+N-1-Si(n+N)xn+N-1‖≤‖xn+N-1-Si(n+N)p(n+N)xn+N-1‖+L2‖xn+N-1-xn-1‖+L‖Si(n)p(n)xn-1-xn-1‖+L‖xn-1-xn+N-1‖.
From (3.30) and (3.14), we have
(3.37)lim⁡n→∞‖xn+N-1-Si(n+N)xn+N-1‖=0.
Also,
(3.38)‖xn+N-Si(n+N)xn+N-1‖≤‖xn+N-xn+N-1‖+‖xn+N-1-Si(n+N)xn+N-1‖,
so that
(3.39)xn+N-1-Si(n+N)xn+N-1⟶0asn⟶∞.
Indeed, noting that each Si(n) is Lipschitzian and using (3.39), we can calculate the following:
(3.40)xn+N-Si(n+N)xn+N-1⟶0asn⟶∞,Si(n+N)xn+N-1-Si(n+N)Si(n+N-1)xn+N-2asn⟶∞,⋮Si(n+N)Si(n+N-1)⋯Si(n+2)xn+1-Si(n+N)Si(n+N-1)⋯Si(n+2)Si(n+1)xn⟶0asn⟶∞.It follows from (3.40) that(3.41)xn+N-Si(n+N)Si(n+N-1)⋯Si(n+1)xn⟶0asn⟶∞.
Using (3.14), we have
(3.42)xn-Si(n+N)Si(n+N-1)⋯Si(n+1)xn⟶0asn⟶∞.
Hence (3.26) is proved. Let Φ=PΩ. Then, Φ(I-F-γf) is a contraction on C. In fact, from Lemma 2.7(i), we have
(3.43)‖Φ(I-F-γf)x-Φ(I-F-γf)y‖≤‖(I-F-γf)x-(I-F-γf)y‖≤‖(I-F)x-(I-F)y‖+γ‖f(x)-f(y)‖≤1-δλ‖x-y‖+αγ‖x-y‖=(1-δλ+αγ)‖x-y‖,∀x,y∈C.
Therefore, Φ(I-F-γf) is a contraction on C with coefficient ((1-δ)/λ+αγ)∈(0,1). Thus, by Banach contraction principal, PΩ(I-F-γf) has a uninique fixed point x*, that is PΩ(I-F-γf)x*=x* which mean that x* is the unique solution in Ω of the variational inequality (3.2). Next, we show that
(3.44)lim⁡⁡sup⁡n→∞〈γf(x*)-Fx*,xn-x*〉≤0.
Let {xnj} be a subsequence of {xn} such that
(3.45)lim⁡⁡sup⁡n→∞〈γf(x*)-Fx*,xn-x*〉=lim⁡j→∞〈γf(x*)-Fx*,xnj-x*〉.
Since {xn} is bounded, we may also assume that there exists some x̃∈H such that xnj⇀x̃. Since the family {Si}i=1N is finite, passing to a further subsequence if necessary, we may further assume, for some i(n)∈{1,2,…,N}, it follows that
(3.46)xnj-Si(n+N)Si(n+N-1)⋯Si(n+1)xnj⟶0asj⟶∞.
By Lemma 2.4, we obtain
(3.47)x̃∈F(Si(n+N)Si(n+N-1)⋯Si(n+1)),
so this implies that x̃∈Γ. Next, we show x̃∈EP. Since un=Tr(xn-rAxn), for any y∈C, we have
(3.48)ϕ(un,y)+1r〈y-un,un-(xn-rAxn)〉≥0.
From the monotonicity of F, we have
(3.49)1r〈y-un,un-(xn-rAxn)〉≥ϕ(y,un),∀y∈C.
Hence,
(3.50)〈y-un,uni-xnir+Axni〉≥ϕ(y,uni),∀y∈C.
Put zt=ty+(1-t)x̃ for all t∈(0,1] and y∈C. Then, we have zt∈C. So, from (3.50), we have
(3.51)〈zt-uni,Azt〉≥〈zt-uni,Azt〉-〈zt-uni,uni-xnir+Axni〉+ϕ(zt,uni)=〈zt-uni,Azt-Auni〉+〈zt-uni,Auni-Axni〉+〈zt-uni,uni-xnir〉+ϕ(zt,uni).
Note that ∥Auni-Axni∥≤(1/α)∥uni-xni∥→0. Further, from monotonicity of A, we have 〈zt-uni,Azt-Auni〉≥0. Letting i→∞ in (3.51), we have
(3.52)〈zt-x̃,Azt〉≥ϕ(zt,x̃).
From (A1), (A4), and (3.52), we also have
(3.53)0=ϕ(zt,zt)≤tϕ(zt,y)+(1-t)ϕ(zt,x̃)≤tϕ(zt,y)+(1-t)〈zt-x̃,Azt〉=tϕ(zt,y)+(1-t)t〈y-x̃,Azt〉
and, hence,
(3.54)0≤ϕ(zt,y)+(1-t)〈Azt,y-x̃〉.
Letting t→0 in (3.54) and using (A3), we have, for each y∈C,
(3.55)0≤ϕ(x̃,y)+〈y-x̃,Ax̃〉.
This implies that x̃∈EP. Therefore, x̃∈Ω. Therefore,
(3.56)lim⁡⁡sup⁡n→∞〈γf(x*)-Fx*,xn-x*〉=〈γf(x*)-Fx*,x̃-x*〉≤0.
Finally, we prove that xn→x* as n→∞. From Lemma 2.7 and (1.23), we obtain
(3.57)‖xn+1-x*‖2=‖μn(PC[yn]-x*)+(1-μn)(un-x*)‖2≤μn‖PC[yn]-x*‖2+(1-μn)‖un-x*‖2≤μn‖yn-x*‖2+(1-μn)‖un-x*‖2=μ‖αnγf(xn)-αnF(x*)+(I-αnF)Si(n+1)p(n+1)xn-(I-αnF)x*‖2+(1-μn)‖un-x*‖2=(1-μn)‖xn-x*‖2+μn‖αn(γf(xn)-F(x*))+(I-αnF)(Si(n+1)p(n+1)xn-x*)‖2≤(1-μn)‖xn-x*‖2+μn(1-αnγ¯)2(1+hn+1)2‖xn-x*‖2+2μnαn〈γf(xn)-F(x*),xn+1-x*〉≤(1-μn)‖xn-x*‖2+μn(1-αnγ¯)(1+2hn+1+hn+12)‖xn-x*‖2+2μnαn〈γf(xn)-F(x*),xn+1-x*〉=(1-μn)‖xn-x*‖2+μn(1-αnγ¯)(1+hn+1*)‖xn-x*‖2+2μnαn〈γf(xn)-F(x*),xn+1-x*〉≤(1-μnαn(γ¯-hn+1*αn))‖xn-x*‖2+2μnαn(γ¯-hn+1*αn)[1(γ¯-hn+1*/αn)〈γf(xn)-F(x*),xn+1-x*〉],
where hn+1*=2hn+1+hn+12. Hence, all conditions of Lemma 2.5 are satisfied. Therefore, xn→x*. This completes the proof.The following example shows that there exist the sequences{αn} and {μn} satisfying the conditions (C1) and (C2) of Theorem 3.1.Example 3.2.
For eachn≥0, let αn=1/(n+1) and μn=1/2+1/(n+1). Then, it is easy to obtain limn→∞αn=0, ∑n=0∞αn=∞ and limn→∞(αn+1/αn)=1, 0<1/2=liminfn→∞μn≤limsupn→∞μn=1/2<1 and limn→∞((μn+1-μn)/αn+1)=0. Hence, conditions (C1) and (C2) of Theorem 3.1 are satisfied.Corollary 3.3.
LetC,H,A,ϕ,Ω,f,F,r be as in Theorem 3.1. Let S1,S2,…,SN:C→C be a family of nonexpansive mappings. Let T1,T2,…,TN:C→C be mappings defined by (2.9). For Tn:=TnmodN, let the sequence {xn} be generated by
(3.58)ϕ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,xn+1=μnPC[αnγf(xn)+(1-αnF)Tnxn]+(1-μn)un,n≥0.
Assume that ∑n=1∞supz∈B∥Tn+1z-Tnz∥<∞ for each bounded subset B of C and the sequences {αn} and {μn} satisfy the following conditions:  (C1)
limn→∞αn=0, ∑n=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞μn≤limsupn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=0. Then the sequence {xn} converges strongly to x* of the following variational inequality:
(3.59)〈(F-γf)x*,x-x*〉≥0,x∈Ω,
or equivalently x̃=PΩ(I-F+γf)x̃, where PΩ is the metric projection of H onto Ω.Proof.
By Lemma2.6, we have
(3.60)⋂i=1NFix⁡(Ti)=Fix⁡(TNTN-1TN-2⋯T1)=Fix⁡(T1TN⋯T2)=Fix⁡(TN-1TN-2⋯T1TN).
Therefore, the result follows from Theorem 3.1.Remark 3.4.
As in  [58, Theorem  4.1], we can generate a sequence {Sn} of nonexpansive mappings satisfying the condition ∑n=1∞sup{∥Sn+1z-Snz∥:z∈B}<∞ for any bounded subset B of C by using convex combination of a general sequence {Tk} of nonexpansive mappings with a common fixed point.Settingγ=1, F=I, and Sn≡S, a nonexpansive mapping, in Corollary 3.3, we obtain the following result.Corollary 3.5 ([46], Theorem 3.7).
LetC, H, A, ϕ, f, r be as in Theorem 3.1. Let S:C→C be a nonexpansive mapping such that Ω:=EP∩Fix(S)≠∅. Let the sequence {xn} be generated by
(3.61)ϕ(un,y)+〈Axn,y-un〉+1r〈y-un,un-xn〉≥0,∀y∈C,xn+1=μnPC[αnf(xn)+(1-αn)Sxn]+(1-μn)un,n≥0.
Assume the sequences {αn} and {μn} satisfy the following conditions:  (C1)
limn→∞αn=0, ∑n=0∞αn=∞ and limn→∞(αn+1/αn)=1,(C2)
0<liminfn→∞μn≤limsupn→∞μn<1 and limn→∞((μn+1-μn)/αn+1)=0. Then, the sequence {xn} converges strongly to x* of the following variational inequality:
(3.62)〈(I-f)x*,x-x*〉≥0,x∈Ω.



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*Source: 976412-2011-05-12.xml*