# Some Algorithms for Finding Fixed Points and Solutions of Variational Inequalities

Authors: Jong Soo Jung
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce new implicit and explicit algorithms for finding the fixed point of ak-strictly pseudocontractive mapping and for solving variational inequalities related to the Lipschitzian and strongly monotone operator in Hilbert spaces. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of a k-strictly pseudocontractive mapping. Such a point is also a solution of a variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of a k-strictly pseudocontractive mapping.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥. Let C be a nonempty closed convex subset of H and S:C→C a self-mapping on C. We denote by Fix(S) the set of fixed points of S. We recall that a mapping T:C→H is said to be k-strictly pseudocontractive if there exists a constant k∈[0,1) such that(1.1)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈C.Note that the class ofk-strictly pseudocontractive mappings includes the class of nonexpansive mappings T on C (i.e., ∥Tx-Ty∥≤∥x-y∥,forallx,y∈C) as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudocontractive. Recently, many authors have been devoting their studies to the problems of finding fixed points for k-strictly pseudocontractive mappings; see [1–5] and the references therein.Variational inequalities have been studied widely and are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied sciences; see [6–8]. For general variational inequalities and extended general variational inequalities, we can refer to [9–13] and references therein.A variational inequality (VI) is formulated as finding a pointx̃ with the property(1.2)x̃∈Csuchthat〈Fx̃,p-x̃〉≥0,∀p∈C,
where F:C→H is a nonlinear mapping. It is well known that VI (1.2) is equivalent to the fixed point equation(1.3)x̃=PC(I-λF)x̃,
where λ>0 and PC is the the metric projection of H onto C, which assigns, to each x∈H, the unique point in C, denoted by PCx, such that(1.4)‖x-PCx‖=inf⁡{‖x-y‖:y∈C}.Therefore, fixed point algorithms can be applied to solve VI (1.2). It is also well known that if F is ρ-Lipschitzian and η-strongly monotone with constants ρ,η>0 (i.e., there exist ρ,η≥0 such that ∥Fx-Fy∥≤ρ∥x-y∥ and 〈Fx-Fy,x-y〉≥η∥x-y∥2,x,y∈C, resp.), then, for small enough λ>0, the mapping PC(I-λF) is a contractive mapping on C and so the sequence {xn} of Picard iterates, given by xn=PC(I-λF)xn-1(n≥1), converges strongly to the unique solution of VI (1.2).This sort of VI (1.2) where F is ρ-Lipschitzian and η-strongly monotone and where solutions are sought from the set of fixed points of a nonexpansive mapping is originated from Yamada [14], who provided the hybrid method for solving VI (1.2). In order to find solutions of ceratin variational inequality problems defined on the set of fixed points of nonexpansive mappings, several iterative algorithms were studied by many authors; see [15–24] and the references therein.In this paper, we investigate the following variational inequality (VI) as a special form of VI (1.2), where the constraint set is the fixed points of a k-strictly pseudocontractive mapping T: finding a point x̃ with property(1.5)x̃∈Fix⁡(T)suchthat〈(μF-γV)x̃,p-x̃〉≥0,∀p∈Fix⁡(T),
where T:C→C is a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some k∈[0,1), F:C→H is a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ,η>0, and V:C→H is an l-Lipschitzian mapping with constant l≥0 and μ,γ>0. Indeed, variational inequalities of form (1.5) cover several topics recently considered in the literature, including monotone inclusions, convex optimization, and quadratic minimization over fixed point sets; see [2–4, 15, 18, 19, 21, 22, 24] and the references therein. For some iterative methods and some results related to our approach about VI (1.5), we can refer to [25–31] and references therein.The main purpose of the present paper is to further study the hierarchical fixed point approach to the VI of form (1.5). First, we introduce new implicit and explicit algorithms for finding the fixed point of the k-strictly pseudocontractive mapping T. Then, we establish results on the strong convergence of the sequences generated by the proposed algorithms to a fixed point of the mapping T, which is also a solution of VI (1.5) defined on the set of fixed points of T. As direct consequences, we obtain the unique minimum-norm fixed point of T. Namely, we find the unique solution of the quadratic minimization problem: ∥x̃∥2=min{∥x∥2:x∈Fix(T)}.


## 2. Preliminaries and Lemmas

Throughout this paper, when{xn} is a sequence in H, xn→x (resp., xn⇀x) denotes strong (resp., weak) convergence of the sequence {xn} to x.LetC be a nonempty closed convex subset of a real Hilbert space H. Recall that f:C→H is called a contractive mapping with constant α∈(0,1) if there exists a constant α∈(0,1) such that ∥f(x)-f(y)∥≤α∥x-y∥,forallx,y∈C.For every pointx∈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 to C. It is well known that PC is nonexpansive and that, for x∈H,(2.2)z=PCx⟺〈x-z,y-z〉≤0,∀y∈C.In a Hilbert spaceH, we have(2.3)‖x-y‖2=‖x‖2+‖y‖2-2〈x,y〉,∀x,y∈H.We need the following lemmas for the proof of our main results.Lemma 2.1.
In a real Hilbert spaceH,  the following inequality holds:
(2.4)‖x+y‖2≤‖x‖2+2〈y,x+y〉,∀x,y∈H.Lemma 2.2 (see [32]).
Let{sn} be a sequence of nonnegative real numbers satisfying
(2.5)sn+1≤(1-λn)sn+λnδn,∀n≥0,
where {λn}⊂(0,1) and {δn} satisfy the following conditions:(i)
limn→∞λn=0,(ii)
∑n=0∞λn=∞,(iii)
limsupn→∞δn≤0 or ∑n=0∞λnδn<∞.Then, limn→∞sn=0.Lemma 2.3 (see [33]).
Let{xn} and {zn} be bounded sequences in a Banach space E and {γn} a sequence in [0,1] that satisfies the following condition:
(2.6)0<liminf⁡n→∞γn≤limsup⁡n→∞γn<1.
Suppose that xn+1=γnxn+(1-γn)zn  for all n≥0 and
(2.7)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Then, limn→∞∥zn-xn∥=0.Lemma 2.4 (Demiclosedness principle [34]).
LetC be a nonempty closed convex subset of a real Hilbert space H and S:C→C a nonexpansive mapping with Fix(S)≠∅.  If {xn} is a sequence in C weakly converging to x and  {(I-S)xn} converges strongly to y,  then (I-S)x=y;  in particular, if y=0,  then x∈Fix(S).Lemma 2.5 (see [14, 16]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Assume that the mapping G:C→H is monotone and weakly continuous along segments, that is, G(x+ty)→G(x) weakly as t→0.  Then, the variational inequality
(2.8)x̃∈C,〈Gx̃,p-x̃〉≥0,∀p∈C,
is equivalent to the dual variational inequality
(2.9)x̃∈C,〈Gp,p-x̃〉≥0,∀p∈C.Lemma 2.6 (see [5]).
LetH be a real Hilbert space and C a closed convex subset of H.  If T is a k- strictly pseudocontractive mapping on C,  then the fixed point set Fix(T) is closed convex, so that the projection PFix(T) is well defined.Lemma 2.7 (see [5]).
LetH be a Hilbert space, C a closed convex subset of H,  and T:C→H a k- strictly pseudocontractive mapping. Define a mapping S:C→H by  Sx=λx+(1-λ)Tx for all x∈C.  Then, as λ∈[k,1),  S is a nonexpansive mapping such that Fix(S)=Fix(T).The following lemma can be easily proven, and, therefore, we omit the proof.Lemma 2.8.
LetH be a real Hilbert space. Let V:H→H be an l-Lipschitzian mapping with constant l≥0 and F:H→H a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ,η>0.  Then, for 0≤γl<μη,
(2.10)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)‖x-y‖2,∀x,y∈C.
That is, μF-γV is strongly monotone with constant μη-γl.The following lemma is an improvement of Lemma  2.9 in [4] (see also [14]).Lemma 2.9.
LetH be a real Hilbert space H.  Let F:H→H be a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ,η>0.  Let 0<μ<2η/ρ2 and 0<t<ξ≤1.  Then, R:=ρI-tμF:H→H is a contractive mapping with constant ξ-tτ,  where τ=1-1-μ(2η-μρ2)<1.Proof.
First we show thatI-μF is strictly contractive. In fact, by applying the ρ-Lipschitz continuity and η-strongly monotonicity of F and (2.3), we obtain, for x,y∈H,
(2.11)‖(I-μF)x-(I-μF)y‖2=‖(x-y)-μ(Fx-Fy)‖2=‖x-y‖2-2μ〈Fx-Fy,x-y〉+μ2‖Fx-Fy‖2≤‖x-y‖2-2μη‖x-y‖2+μ2ρ2‖x-y‖2=(1-μ(2η-μρ2))‖x-y‖2,
and so
(2.12)‖(I-μF)x-(I-μF)y‖≤1-μ(2η-μρ2)‖x-y‖.
Now, noting that R:=ξI-tμF=(ξ-t)I-t(μF-I), from (2.12), we have, for x,y∈H,
(2.13)‖Rx-Ry‖=‖(ξ-t)(x-y)-t((μF-I)x-(μF-I)y)‖≤(ξ-t)‖x-y‖+t‖(μF-I)x-(μF-I)y‖≤(ξ-t)‖x-y‖+t1-μ(2η-μρ2)‖x-y‖=(ξ-t(1-1-μ(2η-μρ2)))‖x-y‖=(ξ-tτ)‖x-y‖.
Hence, R is a contractive mapping with constant ξ-tτ.


## 3. Iterative Algorithms

LetH be a real Hilbert space and C a nonempty closed convex subset of H. Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1, F:C→H a ρ-Lipschitzian and η-strongly monotone mapping with constants ρ,η>0, and V:C→H an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/ρ2 and 0<γl<τ, where τ=1-1-μ(2η-μρ2)<1. Let S:C→C be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C.In this section, we introduce the following algorithm that generates a net{xt}t∈(0,1) in an implicit way:(3.1)xt=SPC[tγVxt+(I-tμF)xt].
We prove the strong convergence of {xt} as t→0 to a fixed point x̃ of T, which is a solution of the following variational inequality:(3.2)〈(μF-γV)x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
We also propose the following explicit algorithm, which generates a sequence in an explicit way:(3.3)xn+1=βnxn+(1-βn)SPC[αnγVxn+(I-αnμF)xn],∀n≥0,
where {αn}, {βn}⊂(0,1) and x0∈C is an arbitrary initial guess, and we establish the strong convergence of this sequence to a fixed point x̃ of T, which is also a solution of the variational inequality (3.2).


### 3.1. Strong Convergence of the Implicit Algorithm

Now, fort∈(0,1), consider a mapping Qt:C→C defined by(3.4)Qtx=SPC[tγVx+(I-tμF)x],∀x∈C.
It is easy to see that Qt is a contractive mapping with constant 1-t(τ-γl). Indeed, by Lemma 2.9, we have(3.5)‖Qtx-Qty‖≤tγ‖Vx-Vy‖+‖(I-tμF)x-(I-tμF)y‖≤tγl‖x-y‖+(1-tτ)‖x-y‖=(1-t(τ-γl))‖x-y‖.
Hence, Qt has a unique fixed point, denoted by xt, which uniquely solves the fixed point equation (3.1).We summarize the basic properties of{xt}.Proposition 3.1.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1,  F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0, and V:C→H an l-Lipschitzian mapping with constant l≥0.  Let 0<μ<2η/ρ2 and 0<γl<τ,  where τ=1-1-μ(2η-μρ2)<1.  Let S:C→C be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C. Let {xt} be defined via (3.1). Then,(i)
{xt} is bounded for t∈(0,1),(ii)
limt→0∥xt-Sxt∥=0,(iii)
xt defines a continuous path from (0,1) in C.Proof.
(i) Letp∈Fix(T). Observing Fix(T)=Fix(S) by Lemma 2.7, we have
(3.6)‖xt-p‖=‖SPC[tγVxt+(I-tμF)xt]-SPCp‖≤‖t(γVxt-μFp)+(I-tμF)xt-(I-tμF)p‖≤(1-tτ)‖xt-p‖+t(γl‖xt-p‖+‖(γV-μF)p‖).
So, it follows that
(3.7)‖xt-p‖≤‖(γV-μF)p‖τ-γl.
Hence, {xt} is bounded and so are {Vxt}, {Sxt}, and {Fxt}.
(ii) By the boundedness of{Vxt} and {Fxt} in (i), we have
(3.8)‖xt-Sxt‖=‖SPC[tγVxt+(I-tμF)xt]-SPCxt‖≤t‖(γV-μF)xt‖⟶0ast⟶0.
(iii) Lett,t0∈(0,1), and calculate
(3.9)‖xt-xt0‖=‖SPC[tγVxt+(I-tμF)xt]-SPC[t0γVxt0+(I-t0μF)xt0]‖≤‖(t-t0)γVxt+t0γ(Vxt-Vxt0)-(t-t0)μFxt+(I-t0μF)xt-(I-t0μF)xt0‖≤|t-t0|γ‖Vxt‖+t0γl‖xt-xt0‖+|t-t0|‖μFxt‖+(1-t0τ)‖xt-xt0‖.
It follows that
(3.10)‖xt-xt0‖≤γ‖Vxt‖+μ‖Fxt‖t0(τ-γl)|t-t0|.
This shows that xt is locally Lipschitzian and hence continuous.We establish the strong convergence of the net{xt} as t→0, which guarantees the existence of solutions of the variational inequality (3.2).Theorem 3.2.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1, F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0, and V:C→H an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/ρ2 and 0<γl<τ, where τ=1-1-μ(2η-μρ2)<1. Let S:C→C be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C. The net {xt} defined via (3.1) converges strongly to a fixed point x̃ of T as t→0, which solves the variational inequality (3.2), or, equivalently, one has PF(T)(I-μF+γV)x̃=x̃.Proof.
We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of μF-γV. In fact, noting that 0≤γl<τ and μη≥τ⇔ρ≥η, it follows from Lemma 2.8 that
(3.11)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)‖x-y‖2.
That is, μF-γV is strongly monotone for 0≤γl<τ≤μη. Suppose that x̃∈Fix(T) and x̂∈Fix(T) both are solutions to (3.2). Then, we have
(3.12)〈(μF-γV)x̃,x̂-x̃〉≥0,〈(μF-γV)x̂,x̃-x̂〉≥0.
Adding up (3.12) yields
(3.14)〈(μF-γV)x̃-(μF-γV)x̂,x̃-x̂〉≤0.
The strong monotonicity of μF-γV implies that x̃=x̂ and the uniqueness is proved.
Next, we prove thatxt→x̃ as t→0. To this end, set yt=PC[tγVxt+(I-tμF)xt] for all t∈(0,1). Then, observing Fix(T)=Fix(S) by Lemma 2.7, we have xt=Syt and for any p∈Fix(T)(3.15)‖xt-p‖≤‖yt-p‖.
Also it follows that
(3.16)‖yt-xt‖=‖xt-PC[tγVxt+(I-tμF)xt]‖≤t[γ‖xt‖+μ‖Fxt‖]⟶0ast⟶0.
Since PC is the metric projection from H onto C, we have, for given p∈Fix(T),
(3.17)‖yt-p‖2=〈PC[tγVxt+(I-tμF)xt]-(tγVxt+(I-tμF)xt),yt-p〉+〈tγVxt+(I-tμF)xt-p,yt-p〉≤〈tγVxt+(I-tμF)xt-p,yt-p〉=〈tγVxt-tμFp,yt-p〉+〈(I-tμF)xt-(I-tμF)p,yt-p〉≤(1-tτ)‖xt-p‖‖yt-p‖+t〈γVxt-μFp,yt-p〉.
It follows that
(3.18)‖yt-p‖2≤1τ〈γVxt-μFp,yt-p〉=1τ(γ〈Vxt-Vp,yt-p〉+〈(γV-μF)p,yt-p〉)≤1τ(γl‖xt-p‖‖yt-p‖+〈(γV-μF)p,xt-p〉).
By (3.15), this implies
(3.19)‖yt-p‖2≤1τ-γl〈(γV-μF)p,yt-p〉=1τ-γl〈(μF-γV)p,p-yt〉,
and hence
(3.20)‖xt-p‖2≤1τ-γl〈(μF-γV)p,p-yt〉.
Since {xt} is bounded as t→0 (by Proposition 3.1 (i)), we see that if {tn} is a subsequence in (0,1) such that tn→0 and xtn⇀x̃, then, from (3.16), we have ytn⇀x̃. By Proposition 3.1 (ii), limn→∞(I-S)xtn=0. By Lemmas 2.4 and 2.7, x̃∈Fix(T). Therefore, we can substitute x̃ for p in (3.20) to obtain
(3.21)‖xtn-x̃‖2≤1τ-γl〈(μF-γV)x̃,x̃-ytn〉.
Consequently, the weak convergence of {ytn} to x̃ yields that {xtn}→x̃ strongly. Now we show that x̃ solves the variational inequality (3.2). Again, observe (3.20) and take the limit as n→∞ to obtain
(3.22)‖x̃-p‖2≤1τ-γl〈(μF-γV)p,p-x̃〉,∀p∈Fix⁡(T).
Hence x̃ solves the following variational inequality:
(3.23)〈(μF-γV)p,p-x̃〉≥0,∀p∈Fix⁡(T),
or the equivalent dual variational inequality (see Lemmas 2.5 and 2.8)
(3.24)〈(μF-γV)x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
Moreover, if {tj} is another subsequence in (0,1) such that tj→0 and xtj⇀x̂, then we also have ytj⇀x̂ from (3.16). By the same argument, we can show that x̂∈Fix(T) and x̂ solves the variational inequality (3.2); hence x̂=x̃ by uniqueness. In sum, we have shown that each cluster point of {xt} (at t→0) equals x̃. Therefore xt→x̃ as t→0.
The variational inequality (3.2) can be rewritten as
(3.25)〈(I-μF+γV)x̃-x̃,p-x̃〉≤0,∀p∈Fix⁡(T).
By reminding the reader of (2.2) and Lemma 2.6, this is equivalent to the fixed point equation
(3.26)PF(T)(I-μF+γV)x̃=x̃.From Theorem3.2, we can deduce the following result.Corollary 3.3.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1. For each t∈(0,1), let the net {xt} be defined by
(3.27)xt=SPC(1-t)xt,∀t∈(0,1),
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx and PC is a metric projection of H onto C. Then the net {xt} defined via (3.27) converges strongly, as t→0, to the minimum-norm point x̃∈Fix(T).Proof.
In (3.20) with F≡I, V≡0, l=0, μ=1, and τ=1, letting t→0 yields
(3.28)‖x̃-p‖2≤〈p,p-x̃〉,∀p∈Fix⁡(T).
Equivalently,
(3.29)〈x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
This obviously implies that
(3.30)‖x̃‖2≤〈p,x̃〉≤‖p‖‖x̃‖,∀p∈Fix⁡(T).
It turns out that ∥x̃∥≤∥p∥ for all p∈Fix(T). Therefore, x̃ is minimum-norm point of Fix(T).


### 3.2. Strong Convergence of the Explicit Algorithm

Now, using Theorem3.2, we show the strong convergence of the sequence generated by the explicit algorithm (3.3) to a fixed point x̃ of T, which is also a solution of the variational inequality (3.2).Theorem 3.4.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1,  F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0,  and V:C→H an l-Lipschitzian mapping with constant l≥0.  Let 0<μ<2η/ρ2 and 0<γl<τ,  where τ=1-1-μ(2η-μρ2)<1.  Let S:C→H be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C.  For any given x0∈C,  let {xn} be the sequence generated by the explicit algorithm (3.3), where {αn} and {βn} satisfy the following conditions:(C1)
limn→∞αn=0,(C2)
∑n=0∞αn=∞,(C3)
0<liminfn→∞βn≤limsupn→∞βn<1.
Then,{xn} converges strongly to x̃∈Fix(T), which is the unique solution of the variational inequality (3.2).Proof.
First, from condition (C1), without loss of generality, we assume thatαn(τ-γl)<1 and αn<(1-βn) for n≥0. From now, we put yn=SPCun and un=αnγVxn+(I-αnμF)xn.
We divide the proof several steps.
Step 1.
We show that∥xn-p∥≤max{∥x0-p∥,∥(γV-μF)p∥/(τ-γl)} for all n≥0 and all p∈Fix(T)=Fix(S). Indeed, let p∈Fix(T). Then, from Lemma 2.9, we have
(3.31)‖yn-p‖=‖SPCun-SPCp‖≤‖((1-βn)I-αnμF)αn(γVxn-μFp)+βn(xn-p)+((1-βn)I-αnμF)xn-((1-βn)I-αnμF)p‖≤(1-βn-αnτ)‖xn-p‖+βn‖xn-p‖+αn‖γVxn-μFp‖≤(1-αnτ)‖xn-p‖+αn(‖γVxn-γVp‖+‖(γV-μF)p‖)≤(1-(τ-γl)αn)‖xn-p‖+αn‖(γV-μF)p‖.
Thus, it follows that
(3.32)‖xn+1-p‖≤βn‖xn-p‖+(1-βn)‖yn-p‖≤βn‖xn-p‖+(1-βn)(1-αn(τ-γl))‖xn-p‖+(1-βn)αn‖(γV-μF)p‖=(1-αn(1-βn)(τ-γl))‖xn-p‖+αn(1-βn)(τ-γl)‖(γV-μF)p‖τ-γl≤max⁡{‖xn-p‖,‖(γV-μF)p‖τ-γl}.
Using an induction, we have ∥xn-p∥≤max{∥x0-p∥,∥(γV-μF)p∥/(τ-γl)}. Hence, {xn} is bounded, and so are {yn}, {un}, {Vxn}, {Sxn}, and {Fxn}.Step 2.
We show thatlimn→∞∥xn+1-xn∥=0. Indeed, from (3.3), we observe
(3.33)‖yn+1-yn‖=‖SPCun+1-SPCun‖≤‖αn+1γVxn+1+(I-αn+1μF)xn+1-(αnγVxn+(I-αnμF)xn)‖=‖αn+1γ(Vxn+1-Vxn)+γ(αn+1-αn)Vxn+(I-αn+1μF)xn+1-(I-αn+1μF)xn+μ(αn+1-αn)Fxn‖≤(1-αn+1(τ-γl))‖xn+1-xn‖+M|αn+1-αn|,
where M=sup{γ∥Vxn∥+μ∥Fxn∥:n≥0}. Thus, it follows that
(3.34)‖yn+1-yn‖-‖xn+1-xn‖≤-αn+1(τ-γl)‖xn+1-xn‖+M|αn+1-αn|,
which implies, from condition (C1), that
(3.35)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we have
(3.36)lim⁡n→∞‖yn-xn‖=0.
Consequently, from condition (C3), it follows that
(3.37)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖yn-xn‖=0.Step 3.
We show thatlimn→∞∥xn-Sxn∥=0. Indeed, we have
(3.38)‖xn-Sxn‖≤‖xn-xn+1‖+‖xn+1-Sxn‖=‖xn+1-xn‖+‖βn(xn-Sxn)+(1-βn)(yn-Sxn)‖≤‖xn+1-xn‖+βn‖xn-Sxn‖+(1-βn)‖yn-SPCxn‖≤‖xn-xn+1‖+βn‖xn-Sxn‖+(1-βn)αn‖(γV-μF)xn‖,
that is,
(3.39)‖xn-Sxn‖≤11-βn‖xn+1-xn‖+αn‖(γV-μF)xn‖.
This together with conditions (C1) and (C3) and Step 2 implies
(3.40)lim⁡n→∞‖xn-Sxn‖=0.Step 4.
We show that(3.41)limsup⁡n→∞〈(γV-μF)x̃,un-x̃〉≤0,
where x̃=limt→0xt with xt being defined by (3.1). (We note that, from Theorem 3.2, x̃∈Fix(T) and x̃ is the unique solution of the variational inequality (3.2)). To show this, we can choose a subsequence {xnj} of {xn} such that
(3.42)lim⁡j→∞〈(γV-μF)x̃,xnj-x̃〉=limsup⁡n→∞〈(γV-μF)x̃,xn-x̃〉.
Since {xn} is bounded, there exists a subsequence {xnji} of {xnj}, which converges weakly to w. Without loss of generality, we can assume that xnj⇀w. Since ∥xn-Sxn∥→0 by Step 3, we obtain w=Sw by virtue of Lemma 2.4. From Lemma 2.7, we have w∈Fix(T). Therefore, from (3.2), it follows that
(3.43)limsup⁡n→∞〈(γV-μF)x̃,xn-x̃〉=lim⁡j→∞〈(γV-μF)x̃,xnj-x̃〉=〈(γV-μF)x̃,w-x̃〉≤0.
We notice that, by condition (C1),
(3.44)‖un-xn‖≤αn‖(μF-γV)xn‖⟶0asn⟶∞.
Hence, from (3.43), we obtain
(3.45)limsup⁡n→∞〈(γV-μF)x̃,un-x̃〉≤0.Step 5.
We show thatlimn→∞∥xn-x̃∥=0, where x̃=limt→0xt with xt being defined by (3.1), and x̃ is the unique solution of the variational inequality (3.2). Indeed, we observe that
(3.46)‖un-x̃‖≤‖xn-x̃‖+αn‖(γV-μF)xn‖.
Therefore, from the convexity of ∥·∥2, (3.3), and Lemma 2.1, we have
(3.47)‖xn+1-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖yn-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖un-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖(I-αnμF)xn-(I-αnμF)x̃‖2+2(1-βn)αn〈γVxn-μFx̃,un-x̃〉≤βn‖xn-x̃‖2+(1-βn)(1-αnτ)2‖xn-x̃‖2+2(1-βn)αn[〈γVxn-γVx̃,un-x̃〉+〈(γV-μF)x̃,un-x̃〉]≤[βn+(1-βn)(1-2αnτ+αn2τ2)]‖xn-x̃‖2+2(1-βn)αnγl‖xn-x̃‖‖un-x̃‖+2(1-βn)αn〈γVx̃-μFx̃,un-x̃〉≤[1-(1-βn)(2αnτ-αn2τ2)]‖xn-x̃‖2+2(1-βn)αnγl‖xn-x̃‖2+2(1-βn)αn2γl‖xn-x̃‖‖(γV-μF)xn‖+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉≤[1-2(1-βn)(τ-γl)αn]‖xn-x̃‖2+αn2τ2‖xn-x̃‖2+2αn2γl‖xn-x̃‖‖(γV-μF)xn‖+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉≤[1-2(1-b)(τ-γl)αn]‖xn-x̃‖2+αn(αnτ2M12+2αnγlM1M2)+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉=(1-λn)‖xn-x̃‖2+λnδn,
where 0<b=limsupn→∞βn<1, M1=sup{∥xn-x̃∥:n≥0}, M2=sup{∥γVxn-μFxn∥:n≥0}, λn=2(1-b)(τ-γl)αn, and
(3.48)δn=1(1-b)(τ-γl)(αnτ2M122+αnγlM1M2+(1-βn)〈(γV-μF)x̃,un-x̃〉).
From conditions (C1), (C2), and (C3) and Step 4, it is easy to see that λn→0, ∑n=0∞λn=∞ and limsupn→∞δn≤0. Hence, by Lemma 2.2, we conclude that xn→x̃ as n→∞. This completes the proof.From Theorem3.4, we can also deduce the following result.Corollary 3.5.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1. For each x0∈C, let the sequence {xn} be defined by
(3.49)xn+1=βnxn+(1-βn)SPC(1-αn)xn,∀n≥0,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx and PC is a metric projection of H onto C. If {αn} and {βn} satisfy conditions (C1), (C2), and (C3) in Theorem 3.4, then the sequence {xn} defined via (3.49) converges strongly, as n→∞, to the minimum-norm point x̃∈Fix(T).Proof.
VI (3.2) is reduced to the inequality
(3.50)〈x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
This is equivalent to ∥x̃∥2≤〈p,x̃〉≤∥p∥∥x̃∥ for all p∈Fix(T). It turns out that ∥x̃∥≤∥p∥ for all p∈Fix(T) and x̃ is the minimum-norm point of Fix(T).Remark 3.6.
We point out that our algorithms (3.1) and (3.3) are new ones different from those in the literature (see [2–4, 15, 18, 21, 22] and references therein).


## 3.1. Strong Convergence of the Implicit Algorithm

Now, fort∈(0,1), consider a mapping Qt:C→C defined by(3.4)Qtx=SPC[tγVx+(I-tμF)x],∀x∈C.
It is easy to see that Qt is a contractive mapping with constant 1-t(τ-γl). Indeed, by Lemma 2.9, we have(3.5)‖Qtx-Qty‖≤tγ‖Vx-Vy‖+‖(I-tμF)x-(I-tμF)y‖≤tγl‖x-y‖+(1-tτ)‖x-y‖=(1-t(τ-γl))‖x-y‖.
Hence, Qt has a unique fixed point, denoted by xt, which uniquely solves the fixed point equation (3.1).We summarize the basic properties of{xt}.Proposition 3.1.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1,  F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0, and V:C→H an l-Lipschitzian mapping with constant l≥0.  Let 0<μ<2η/ρ2 and 0<γl<τ,  where τ=1-1-μ(2η-μρ2)<1.  Let S:C→C be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C. Let {xt} be defined via (3.1). Then,(i)
{xt} is bounded for t∈(0,1),(ii)
limt→0∥xt-Sxt∥=0,(iii)
xt defines a continuous path from (0,1) in C.Proof.
(i) Letp∈Fix(T). Observing Fix(T)=Fix(S) by Lemma 2.7, we have
(3.6)‖xt-p‖=‖SPC[tγVxt+(I-tμF)xt]-SPCp‖≤‖t(γVxt-μFp)+(I-tμF)xt-(I-tμF)p‖≤(1-tτ)‖xt-p‖+t(γl‖xt-p‖+‖(γV-μF)p‖).
So, it follows that
(3.7)‖xt-p‖≤‖(γV-μF)p‖τ-γl.
Hence, {xt} is bounded and so are {Vxt}, {Sxt}, and {Fxt}.
(ii) By the boundedness of{Vxt} and {Fxt} in (i), we have
(3.8)‖xt-Sxt‖=‖SPC[tγVxt+(I-tμF)xt]-SPCxt‖≤t‖(γV-μF)xt‖⟶0ast⟶0.
(iii) Lett,t0∈(0,1), and calculate
(3.9)‖xt-xt0‖=‖SPC[tγVxt+(I-tμF)xt]-SPC[t0γVxt0+(I-t0μF)xt0]‖≤‖(t-t0)γVxt+t0γ(Vxt-Vxt0)-(t-t0)μFxt+(I-t0μF)xt-(I-t0μF)xt0‖≤|t-t0|γ‖Vxt‖+t0γl‖xt-xt0‖+|t-t0|‖μFxt‖+(1-t0τ)‖xt-xt0‖.
It follows that
(3.10)‖xt-xt0‖≤γ‖Vxt‖+μ‖Fxt‖t0(τ-γl)|t-t0|.
This shows that xt is locally Lipschitzian and hence continuous.We establish the strong convergence of the net{xt} as t→0, which guarantees the existence of solutions of the variational inequality (3.2).Theorem 3.2.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1, F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0, and V:C→H an l-Lipschitzian mapping with constant l≥0. Let 0<μ<2η/ρ2 and 0<γl<τ, where τ=1-1-μ(2η-μρ2)<1. Let S:C→C be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C. The net {xt} defined via (3.1) converges strongly to a fixed point x̃ of T as t→0, which solves the variational inequality (3.2), or, equivalently, one has PF(T)(I-μF+γV)x̃=x̃.Proof.
We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of μF-γV. In fact, noting that 0≤γl<τ and μη≥τ⇔ρ≥η, it follows from Lemma 2.8 that
(3.11)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)‖x-y‖2.
That is, μF-γV is strongly monotone for 0≤γl<τ≤μη. Suppose that x̃∈Fix(T) and x̂∈Fix(T) both are solutions to (3.2). Then, we have
(3.12)〈(μF-γV)x̃,x̂-x̃〉≥0,〈(μF-γV)x̂,x̃-x̂〉≥0.
Adding up (3.12) yields
(3.14)〈(μF-γV)x̃-(μF-γV)x̂,x̃-x̂〉≤0.
The strong monotonicity of μF-γV implies that x̃=x̂ and the uniqueness is proved.
Next, we prove thatxt→x̃ as t→0. To this end, set yt=PC[tγVxt+(I-tμF)xt] for all t∈(0,1). Then, observing Fix(T)=Fix(S) by Lemma 2.7, we have xt=Syt and for any p∈Fix(T)(3.15)‖xt-p‖≤‖yt-p‖.
Also it follows that
(3.16)‖yt-xt‖=‖xt-PC[tγVxt+(I-tμF)xt]‖≤t[γ‖xt‖+μ‖Fxt‖]⟶0ast⟶0.
Since PC is the metric projection from H onto C, we have, for given p∈Fix(T),
(3.17)‖yt-p‖2=〈PC[tγVxt+(I-tμF)xt]-(tγVxt+(I-tμF)xt),yt-p〉+〈tγVxt+(I-tμF)xt-p,yt-p〉≤〈tγVxt+(I-tμF)xt-p,yt-p〉=〈tγVxt-tμFp,yt-p〉+〈(I-tμF)xt-(I-tμF)p,yt-p〉≤(1-tτ)‖xt-p‖‖yt-p‖+t〈γVxt-μFp,yt-p〉.
It follows that
(3.18)‖yt-p‖2≤1τ〈γVxt-μFp,yt-p〉=1τ(γ〈Vxt-Vp,yt-p〉+〈(γV-μF)p,yt-p〉)≤1τ(γl‖xt-p‖‖yt-p‖+〈(γV-μF)p,xt-p〉).
By (3.15), this implies
(3.19)‖yt-p‖2≤1τ-γl〈(γV-μF)p,yt-p〉=1τ-γl〈(μF-γV)p,p-yt〉,
and hence
(3.20)‖xt-p‖2≤1τ-γl〈(μF-γV)p,p-yt〉.
Since {xt} is bounded as t→0 (by Proposition 3.1 (i)), we see that if {tn} is a subsequence in (0,1) such that tn→0 and xtn⇀x̃, then, from (3.16), we have ytn⇀x̃. By Proposition 3.1 (ii), limn→∞(I-S)xtn=0. By Lemmas 2.4 and 2.7, x̃∈Fix(T). Therefore, we can substitute x̃ for p in (3.20) to obtain
(3.21)‖xtn-x̃‖2≤1τ-γl〈(μF-γV)x̃,x̃-ytn〉.
Consequently, the weak convergence of {ytn} to x̃ yields that {xtn}→x̃ strongly. Now we show that x̃ solves the variational inequality (3.2). Again, observe (3.20) and take the limit as n→∞ to obtain
(3.22)‖x̃-p‖2≤1τ-γl〈(μF-γV)p,p-x̃〉,∀p∈Fix⁡(T).
Hence x̃ solves the following variational inequality:
(3.23)〈(μF-γV)p,p-x̃〉≥0,∀p∈Fix⁡(T),
or the equivalent dual variational inequality (see Lemmas 2.5 and 2.8)
(3.24)〈(μF-γV)x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
Moreover, if {tj} is another subsequence in (0,1) such that tj→0 and xtj⇀x̂, then we also have ytj⇀x̂ from (3.16). By the same argument, we can show that x̂∈Fix(T) and x̂ solves the variational inequality (3.2); hence x̂=x̃ by uniqueness. In sum, we have shown that each cluster point of {xt} (at t→0) equals x̃. Therefore xt→x̃ as t→0.
The variational inequality (3.2) can be rewritten as
(3.25)〈(I-μF+γV)x̃-x̃,p-x̃〉≤0,∀p∈Fix⁡(T).
By reminding the reader of (2.2) and Lemma 2.6, this is equivalent to the fixed point equation
(3.26)PF(T)(I-μF+γV)x̃=x̃.From Theorem3.2, we can deduce the following result.Corollary 3.3.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1. For each t∈(0,1), let the net {xt} be defined by
(3.27)xt=SPC(1-t)xt,∀t∈(0,1),
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx and PC is a metric projection of H onto C. Then the net {xt} defined via (3.27) converges strongly, as t→0, to the minimum-norm point x̃∈Fix(T).Proof.
In (3.20) with F≡I, V≡0, l=0, μ=1, and τ=1, letting t→0 yields
(3.28)‖x̃-p‖2≤〈p,p-x̃〉,∀p∈Fix⁡(T).
Equivalently,
(3.29)〈x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
This obviously implies that
(3.30)‖x̃‖2≤〈p,x̃〉≤‖p‖‖x̃‖,∀p∈Fix⁡(T).
It turns out that ∥x̃∥≤∥p∥ for all p∈Fix(T). Therefore, x̃ is minimum-norm point of Fix(T).


## 3.2. Strong Convergence of the Explicit Algorithm

Now, using Theorem3.2, we show the strong convergence of the sequence generated by the explicit algorithm (3.3) to a fixed point x̃ of T, which is also a solution of the variational inequality (3.2).Theorem 3.4.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1,  F:C→H a ρ-Lipschitzain and η-strongly monotone mapping with constants ρ,η>0,  and V:C→H an l-Lipschitzian mapping with constant l≥0.  Let 0<μ<2η/ρ2 and 0<γl<τ,  where τ=1-1-μ(2η-μρ2)<1.  Let S:C→H be a mapping defined by Sx=kx+(1-k)Tx and PC a metric projection of H onto C.  For any given x0∈C,  let {xn} be the sequence generated by the explicit algorithm (3.3), where {αn} and {βn} satisfy the following conditions:(C1)
limn→∞αn=0,(C2)
∑n=0∞αn=∞,(C3)
0<liminfn→∞βn≤limsupn→∞βn<1.
Then,{xn} converges strongly to x̃∈Fix(T), which is the unique solution of the variational inequality (3.2).Proof.
First, from condition (C1), without loss of generality, we assume thatαn(τ-γl)<1 and αn<(1-βn) for n≥0. From now, we put yn=SPCun and un=αnγVxn+(I-αnμF)xn.
We divide the proof several steps.
Step 1.
We show that∥xn-p∥≤max{∥x0-p∥,∥(γV-μF)p∥/(τ-γl)} for all n≥0 and all p∈Fix(T)=Fix(S). Indeed, let p∈Fix(T). Then, from Lemma 2.9, we have
(3.31)‖yn-p‖=‖SPCun-SPCp‖≤‖((1-βn)I-αnμF)αn(γVxn-μFp)+βn(xn-p)+((1-βn)I-αnμF)xn-((1-βn)I-αnμF)p‖≤(1-βn-αnτ)‖xn-p‖+βn‖xn-p‖+αn‖γVxn-μFp‖≤(1-αnτ)‖xn-p‖+αn(‖γVxn-γVp‖+‖(γV-μF)p‖)≤(1-(τ-γl)αn)‖xn-p‖+αn‖(γV-μF)p‖.
Thus, it follows that
(3.32)‖xn+1-p‖≤βn‖xn-p‖+(1-βn)‖yn-p‖≤βn‖xn-p‖+(1-βn)(1-αn(τ-γl))‖xn-p‖+(1-βn)αn‖(γV-μF)p‖=(1-αn(1-βn)(τ-γl))‖xn-p‖+αn(1-βn)(τ-γl)‖(γV-μF)p‖τ-γl≤max⁡{‖xn-p‖,‖(γV-μF)p‖τ-γl}.
Using an induction, we have ∥xn-p∥≤max{∥x0-p∥,∥(γV-μF)p∥/(τ-γl)}. Hence, {xn} is bounded, and so are {yn}, {un}, {Vxn}, {Sxn}, and {Fxn}.Step 2.
We show thatlimn→∞∥xn+1-xn∥=0. Indeed, from (3.3), we observe
(3.33)‖yn+1-yn‖=‖SPCun+1-SPCun‖≤‖αn+1γVxn+1+(I-αn+1μF)xn+1-(αnγVxn+(I-αnμF)xn)‖=‖αn+1γ(Vxn+1-Vxn)+γ(αn+1-αn)Vxn+(I-αn+1μF)xn+1-(I-αn+1μF)xn+μ(αn+1-αn)Fxn‖≤(1-αn+1(τ-γl))‖xn+1-xn‖+M|αn+1-αn|,
where M=sup{γ∥Vxn∥+μ∥Fxn∥:n≥0}. Thus, it follows that
(3.34)‖yn+1-yn‖-‖xn+1-xn‖≤-αn+1(τ-γl)‖xn+1-xn‖+M|αn+1-αn|,
which implies, from condition (C1), that
(3.35)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we have
(3.36)lim⁡n→∞‖yn-xn‖=0.
Consequently, from condition (C3), it follows that
(3.37)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-βn)‖yn-xn‖=0.Step 3.
We show thatlimn→∞∥xn-Sxn∥=0. Indeed, we have
(3.38)‖xn-Sxn‖≤‖xn-xn+1‖+‖xn+1-Sxn‖=‖xn+1-xn‖+‖βn(xn-Sxn)+(1-βn)(yn-Sxn)‖≤‖xn+1-xn‖+βn‖xn-Sxn‖+(1-βn)‖yn-SPCxn‖≤‖xn-xn+1‖+βn‖xn-Sxn‖+(1-βn)αn‖(γV-μF)xn‖,
that is,
(3.39)‖xn-Sxn‖≤11-βn‖xn+1-xn‖+αn‖(γV-μF)xn‖.
This together with conditions (C1) and (C3) and Step 2 implies
(3.40)lim⁡n→∞‖xn-Sxn‖=0.Step 4.
We show that(3.41)limsup⁡n→∞〈(γV-μF)x̃,un-x̃〉≤0,
where x̃=limt→0xt with xt being defined by (3.1). (We note that, from Theorem 3.2, x̃∈Fix(T) and x̃ is the unique solution of the variational inequality (3.2)). To show this, we can choose a subsequence {xnj} of {xn} such that
(3.42)lim⁡j→∞〈(γV-μF)x̃,xnj-x̃〉=limsup⁡n→∞〈(γV-μF)x̃,xn-x̃〉.
Since {xn} is bounded, there exists a subsequence {xnji} of {xnj}, which converges weakly to w. Without loss of generality, we can assume that xnj⇀w. Since ∥xn-Sxn∥→0 by Step 3, we obtain w=Sw by virtue of Lemma 2.4. From Lemma 2.7, we have w∈Fix(T). Therefore, from (3.2), it follows that
(3.43)limsup⁡n→∞〈(γV-μF)x̃,xn-x̃〉=lim⁡j→∞〈(γV-μF)x̃,xnj-x̃〉=〈(γV-μF)x̃,w-x̃〉≤0.
We notice that, by condition (C1),
(3.44)‖un-xn‖≤αn‖(μF-γV)xn‖⟶0asn⟶∞.
Hence, from (3.43), we obtain
(3.45)limsup⁡n→∞〈(γV-μF)x̃,un-x̃〉≤0.Step 5.
We show thatlimn→∞∥xn-x̃∥=0, where x̃=limt→0xt with xt being defined by (3.1), and x̃ is the unique solution of the variational inequality (3.2). Indeed, we observe that
(3.46)‖un-x̃‖≤‖xn-x̃‖+αn‖(γV-μF)xn‖.
Therefore, from the convexity of ∥·∥2, (3.3), and Lemma 2.1, we have
(3.47)‖xn+1-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖yn-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖un-x̃‖2≤βn‖xn-x̃‖2+(1-βn)‖(I-αnμF)xn-(I-αnμF)x̃‖2+2(1-βn)αn〈γVxn-μFx̃,un-x̃〉≤βn‖xn-x̃‖2+(1-βn)(1-αnτ)2‖xn-x̃‖2+2(1-βn)αn[〈γVxn-γVx̃,un-x̃〉+〈(γV-μF)x̃,un-x̃〉]≤[βn+(1-βn)(1-2αnτ+αn2τ2)]‖xn-x̃‖2+2(1-βn)αnγl‖xn-x̃‖‖un-x̃‖+2(1-βn)αn〈γVx̃-μFx̃,un-x̃〉≤[1-(1-βn)(2αnτ-αn2τ2)]‖xn-x̃‖2+2(1-βn)αnγl‖xn-x̃‖2+2(1-βn)αn2γl‖xn-x̃‖‖(γV-μF)xn‖+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉≤[1-2(1-βn)(τ-γl)αn]‖xn-x̃‖2+αn2τ2‖xn-x̃‖2+2αn2γl‖xn-x̃‖‖(γV-μF)xn‖+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉≤[1-2(1-b)(τ-γl)αn]‖xn-x̃‖2+αn(αnτ2M12+2αnγlM1M2)+2(1-βn)αn〈(γV-μF)x̃,un-x̃〉=(1-λn)‖xn-x̃‖2+λnδn,
where 0<b=limsupn→∞βn<1, M1=sup{∥xn-x̃∥:n≥0}, M2=sup{∥γVxn-μFxn∥:n≥0}, λn=2(1-b)(τ-γl)αn, and
(3.48)δn=1(1-b)(τ-γl)(αnτ2M122+αnγlM1M2+(1-βn)〈(γV-μF)x̃,un-x̃〉).
From conditions (C1), (C2), and (C3) and Step 4, it is easy to see that λn→0, ∑n=0∞λn=∞ and limsupn→∞δn≤0. Hence, by Lemma 2.2, we conclude that xn→x̃ as n→∞. This completes the proof.From Theorem3.4, we can also deduce the following result.Corollary 3.5.
LetH be a real Hilbert space and C a nonempty closed convex subset of H.  Let T:C→C be a k-strictly pseudocontractive mapping with Fix(T)≠∅ for some 0≤k<1. For each x0∈C, let the sequence {xn} be defined by
(3.49)xn+1=βnxn+(1-βn)SPC(1-αn)xn,∀n≥0,
where S:C→C is a mapping defined by Sx=kx+(1-k)Tx and PC is a metric projection of H onto C. If {αn} and {βn} satisfy conditions (C1), (C2), and (C3) in Theorem 3.4, then the sequence {xn} defined via (3.49) converges strongly, as n→∞, to the minimum-norm point x̃∈Fix(T).Proof.
VI (3.2) is reduced to the inequality
(3.50)〈x̃,p-x̃〉≥0,∀p∈Fix⁡(T).
This is equivalent to ∥x̃∥2≤〈p,x̃〉≤∥p∥∥x̃∥ for all p∈Fix(T). It turns out that ∥x̃∥≤∥p∥ for all p∈Fix(T) and x̃ is the minimum-norm point of Fix(T).Remark 3.6.
We point out that our algorithms (3.1) and (3.3) are new ones different from those in the literature (see [2–4, 15, 18, 21, 22] and references therein).


## 4. Conclusion and Future Directions

In this paper, we have introduced new implicit and explicit algorithms for finding fixed points of ak-strictly pseudocontractive mapping and for solving a certain variational inequality and have established strong convergence of the proposed algorithms to a fixed point of the mapping, which is a solution of a certain variational inequality, where the constraint set is the fixed points of the mapping. As direct consequences, we have considered the quadratic minimization problem on the set of fixed points of the mapping.In forthcoming studies, we will consider implicit and explicit algorithms for solving some variational inequalities, where the constraint set is the common set of the set of fixed points of the mapping and the set of solutions of the equilibrium problem.We hope that the ideas and techniques of this paper may stimulate further research in this field.



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*Source: 153456-2012-03-28.xml*