# Hybrid Steepest Descent Viscosity Method for Triple Hierarchical Variational Inequalities

Authors: L.-C. Ceng; Q. H. Ansari; C.-F. Wen
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a triple hierarchical variational inequality problem (in short, THVIP). By combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP.


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## Body

## 1. Introduction and Formulations

LetH be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed convex subset of H and let Γ:C→H be a nonlinear mapping. The variational inequality problem (for short, VIP) is to find x*∈C such that
(1.1)〈Γx*,y-x*〉≥0,∀y∈C.
The inequality (1.1) is called a variational inequality (in short, VI). If the mapping Γ is a monotone operator, then the inequality (1.1) is called a monotone variational inequality. The theory of variational inequalities is well established in the literature because of its applications in science, engineering, social sciences, and so forth. For further detail on variational inequalities and their applications, we refer to [1–10] and the references therein.It is well known that the VI (1.1) is equivalent to the fixed point equation
(1.2)x*=PC(I-λΓ)x*,
where λ>0 and PC is the metric projection of H onto C which assigns to each x∈H the only point in C, denoted by PCx, such that
(1.3)∥x-PCx∥=infy∈C∥x-y∥.
It is known that the fixed point methods can be implemented to find a solution of the VI (1.1) provided Γ satisfies some conditions and λ>0 is chosen appropriately. For instance, if Γ is Lipschitzian and strongly monotone (i.e., 〈Γx-Γy,x-y〉≥η∥x-y∥2,forallx,y∈C for some η>0) and λ>0 is small enough, then the mapping determined by the right-hand side of (1.2) is a contraction. Hence, the Banach contraction principle guarantees that the sequence {xn} of Picard iterates, given by xn=PC(I-λΓ)xn-1(n≥1), converges strongly to a unique solution of the VI (1.1).Furthermore, it is also known that ifΓ is inverse strongly monotone (i.e., there is a constant α>0 such that 〈Γx-Γy,x-y〉≥α∥Γx-Γy∥2,forallx,y∈C), then the mapping PC(I-λΓ) is an averaged mapping (namely, there are β∈(0,1) and a nonexpansive mapping T such that PC(I-λΓ)=(1-β)I+βT), then the sequence of Picard iterates, {(PC(I-λΓ))nx0}, converges weakly to a solution of the VI (1.1) (if such a solution exists).In the last decade, the variational inequality problem is considered over the set of fixed points of a nonexpansive mapping; see, for example, [11–15] and the reference therein. In particular, Moudafi and Maingé [12] and Xu [14] considered the following VIP over the set Fix(T) of fixed points of a nonexpansive mapping T:C→C (i.e., C=Fix(T)) with Γ=I-V, where V is another nonexpansive self-mapping on C: find x*∈Fix(T) such that
(1.4)〈(I-V)x*,x-x*〉≥0,∀x∈Fix(T),
where we assume that Fix(T)≠∅. It is called hierarchical variational inequality problem (in short, HVIP). The HVIP (1.4) is equivalent to the following fixed point problem:
(1.5)findx*∈Csuchthatx*=PFix(T)∘V(x*).
Let S denote the solution set of the HVIP (1.4). It has been shown in [12] that the HVIP (1.4) contains the HVIP considered in [15], monotone inclusion problem, convex programming problem, minimization problem over a set of fixed points, and so forth, as special cases; see, for example, [12, 14] and the references therein. In the recent past, several kinds of approximation methods for computing the approximate solutions of HVIP are proposed; see, for example, [11–15] and the reference therein. Yamada [15] considered the so-called hybrid steepest descent method for solving the VIP over the set of fixed points of a nonexpansive mapping. Moudafi [11] proposed the viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping which is also a solution of a variational inequality problem. Subsequently, this method was developed by Xu [13]. Moudafi and Maingé [12] and Xu [14] further studied the viscosity method for HVIP.Very recently, Iiduka [16, 17] considered a variational inequality problem with variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with hierarchical constrained optimization problems or hierarchical fixed point problem, it is referred as triple hierarchical-constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms is also studied in [16, 17]. Since the original problem is a variational inequality problem, in this paper, we call it the triple hierarchical variational inequality problem (THVIP).LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be L-Lipschitzian with constant L≥0 and let T,V:C→C be nonexpansive mappings with Fix(T)≠∅. Let 0<μ<2η/κ2 and 0≤γL<τ, where τ=1-1-μ(2η-μκ2). We consider the following triple hierarchical variational inequality problem (for short, THVIP): find x*∈S such that
(1.6)〈(μF-γf)x*,x-x*〉≥0,∀x∈S,
where S denotes the solution set of the hierarchical variational inequality problem (1.4) which is assumed to be nonempty.Recall the functiong:C→ℝ is said to be convex if for all x,y∈C and for all λ∈[0,1],g(λx+(1-λ)y)≤λg(x)+(1-λ)g(y). It is said to be α-strongly convex if there exists α>0 such that for all x,y∈K and for all λ∈[0,1], g(λx+(1-λ)y)≤λg(x)+(1-λ)g(y)-(1/2)αλ(1-λ)∥x-y∥2. It is easy to see that if g is Fréchet differential and α-strongly convex, then the gradient ∇g is α-strongly monotone.Now, we illustrate the triple hierarchical variational inequality problem (for short, THVIP) by an example which is closely related to [17, Example 3.1].Example 1.1.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f:C→H be L-Lipschitz continuous with constant L>0. Suppose that g0:H→ℝ is a convex function with a 1/α0-Lipschitz continuous gradient, g1:H→ℝ is a convex function with a 1/α1-Lipschitz continuous gradient, and g2:H→ℝ is an α-strongly convex function with an α2-Lipschitz continuous gradient. Define T:=PC(I-λ∇g0)(λ∈(0,2α0]), V:=PC(I-λ~∇g1)(λ~∈(0,2α1]) and F:=∇g2. Then T,V:C→C are nonexpansive mappings with Fix(T)=Argminz∈Cg0(z) and Fix(V)=Argminz∈Cg1(z), and F is κ-Lipschitzian and η-strongly monotone with κ=1/α2 and η=α. Assume that Argminz∈Cg0(z)∩Argminz∈Cg1(z)≠∅. Then for the solution set S of the hierarchical variational inequality problem (for short, HVIP), we have
(1.7)∅≠Argminz∈Cg0(z)∩Argminz∈Cg1(z)=Fix(T)∩Fix(V)⊂{z*∈Fix(T):〈(I-V)z*,z-z*〉≥0,∀z∈Fix(T)}=S.
When 0<μ<2αα22 and 0≤γL<τ, where τ=1-1-μ(2α-μ/α22), we have
(1.8)0<μ<2ηκ2,0≤γL<τ,
where τ=1-1-μ(2η-μκ2). In particular, when μ=η/κ2=αα22, we have
(1.9)0≤γL<τ=1-1-μ(2α-μα22)=1-1-α2α22.
In this case, when γ=(1/2L)α2α22(obviously,1-α2α22<1-(1/2)α2α22), the following triple hierarchical variational inequality problem (for short, THVIP): find x*∈S such that
(1.10)〈(μF-γf)x*,x-x*〉≥0,∀x∈S
reduces to the following THVIP: find x*∈S such that
(1.11)〈(∇g2-α2Lf)x*,x-x*〉≥0,∀x∈S.In this paper, by combining hybrid steepest descent method, viscosity method, and projection method, we propose an approximation method to compute the approximate solution of THVIP. We also study the strong convergence of the sequences generated by the proposed method to a solution of THVIP. The results of this paper extend and generalize the results given in [12, 14] and several others given in the literature.


## 2. Preliminaries

Throughout the paper, unless other specified, we assume thatC is a nonempty closed convex subset of a real Hilbert space H. We use xn→x and xn⇀x to denote strong and weak convergence to x of the sequence {xn}, respectively.Recall that a mappingf:C→H is called L-Lipschitzian on C if there exists L∈[0,∞) such that ∥f(x)-f(y)∥≤L∥x-y∥,forallx,y∈C. In particular, if L∈[0,1) then f is called a contraction on C; if L=1 then f is called a nonexpansive mapping on C.We present some basic facts and results which will be used in the sequel.Lemma 2.1 (see [18]).
LetH be a real Hilbert space. Then,
(2.1)∥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].The following lemma can be easily proved, and therefore, we omit the proof.Lemma 2.2.
Letf:C→H be an L-Lipschitzian mapping with constant L∈[0,∞), and let F:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Then, for 0≤γL<μη,
(2.2)〈x-y,(μF-γf)x-(μF-γf)y〉≥(μη-γL)∥x-y∥2,∀x,y∈C.
That is, μF-γf is strongly monotone with constant μη-γL>0.Lemma 2.3 (see [18, Demiclosedness Principle]).
LetT:C→C be a nonexpansive mapping with Fix(T)≠∅. If {xn} weakly converges to x in C and if {(I-T)xn} strongly converges to y, then (I-T)x=y; in particular, if y=0, then x∈Fix(T).In the following lemma, we present some properties of the projection.Lemma 2.4.
Givenx∈H and z∈C. Then  (a)
z=PCx if and only if there holds the relation:
(2.3)〈x-z,y-z〉≤0,∀y∈C.(b)
z=PCx if and only if there holds the relation:
(2.4)∥x-z∥2≤∥x-y∥2-∥y-z∥2,∀y∈C.(c)
PC is nonexpansive and monotone, that is,
(2.5)〈PCx-PCy,x-y〉≥∥PCx-PCy∥2,∀x,y∈H.Lemma 2.5.
LetH be a real Hilbert space. Then, for all x,y∈H,
(2.6)∥x+y∥2≤∥x∥2+2〈y,x+y〉.The following lemma plays a key role in proving the main results of this paper.Lemma 2.6 (see [19, Lemma 3.1]).
Letλ∈(0,1] and μ>0. Let F:C→H be an operator on C such that, for some constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone. Associating with a nonexpansive mapping T:C→C, define the mapping Tλ:C→H by
(2.7)Tλx:=Tx-λμF(Tx),∀x∈C.
Then Tλ is a contraction provided μ<2η/κ2, that is,
(2.8)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Remark 2.7.
IfF=I, where I is the identity operator of H. Then κ=η=1 and hence μ<2η/κ2=2. Also, if μ=1, then it is easy to see that
(2.9)τ=1-1-μ(2η-μκ2)=1.
In particular, whenever λ>0, we have Tλx:=Tx-λμF(Tx)=(1-λ)Tx.


## 3. Approximation Methods and Convergence Results

LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be a L-Lipschitzian mapping with constant L≥0 and let T:C→C be a nonexpansive mapping with Fix(T)≠∅. Let 0<μ<2η/κ2 and 0≤γL≤τ, where τ=1-1-μ(2η-μκ2). We consider the hierarchical variational inequality problem (in short, HVIP) of finding z*∈Fix(T) such that
(3.1)〈(μF-γf)z*,z-z*〉≥0,∀z∈Fix(T).
We denote by Ω the solution set of the HVIP (3.1).Whenμ=1, F=I, γ=τ=1 and f=V are a nonexpansive self-mapping on C, the HVIP (3.1) reduces to the following hierarchical variational inequality problem of finding z*∈Fix(T) such that
(3.2)〈(I-V)z*,z-z*〉≥0,∀z∈Fix(T).
It is considered and studied in [12, 14].We consider a mappingΘt on C defined by
(3.3)Θtx=PC[tγf(x)+(I-tμF)Tx],∀x∈C.
It is easy to see that Θt is a nonexpansive mapping. Indeed, we have
(3.4)∥Θtx-Θty∥=∥PC[tγf(x)+(I-tμF)Tx]-PC[tγf(y)+(I-tμF)Ty]∥≤tγ∥f(x)-f(y)∥+∥(I-tμF)Tx-(I-tμF)Ty∥≤tγL∥x-y∥+(1-tτ)∥x-y∥=(1-(τ-γL)t)∥x-y∥.
Since 0≤γL≤τ, it is known that Θt is nonexpansive on C.Proposition 3.1.
LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be L-Lipschitzian with constant L≥0 and let T:C→C be nonexpansive with Fix(T)≠∅. Let 0<μ<2η/κ2 and 0≤γL≤τ, where τ=1-1-μ(2η-μκ2). Let t∈(0,1) and zt be a fixed point of the mapping Θt=PC[tγf+(I-tμF)T], that is, zt=PC[tγf(zt)+(I-tμF)Tzt]. Assume {zt} remains bounded as t→0, then the following conclusions hold.  (a)
The solution setΩ of the HVIP (3.1) is nonempty and each weak limit point (as t→0) of {zt} solves the HVIP (3.1).(b)
IfμF-γf is strictly monotone, then the net {zt} converges weakly to the (unique) solution of the HVIP (3.1).(c)
IfμF-γf is strongly monotone (e.g., μη>γL), then the net {zt} converges strongly to a solution of the HVIP (3.1).Proof.
LetW be the set of all weak accumulation points of {zt} as t→0; that is,
(3.5)W={z:ztn⇀zforsomesequence{tn}in(0,1)suchthattn→0}.
Then, W≠∅ because {zt} is bounded.
To prove (a), we notice that the boundedness of{zt} implies that W≠∅ and
(3.6)∥zt-Tzt∥=∥PC[tγf(zt)+(I-tμF)Tzt]-PCTzt∥≤∥tγf(zt)+(I-tμF)Tzt-Tzt∥=t∥γf(zt)-μF(Tzt)∥→0ast→0.
It thus follows from Lemma 2.3 that W⊂Fix(T). Take a fixed x^∈Fix(T) arbitrarily and set
(3.7)wt=tγf(zt)+(I-tμF)Tzt,∀t∈(0,1).
Then zt=PCwt and
(3.8)zt-x^=PCwt-wt+tγf(zt)+(I-tμF)Tzt-x^=PCwt-wt+t(γf(zt)-μFx^)+(I-tμF)Tzt-(I-tμF)x^=PCwt-wt+tγ(f(zt)-f(x^))+t(γf(x^)-μFx^)+(I-tμF)Tzt-(I-tμF)Tx^.
Since PC is the metric projection from H onto C, utilizing Lemma 2.4, we have
(3.9)〈PCwt-wt,PCwt-x^〉≤0.
Hence, utilizing κ-Lipschitzian property of F, we get
(3.10)∥zt-x^∥2=〈PCwt-wt,PCwt-x^〉+〈wt-x^,zt-x^〉≤〈wt-x^,zt-x^〉=t〈γf(zt)-μFx^,zt-x^〉+〈(I-tμF)Tzt-(I-tμF)x^,zt-x^〉≤t(〈(γf-μF)zt,zt-x^〉+μ〈Fzt-Fx^,zt-x^〉)+12(∥(I-tμF)Tzt-(I-tμF)x^∥2+∥zt-x^∥2)=t(〈(γf-μF)zt,zt-x^〉+μ〈Fzt-Fx^,Tzt-x^〉+μ〈Fzt-Fx^,zt-Tzt〉)+12[∥Tzt-x^∥2-2tμ〈FTzt-Fx^,Tzt-x^〉+t2μ2∥FTzt-Fx^∥2+∥zt-x^∥2]=t(〈(γf-μF)zt,zt-x^〉+μ〈FTzt-Fx^,Tzt-x^〉+μ〈Fzt-FTzt,Tzt-x^〉+μ〈Fzt-Fx^,zt-Tzt〉)+12[∥Tzt-x^∥2-2tμ〈FTzt-Fx^,Tzt-x^〉+t2μ2∥FTzt-Fx^∥2+∥zt-x^∥2]=t(〈(γf-μF)zt,zt-x^〉+μ〈Fzt-FTzt,Tzt-x^〉+μ〈Fzt-Fx^,zt-Tzt〉)+12[∥Tzt-x^∥2+t2μ2∥FTzt-Fx^∥2+∥zt-x^∥2]≤t〈(γf-μF)zt,zt-x^〉+2tμκ∥zt-Tzt∥∥zt-x^∥+(1+t2μ2κ22)∥zt-x^∥2.
It follows that
(3.11)〈(μF-γf)zt,zt-x^〉≤tμ2κ22∥zt-x^∥2+2μκ∥zt-Tzt∥∥zt-x^∥.
Note that 0≤γL≤τ and
(3.12)μη≥τ⟺μη≥1-1-μ(2η-μκ2)⟺1-μ(2η-μκ2)≥1-μη⟺1-2μη+μ2κ2≥1-2μη+μ2η2⟺κ2≥η2⟺κ≥η.
Since 0≤γL≤τ≤μη, we have μη-γL≥0. Thus, utilizing the η-strong monotonicity of F and L-Lipschitzian property of f, we know that μF-γf is monotone because the following inequality holds:
(3.13)〈(μF-γf)x-(μF-γf)y,x-y〉≥(μη-γL)∥x-y∥2,∀x,y∈C.
Consequently, we have
(3.14)〈(μF-γf)zt,zt-x^〉≥〈(μF-γf)x^,zt-x^〉.
This together with (3.11) implies that
(3.15)〈(μF-γf)x^,zt-x^〉≤tμ2κ22∥zt-x^∥2+2μκ∥zt-Tzt∥∥zt-x^∥.
Now, if x~∈W⊂Fix(T) and if tn→0 is such that ztn⇀x~, then we obtain from (3.15) and ∥zt-Tzt∥→0 that
(3.16)〈(μF-γf)x^,x~-x^〉≤0,∀x^∈Fix(T).
Replacing x^ by x~+λ(x-x~)∈Fix(T) in (3.16), where λ∈(0,1) and x∈Fix(T), we get
(3.17)〈(μF-γf)(x~+λ(x-x~)),x~-x〉≤0.
Letting λ→0 yields
(3.18)〈(μF-γf)x~,x~-x〉≤0,∀x∈Fix(T).
Consequently, x~∈Ω.
To see (b), we assume that{tn′} is another null sequence in (0,1) such that xtn′x^. Then x^∈Fix(T) and by replacing x by x^ in (3.18), we get
(3.19)〈(μF-γf)x~,x~-x^〉≤0.
By interchanging x~ and x^, we get
(3.20)〈(μF-γf)x^,x^-x~〉≤0.
Adding up (3.19) and (3.20) yields
(3.21)〈(μF-γf)x~-(μF-γf)x^,x~-x^〉≤0.
So the strict monotonicity of μF-γf implies that x~=x^ and {zt} converges weakly.
Finally, to prove (c), we observe that the strong monotonicity ofμF-γf and (3.11) implies that
(3.22)α∥zt-x^∥2+〈(μF-γf)x^,zt-x^〉≤tμ2κ22∥zt-x^∥2+2μκ∥zt-Tzt∥∥zt-x^∥,
where α>0 is the strong monotonicity constant of μF-γf; that is,
(3.23)〈(μF-γf)x-(μF-γf)y,x-y〉≥α∥x-y∥2,∀x,y∈C.
A straightforward consequence of (3.22) is that if x^∈W and if ztn⇀x^ for some null sequence {tn} in (0,1), then we must have ztn→x^. This shows that {zt} is relatively compact in the norm topology, and each of its limit points solves the HVIP (3.1). Finally repeating the argument in the weak convergence case of (b), we see that {zt} can have exactly one limit point; hence, {zt} converges in norm.Corollary 3.2 (see [14, Proposition 3.1]).
LetV,T:C→C be nonexpansive mappings with Fix(T)≠∅. Let t∈(0,1) and zt be a fixed point of the mapping Wt=tV+(1-t)T, that is, zt=tVzt+(1-t)Tzt. Assume {zt} remains bounded as t→0, then the following conclusions hold.  (a)
The solution setS of the HVIP (1.4) is nonempty and each weak limit point (as t→0) of {zt} solves the HVIP (1.4).(b)
IfI-V is strictly monotone, then the net {zt} converges weakly to the solution of the HVIP (1.4).(c)
IfI-V is strongly monotone (e.g., V is a contraction), then the net {zt} converges strongly to a solution of the HVIP (1.4).Now by combining hybrid steepest descent method, viscosity method, and projection method, we define, for eachs,t∈(0,1), two mappings Wt and fs,t by
(3.24)Wt=tV+(1-t)T,fs,t=PC[sγf+(I-sμF)Wt].
It is easy to see that Wt is a nonexpansive self-mapping on C. Moreover, utilizing Lemma 2.6, we can see that fs,t is a (1-(τ-γL)s)-contraction. Indeed, observe that
(3.25)∥fs,t(x)-fs,t(y)∥=∥PC[sγf(x)+(I-sμF)Wtx]-PC[sγf(y)+(I-sμF)Wty]∥≤∥[sγf(x)+(I-sμF)Wtx]-[sγf(y)+(I-sμF)Wty]∥≤sγ∥f(x)-f(y)∥+∥(I-sμF)Wtx-(I-sμF)Wty∥≤sγL∥x-y∥+(1-sτ)∥x-y∥=(1-s(τ-γL))∥x-y∥.
Let xs,t be the unique fixed point of fs,t. Namely, xs,t is the unique solution in C to the following:
(3.26)xs,t=PC[sγf(xs,t)+(I-sμF)Wtxs,t]=PC[sγf(xs,t)+(I-sμF)(tVxs,t+(1-t)Txs,t)].Theorem 3.3.
LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be L-Lipschitzian with constant L≥0 and let V,T:C→C be nonexpansive with Fix(T)≠∅. Let 0<μ<2η/κ2 and 0≤γL<τ, where τ=1-1-μ(2η-μκ2). For each s,t∈(0,1), let xs,t be the unique solution to (3.26). Assume also that, for each t∈(0,1), Fix(Wt) is nonempty (but not necessarily bounded), and the following assumption holds:
(A)∅≠S⊂∥·∥-liminft→0Fix(Wt):={z:∃zt∈Fix(Wt)suchthatzt→z}.
Then the strong lims→0xs,t=:xt exists for each t∈(0,1). Moreover, the strong limt→0xt=:x∞ exists and solves the THVIP (1.6). Hence, for any null sequence {sn} in (0,1), there is another null sequence {tn} in (0,1) such that xsn,tn→x∞ in norm, as n→∞.Proof.
Observe that the condition0≤γL<τ and the fact τ≤μη imply that
(3.27)0≤γL<τ≤μη.
Therefore, μF-γf is a strongly monotone operator with constant μη-γL>0. Since, for each fixed t∈(0,1), the fixed point set Fix(Wt) of Wt is nonempty, we can apply Proposition 3.1 (c) to get that
(3.28)xt:=∥·∥-lims→0xs,t
exists in Fix(Wt) and solves the following hierarchical variational inequality problem of finding xt∈Fix(Wt) such that
(3.29)〈(μF-γf)xt,x-xt〉≥0,∀x∈Fix(Wt).
Equivalently, xt=PFix(Wt)(I-μF+γf)xt, where PFix(Wt) is the metric projection from H onto Fix(Wt).
Utilizing the strong monotonicity ofμF-γf, we conclude from (3.29) that for each z∈Fix(Wt),
(3.30)(μη-γL)∥xt-z∥2≤〈(μF-γf)xt-(μF-γf)z,xt-z〉=-〈(μF-γf)xt,z-xt〉+〈(γf-μF)z,xt-z〉≤〈(γf-μF)z,xt-z〉.
Hence,
(3.31)∥xt-z∥2≤1μη-γL〈(γf-μF)z,xt-z〉,∀z∈Fix(Wt).
This implies that
(3.32)∥xt-z∥≤1μη-γL∥(γf-μF)z∥,∀z∈Fix(Wt).
The inequality (3.32) is yet to imply the boundedness of {xt} since z may depend on t. However, since the solution set S of the HVIP (1.4) is nonempty, we can take (an arbitrary) v∈S and use assumption (A) to find zt∈Fix(Wt) such that zt→v in norm as t→0. Hence, {zt} must be bounded (as t→0). The inequality (3.32) implies
(3.33)∥xt∥≤∥xt-zt∥+∥zt-v∥+∥v∥≤1μη-γL∥(γf-μF)zt∥+∥zt-v∥+∥v∥,
and this is sufficient to ensure that {xt} is bounded (as t closes 0).
Now, the boundedness of{xt} allows us to apply Corollary 3.2 (a) to conclude that every weak limit point x~ of {xt} belongs to the solution set S of the HVIP (1.4). Then (3.31) guarantees that every such weak limit point x~ of {xt} is also a strong limit point of {xt}. Indeed, if {tn} is a null sequence in (0,1) and if xtn⇀x~, then x~∈S. By assumption (A), we get a sequence {zn} such that zn∈Fix(Wtn) for all n and zn→x~ in norm. From (3.31) we derive
(3.34)∥xtn-x~∥2=∥(xtn-zn)+(zn-x~)∥2≤2(∥xtn-zn∥2+∥zn-x~∥2)≤2μη-γL〈(γf-μF)zn,xtn-zn〉+2∥zn-x~∥2.
However, 〈(γf-μF)zn,xtn-zn〉→0 since (γf-μF)zn→(γf-μF)x~ in norm and xtn-zn→0 weakly, and we find that the right-hand side of (3.34) tends to zero. Hence, xtn→x~ in norm.
So to prove the strong convergence of the entire net{xt}, it remains to prove that {xt} can have only one strong limit point. Let x~ and x~′ be two strong limit points of {xt} and assume that xtn→x~ and xtn′→x~′ both in norm, where {tn} and {tn′} are null sequences in (0,1). It remains to verify that x~=x~′.
Sincex~′∈S, by assumption (A), we can find zt∈Fix(Wt) such that zt→x~′ in norm as t→0. The HVIP (3.29) implies
(3.35)〈(μF-γf)xtn,ztn-xtn〉≥0.
Taking the limit as n→∞ yields
(3.36)〈(μF-γf)x~,x~′-x~〉≥0.
Similarly, we have
(3.37)〈(μF-γf)x~′,x~-x~′〉≥0.
Adding up (3.36) and (3.37) gives
(3.38)〈(μF-γf)x~-(μF-γf)x~′,x~-x~′〉≤0.
Utilizing Lemma 2.2, we know that μF-γf is strongly monotone with constant μη-γL>0. Hence, from (3.38) it follows that x~=x~′ and so {xt} converges in norm to (say) x∞.
Now, for anyv∈S, since by assumption (A), we can find zt∈Fix(Wt) such that zt→v in norm, (3.29) then implies
(3.39)〈(μF-γf)xt,v-xt〉≥〈(μF-γf)xt,v-zt〉→0
which in turns implies
(3.40)〈(μF-γf)x∞,v-x∞〉≥0,∀v∈S,
that is, x∞=PS(I-μF+γf)x∞, the unique fixed point of the contraction PS(I-μF+γf). Finally, for any null sequence {sn} in (0,1), using a diagonalization argument (cf. [1]), we can find another null sequence {tn} in (0,1) such that xsn,tn→x∞ in norm, as n→∞.Remark 3.4.
Theorem3.3 shows that for any null sequence {sn} in (0,1), there is another null sequence {tn} in (0,1) such that xsn,tn→x∞ in norm, as n→∞, and x∞ is a solution to the HVIP (3.40). Theorem 3.3 is the main result of the present paper in which we improve the result of Moudafi and Maingé [12] by proving that {xt} actually converges strongly and also by removing the boundedness of the set {Fix(Wt):0<t<1}. Our proof is different from that of [12]. In the meantime, Theorem 3.3 covers [14, Theorem 3.2] as a special case. For instance, whenever we put μ=1, F=I, γ=τ=1, and let the L-Lipschitzian mapping f:C→H be a (self-) contraction with coefficient ρ∈[0,1), our Theorem 3.3 reduces to [14, Theorem 3.2].
Now, we present a general result. We show that as long asts is taken so that ts=o(s) (i.e., lims→0ts/s=0), then xs,ts→z∞ in norm, and moreover, z∞ solves the HVIP (3.40) on the larger set Fix(T) (i.e., z∞ is the unique fixed point in Fix(T) of the contraction PFix(T)(I-μF+γf)), without the assumption (A). However, for such a general choice of {ts}, this solution z∞ may differ from the solution x∞ of the HVIP (3.40) on the smaller set S (i.e., x∞ is the unique fixed point in S of the contraction PS(I-μF+γf)). We will verify this by taking ts=s2 for simplicity (the argument, however, works for any net {ts} in (0,1) such that lims→0ts/s=0).Theorem 3.5.
LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be L-Lipschitzian with constant L≥0 and let V,T:C→C be nonexpansive with Fix(T)≠∅. Let 0<μ<2η/κ2 and 0≤γL<τ, where τ=1-1-μ(2η-μκ2). For each s∈(0,1), let xs be the unique solution in C to the following:
(3.41)xs=PC[sγf(xs)+(I-sμF)(s2Vxs+(1-s2)Txs)].
Then, as s→0, xs converges in norm to the solution of the HVIP of finding z∞∈Fix(T) such that
(3.42)〈(μF-γf)z∞,z-z∞〉≥0,∀z∈Fix(T);
equivalently, z∞=PFix(T)(I-μF+γf)z∞.Proof.
WriteWs (instead of Ws2) for s2V+(1-s2)T; then
(3.43)xs=PC[sγf(xs)+(I-sμF)Wsxs].
Take a fixed z∈Fix(T) arbitrarily and put
(3.44)ys=sγf(xs)+(I-sμF)Wsxs,∀s∈(0,1).
Then from (3.41) we get xs=PCys. Since PC is the metric projection from H onto C, we have
(3.45)〈PCys-ys,PCys-z〉≤0.
Also, observe that
(3.46)ys-z=sγf(xs)+(I-sμF)Wsxs-z=s(γf(xs)-μFWsz)+(I-sμF)Wsxs-(I-sμF)Wsz+Wsz-z=sγ(f(xs)-f(z))+s(γf(z)-μFWsz)+(I-sμF)Wsxs-(I-sμF)Wsz+s2(V-I)z.
Utilizing Lemma 2.6, we deduce from (3.41) that
(3.47)∥xs-z∥2=〈PCys-ys,PCys-z〉+〈ys-z,xs-z〉≤〈ys-z,xs-z〉=sγ〈f(xs)-f(z),xs-z〉+s〈γf(z)-μFWsz,xs-z〉+〈(I-sμF)Wsxs-(I-sμF)Wsz,xs-z〉+s2〈(V-I)z,xs-z〉≤sγ∥f(xs)-f(z)∥∥xs-z∥+s〈γf(z)-μFWsz,xs-z〉+∥(I-sμF)Wsxs-(I-sμF)Wsz∥∥xs-z∥+s2〈(V-I)z,xs-z〉≤sγL∥xs-z∥2+s〈γf(z)-μFWsz,xs-z〉+(1-sτ)∥xs-z∥2+s2〈(V-I)z,xs-z〉=(1-s(τ-γL))∥xs-z∥2+s〈γf(z)-μFWsz,xs-z〉+s2〈(V-I)z,xs-z〉.
It follows that, for any fixed z∈Fix(T),
(3.48)∥xs-z∥2≤1τ-γL(〈γf(z)-μFWsz,xs-z〉+s〈(V-I)z,xs-z〉).
This implies that
(3.49)∥xs-z∥≤1τ-γL(∥γf(z)-μFWsz∥+∥(V-I)z∥).
In particular, {xs} is bounded, and from (3.41), we further get
(3.50)∥xs-Txs∥=∥PC[sγf(xs)+(I-sμF)(s2Vxs+(1-s2)Txs)]-PCTxs∥≤∥sγf(xs)+(I-sμF)(s2Vxs+(1-s2)Txs)-Txs∥≤s∥γf(xs)-μF(s2Vxs+(1-s2)Txs)∥+s2∥Vxs-Txs∥→0,
as s→0. Lemma 2.3 ensures that every weak limit point, as s→0, of {xs} is a fixed point of T. Going back to (3.48), we find that each weak limit point of {xs} is actually a strong limit point of {xs} because
(3.51)|〈γf(z)-μFWsz,xs-z〉|=|〈γf(z)-μFz,xs-z〉+〈μFz-μFWsz,xs-z〉|≤|〈γf(z)-μFz,xs-z〉|+μ∥Fz-FWsz∥∥xs-z∥≤|〈γf(z)-μFz,xs-z〉|+μκs2∥(V-I)z∥∥xs-z∥.
So to prove the strong convergence of {xs}, we need only to show the uniqueness of strong limit points of {xs}. Assuming {sn} and {sn′} are null sequences in (0,1) such that xsn→v and xsn′→v', both in norm. Observing that (3.41) implies
(3.52)(μF-γf)xs=1s(PCys-ys)-1s(I-Ws)xs+μ(Fxs-FWsxs),
where ys=sγf(xs)+(I-sμF)Wsxs and xs=PCys. Utilizing Lemmas 2.4 and 2.6, we deduce from the monotonicity of I-Ws that for any fixed z∈Fix(T),
(3.53)〈(μF-γf)xs,xs-z〉=1s〈PCys-ys,PCys-z〉-1s〈(I-Ws)xs,xs-z〉+μ〈Fxs-FWsxs,xs-z〉≤1s〈PCys-ys,PCys-z〉-1s[〈(I-Ws)xs-(I-Ws)z,xs-z〉≤+〈(I-Ws)z,xs-z〉]+μ〈Fxs-FWsxs,xs-z〉≤-s〈(I-V)z,xs-z〉+μκ∥xs-Wsxs∥∥xs-z∥=-s〈(I-V)z,xs-z〉+μκ∥(xs-Txs)+s2(T-V)xs∥∥xs-z∥.
In particular, we have from I-Ws that for any fixed z∈Fix(T),
(3.54)〈(μF-γf)xsn,xsn-v′〉≤-sn〈(I-V)v′,xsn-v′〉+μκ∥(xsn-Txsn)+sn2(T-V)xsn∥∥xsn-v'∥.
So letting n→∞ yields
(3.55)〈(μF-γf)v,v-v′〉≤0.
Repeating the above argument obtains
(3.56)〈(μF-γf)v′,v′-v〉≤0.
Adding up (3.55) and (3.56) gives us that
(3.57)〈(μF-γf)v-(μF-γf)v′,v-v′〉≤0.
The strong monotonicity of μF-γf (Lemma 2.2) then implies v=v'. Finally, taking the limit as s→0 in (3.53) and letting z∞=∥·∥-lims→0xs, we conclude immediately that z∞ solves the variational inequality of finding z∞∈Fix(T) such that
(3.58)〈(μF-γf)z∞,z∞-z〉≤0,∀z∈Fix(T).
Equivalently, z∞=PFix(T)(I-μF+γf)z∞. The proof is therefore complete.Remark 3.6.
IfT and V have a common fixed point, then it is not hard to see that Fix(Wt)=Fix(T)∩Fix(V) for all t∈(0,1). Indeed, it suffices to show the inclusion Fix(Wt)⊂Fix(T)∩Fix(V). Let z∈Fix(Wt). Then for any fixed p∈Fix(T)∩Fix(V) we have
(3.59)∥z-p∥2=t∥Vz-p∥2+(1-t)∥Tz-p∥2-t(1-t)∥Vz-Tz∥2≤∥z-p∥2-t(1-t)∥Vz-Tz∥2.
This implies Vz=Tz=z; that is z∈Fix(T)∩Fix(V). Furthermore, it is clear that Fix(T)∩Fix(V)⊂S. In this case, assumption (A) is reduced to the assumption S⊂Fix(T)∩Fix(V). Therefore, assumption (A) is equivalent to the assumption S=Fix(T)∩Fix(V).Corollary 3.7.
LetF:C→H be a κ-Lipschitzian and η-strongly monotone operator with constants κ and η>0, respectively. Let f:C→H be L-Lipschitzian with constant L≥0 and let V,T:C→C be nonexpansive with Fix(T)∩Fix(V)≠∅. Let 0<μ<2η/κ2 and 0≤γL<τ, where τ=1-1-μ(2η-μκ2). For each s,t∈(0,1), let xs,t be the unique solution to (3.26). Then the conclusion of Theorem 3.3 holds. Namely, the strong lims→0xs,t=:xt exists for each fixed t∈(0,1), and moreover, the strong limt→0xt=:x∞ exists and solves the THVIP (1.6).Proof.
SinceFix(Wt)=Fix(T)∩Fix(V) is independent of t; the z in both (3.31) and (3.32) does not depend on t. Hence, it is immediately clear that {xt} is bounded, which then implies via (3.31) that every weak accumulation point of {xt} is also a strong accumulation point of {xt}. Eventually, {xt} converges in norm as shown in the final part of the proof of Theorem 3.5.Remark 3.8.
Theorems3.3 and 3.5 improve and extend [14, Theorems 3.2 and 3.4], respectively, in the following ways.  (a)
The contraction mappingf:C→C in [14, Theorems 3.2 and 3.4] is extended to the case of (possibly nonself) L-Lipschitzian mapping f:C→H from a nonempty closed convex subset C to H.(b)
The convex combination of (self) contraction mappingf and nonexpansive mapping Wt in the implicit scheme in [14, Theorem 3.2] is extended to the linear combination of (possibly nonself) L-Lipschitzian mapping f and hybrid steepest descent method involving Wt. In particular, if t=s2, Theorem 3.5 is an extension of [14, Theorem 3.4].(c)
In order to guarantee that the net{xs,t} generated by the implicit scheme still lies in C, the implicit scheme in [14, Theorem 3.2] is extended to develop our new implicit scheme (3.26) by virtue of the projection method. In particular, if t=s2, [14, Theorem 3.4] is extended to the corresponding case in our Theorem 3.5.(d)
The new technique of argument is applied to derive our Theorems3.3 and 3.5. For instance, the characteristic properties (Lemma 2.4) of the metric projection play a key role in proving the strong convergence of the nets {xs,t}s,t∈(0,1) and {xs}s∈(0,1) in our Theorems 3.3 and 3.5, respectively.(e)
If we putμ=1,F=I and γ=τ=1 and let f be a contractive self-mapping on C with coefficient ρ∈[0,1), then our Theorems 3.3 and 3.5 reduce to [14, Theorems 3.2 and 3.4], respectively. Thus, our Theorems 3.3 and 3.5 cover [14, Theorems 3.2 and 3.4] as special cases, respectively.



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*Source: 907105-2012-09-20.xml*