# An Implicit Algorithm for Maximal Monotone Operators and Pseudocontractive Mappings

Authors: Hong-Jun Li; Yeong-Cheng Liou; Cun-Lin Li; Muhammad Aslam Noor; Yonghong Yao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to construct an implicit algorithm for finding the
common solution of maximal monotone operators and strictly pseudocontractive
mappings in Hilbert spaces. Some applications are also included.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H.Recall thatS is said to be a strictly pseudo contractive mapping if there exists a constant 0≤ρ<1 such that(1.1)‖Sx-Sy‖2≤‖x-y‖2+ρ‖(I-S)x-(I-S)y‖2,∀x,y∈C.
For such case, we also say that S is a ρ-strictly pseudo-contractive mapping. When ρ=0, T is said to be nonexpansive. It is clear that (1.1) is equivalent to(1.2)〈Sx-Sy,x-y〉≤‖x-y‖2-1-ρ2‖(I-S)x-(I-S)y‖2,∀x,y∈C.
We denote by F(S) the set of fixed points of S.A mappingA:C→H is said to be α-inverse strongly monotone if(1.3)〈Ax-Ay,x-y〉≥α‖Ax-Ay‖2,
for some α>0 and for all x, y∈C. It is known that if A is an α-inverse strongly monotone, then ∥Ax-Ay∥≤1/α∥x-y∥for all x, y∈C.LetB be a mapping of H into 2H. The effective domain of B is denoted by dom(B), that is, dom(B)={x∈H:Bx≠∅}. A multi valued mapping B is said to be a monotone operator on H iff(1.4)〈x-y,u-v〉≥0,
for all x, y∈dom(B), u∈Bx, and v∈By. A monotone operator B on H is said to be maximal if its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H, and let B-10={x∈H:0∈Bx}.For a maximal monotone operatorB on H and λ>0, we may define a single-valued operator JλB=(I+λB)-1:H→dom(B), which is called the resolvent of B for λ. It is known that the resolvent JλB is firmly nonexpansive, that is,(1.5)‖JλBx-JλBy‖2≤〈JλBx-JλBy,x-y〉,
for all x, y∈C and B-10=F(JλB) for all λ>0.Algorithms for finding the fixed points of nonlinear mappings or for finding the zero points of maximal monotone operators have been studied by many authors. The reader can refer to [1–24]. Especially, Takahashi et al. [6] recently gave the following convergence result.Theorem 1.1.
LetC be a closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let JλB=(I+λB)-1 be the resolvent of B for λ>0, and let S be a nonexpansive mapping of C into itself, such that F(S)∩(A+B)-10≠∅. Let x1=x∈C and let {xn}⊂C, be a sequence generated by
(1.6)xn+1=βnxn+(1-βn)S(αnx+(1-αn)JλnB(xn-λnAxn)),
for all n≥0, where {λn}⊂(0,2α), {αn}⊂(0,1) and {βn}⊂(0,1) satisfy
(1.7)0<a≤λn≤b<2α,<c≤βn≤d<1,lim⁡n→∞(λn+1-λn)=0,lim⁡n→∞αn=0,∑nαn=∞,
then {xn} generated by (1.6) converges strongly to a point of F(S)∩(A+B)-10.Motivated and inspired by the works in this field, the purpose of this paper is to construct an implicit algorithm for finding the common solution of maximal monotone operators and strictly-pseudocontractive mappings in Hilbert spaces. Some applications are also included.


## 2. Preliminaries

The following resolvent identity is well known: forλ>0 and μ>0, there holds the identity(2.1)JλBx=JμB(μλx+(1-μλ)JλBx),x∈H.
We use the following notation:(i)
xn⇀x stands for the weak convergence of {xn} to x;(ii)
xn→x stands for the strong convergence of {xn} to x.We need the following lemmas for the next section.Lemma 2.1 (see[14]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let S:C→H be a ρ-strict pseudo contraction. Define T:C→H by Tx=αx+(1-α)Tx for each x∈C, then, as α∈[ρ,1), T is nonexpansive such that F(S)=F(T).Lemma 2.2 (see[15]).
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 λ>0 a constant, then one has
(2.2)‖(I-λA)x-(I-λA)y‖2≤‖x-y‖2+λ(λ-2α)‖Ax-Ay‖2,∀x,y∈C.
In particular, if 0≤λ≤2α, then I-λA is nonexpansive.Lemma 2.3 (see[14]).
LetC be a nonempty, closed and convex of a real Hilbert space H. Let T:C→C be a λ-strictly pseudo-contractive mapping, then I-T is demi closed at 0, that is, if xn⇀x∈C and xn-Txn→0, then x=Tx.Lemma 2.4 (see[16]).
Let{xn} and {yn} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)yn+βnxn for all n≥0 and limsupn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0, then limn→∞∥yn-xn∥=0.Lemma 2.5 (see[17]).
Assume that{an} is a sequence of nonnegative real numbers such that
(2.3)an+1≤(1-γn)an+δnγn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that(1)
∑n=1∞γn=∞,(2)
limsupn→∞δn≤0 or ∑n=1∞|δnγn|<∞, then limn→∞an=0.


## 3. Main Results

In this section, we will prove our main results.Theorem 3.1.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H, and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let JλB=(I+λB)-1 be the resolvent of B for λ which satisfies a≤λ≤b where [a,b]⊂(0,2α). Let κ∈(0,1) be a constant and S:C→C a ρ-strict pseudocontraction with ρ∈[0,1) such that F(S)∩(A+B)-10≠∅. For t∈(0,1-λ/2α), let {xt}⊂C be a net defined by
(3.1)xt=κ(1-ρ)1-κρSxt+1-κ1-κρJλB((1-t)xt-λAxt),
then the net {xt} converges strongly, as t→0+, to a point x̃=PF(S)∩(A+B)-10(0), where P is the metric projection.Remark 3.2.
Now, we show that the net{xt} defined by (3.1) is well defined. For any t∈(0,1-λ/2α), we define a mapping W:=κ(ρI+(1-ρ)S)+(1-κ)JλB((1-t)I-λA). Note that ρI+(1-ρ)S (by Lemma 2.1), JλB, and I-λ/(1-t)A (by Lemma 2.2) are nonexpansive. For any x, y∈C, we have
(3.2)‖Wx-Wy‖=‖κ(ρx+(1-ρ)Sx)+(1-κ)JλB((1-t)x-λAx)-κ(ρy+(1-ρ)Sy)-(1-κ)JλB((1-t)y-λAy)‖≤κ‖ρ(x-y)+(1-ρ)(Sx-Sy)‖+(1-κ)‖(1-t)(x-λ1-tAx)-(1-t)(y-λ1-tAy)‖≤[1-(1-κ)t]‖x-y‖,
which implies the mapping T is a contraction on C. We use xt to denote the unique fixed point of W in C. Therefore, {xt} is well defined. We can rewrite (3.1) as
(3.3)xt=κ(ρxt+(1-ρ)Sxt)+(1-κ)JλB((1-t)xt-λAxt).In order to prove Theorem3.1, we need the following propositions.Proposition 3.3.
Under the assumptions of Theorem3.1, the net {xt} defined by (3.1) and hence (3.3) is bounded.Proof.
Letz∈F(S)∩(A+B)-10. It follows that
(3.4)z=Sz=ρz+(1-ρ)Sz=JλB(z-λAz),
for all λ>0. We can write JλB(z-λAz) as JλB(tz+(1-t)(z-λAz/(1-t))), for all t∈(0,1). Since JλB is nonexpansive for all λ>0, we have
(3.5)‖JλB((1-t)xt-λAxt)-z‖2=‖JλB((1-t)(xt-λAxt/(1-t)))-JλB(tz+(1-t)(z-λAz/(1-t)))‖2≤‖((1-t)(xt-λAxt/(1-t)))-(tz+(1-t)(z-λAz/(1-t)))‖2=‖(1-t)((xt-λAxt/(1-t))-(z-λAz/(1-t)))+t(-z)‖2.
By using the convexity of ∥·∥ and the α-inverse strong monotonicity of A, we derive
(3.6)‖(1-t)((xt-λAxt/(1-t))-(z-λAz/(1-t)))+t(-z)‖2≤(1-t)‖(xt-λAxt/(1-t))-(z-λAz/(1-t))‖2+t‖z‖2=(1-t)‖(xt-z)-λ(Axt-Az)/(1-t)‖2+t‖z‖2=(1-t)(‖xt-z‖2-2λ1-t〈Axt-Az,xt-z〉+λ2(1-t)2‖Axt-Az‖2)+t‖z‖2≤(1-t)(‖xt-z‖2-2αλ1-t‖Axt-Az‖2+λ2(1-t)2‖Axt-Az‖2)+t‖z‖2=(1-t)(‖xt-z‖2+λ(1-t)2(λ-2(1-t)α)‖Axt-Az‖2)+t‖z‖2.
By the assumption, we have λ-2(1-t)α≤0, for all t∈(0,1-λ/2α). Then, from (3.5) and (3.6), we obtain
(3.7)‖JλB((1-t)xt-λAxt)-z‖2≤(1-t)(‖xt-z‖2+λ(1-t)2(λ-2(1-t)α)‖Axt-Az‖2)+t‖z‖2≤(1-t)‖xt-z‖2+t‖z‖2.
It follows from (3.3) and (3.7) that
(3.8)‖xt-z‖2≤κ‖(ρI+(1-ρ)S)xt-z‖2+(1-κ)‖JλB((1-t)xt-λAxt)-z‖2≤κ‖xt-z‖2+(1-κ)‖JλB((1-t)xt-λAxt)-z‖2≤κ‖xt-z‖2+(1-κ)[(1-t)‖xt-z‖2+t‖z‖2].
It follows that
(3.9)‖xt-z‖≤‖z‖.
Therefore, {xt} is bounded.Remark 3.4.
SinceA is α-inverse strongly monotone, it is 1/α-Lipschitz continuous. At the same time, S is nonexpansive. So, from the boundedness, we deduce immediately that {Axt}, JλB((1-t)xt-λAxt), and {Sxt} are also bounded.Proposition 3.5.
Assume that all conditions in Theorem3.1 hold. Let {xt} be the net defined by (3.1), then one has limt→0+∥xt-Sxt∥=0 and limt→0+∥xt-JλB((1-t)xt-λAxt)∥=0.Proof.
By (3.7) and (3.8), we obtain
(3.10)‖xt-z‖2≤[1-(1-κ)t]‖xt-z‖2+λ(1-κ)(1-t)(λ-2(1-t)α)‖Axt-Az‖2+(1-κ)t‖z‖2.
So,
(3.11)λ(1-t)(2(1-t)α-λ)‖Axt-Az‖2≤t‖z‖2-t‖xt-z‖2⟶0.
Since liminft→0+(λ/(1-t))(2(1-t)α-λ)>0, we obtain
(3.12)lim⁡t→0+‖Axt-Az‖=0.
Next, we show ∥xt-Sxt∥→0. By using the firm nonexpansivity of JλB, we have
(3.13)‖JλB((1-t)xt-λAxt)-z‖2=‖JλB((1-t)xt-λAxt)-JλB(z-λAz)‖2≤〈(1-t)xt-λAxt-(z-λAz),JλB((1-t)xt-λAxt)-z〉=12(‖(1-t)xt-λAxt-(z-λAz)‖2+‖JλB((1-t)xt-λAxt)-z‖2-‖(1-t)xt-λ(Axt-λAz)-JλB((1-t)xt-λAxt)‖2).
By the nonexpansivity of I-λA/(1-t), we have
(3.14)‖(1-t)xt-λAxt-(z-λAz)‖2=‖(1-t)(xt-λAxt/(1-t)-(z-λAz/(1-t)))+t(-z)‖2≤(1-t)‖(xt-λAxt/(1-t)-(z-λAz/(1-t)))‖2+t‖z‖2≤(1-t)‖xt-z‖2+t‖z‖2.
It follows that
(3.15)‖JλB((1-t)xt-λAxt)-z‖2≤12((1-t)‖xt-z‖2+t‖z‖2+‖JλB((1-t)xt-λAxt)-z‖2-‖(1-t)xt-JλB((1-t)xt-λAxt)-λ(Axt-Az)‖2).
Thus,
(3.16)‖JλB((1-t)xt-λAxt)-z‖2≤(1-t)‖xt-z‖2+t‖z‖2-‖(1-t)xt-JλB((1-t)xt-λAxt)-λ(Axt-Az)‖2=(1-t)‖xt-z‖2+t‖z‖2-‖(1-t)xt-JλB((1-t)xt-λAxt)‖2+2λ〈(1-t)xt-JλB((1-t)xt-λAxt),Axt-Az〉-λ2‖Axt-Az‖2≤(1-t)‖xt-z‖2+t‖z‖2-‖(1-t)xt-JλB((1-t)xt-λAxt)‖2+2λ‖(1-t)xt-JλB((1-t)xt-λAxt)‖‖Axt-Az‖.
This together with (3.8) implies that
(3.17)‖xt-z‖2≤‖JλB((1-t)xt-λAxt)-z‖2≤(1-t)‖xt-z‖2+t‖z‖2-‖(1-t)xt-JλB((1-t)xt-λAxt)‖2+2λ‖(1-t)xt-JλB((1-t)xt-λAxt)‖‖Axt-Az‖.
Hence,
(3.18)‖(1-t)xt-JλB((1-t)xt-λAxt)‖2≤t(‖z‖2-‖xt-z‖2)+2λ‖(1-t)xt-JλB((1-t)xt-λAxt)‖‖Axt-Az‖.
Since ∥Axt-Az∥→0 (by (3.12)), we deduce
(3.19)lim⁡t→0+‖(1-t)xt-JλB((1-t)xt-λAxt)‖=0.
Therefore,
(3.20)lim⁡t→0+‖xt-JλB((1-t)xt-λAxt)‖=0.
Hence,
(3.21)lim⁡t→0+‖xt-Sxt‖=lim⁡t→0+‖xt-JλB((1-t)xt-λAxt)‖=0.Finally, we prove Theorem3.1.Proof.
From (3.5) and (3.8), we have
(3.22)‖xt-z‖2≤‖(1-t)((xt-λ1-tAxt)-(z-λ1-tAz))-tz‖2=(1-t)2‖(xt-λ1-tAxt)-(z-λ1-tAz)‖2-2t(1-t)〈z,(xt-λ1-tAxt)-(z-λ1-tAz)〉+t2‖z‖2≤(1-t)2‖xt-z‖2-2t(1-t)〈z,xt-λ1-t(Axt-Az)-z〉+t2‖z‖2=(1-2t)‖xt-z‖2+2t{-(1-t)〈z,xt-λ1-t(Axt-Az)-z〉+t2(‖z‖2+‖xt-z‖2)}.
It follows that
(3.23)‖xt-z‖2≤-〈z,xt-λ1-t(Axt-Az)-z〉+t2(‖z‖2+‖xt-z‖2)+t‖z‖‖xt-λ1-t(Axt-Az)-z‖≤-〈z,xt-λ1-t(Axt-Az)-z〉+tM,
where M is some constant such that
(3.24)sup⁡t∈(0,1-λ/2α){‖z‖2+‖xt-z‖2+‖z‖‖xt-λ1-t(Axt-Az)-z‖}≤M.
Next we show that {xt} is relatively norm compact as t→0+. Assume that {tn}⊂(0,1-λ/2α) is such that tn→0+ as n→∞. Put xn:=xtn. From (3.23), we have
(3.25)‖xn-z‖2≤-〈z,xn-λ1-tn(Axn-Az)-z〉+tnM,z∈F(S)∩(A+B)-10.
Since {xn} is bounded, without loss of generality, we may assume that xn⇀x̃∈C. From (3.21), we have
(3.26)lim⁡n→∞‖xn-Sxn‖=0.
We can apply Lemma 2.3 to (3.26) to deduce x̃∈F(S). Further, we show that x̃ is also in (A+B)-10. Let v∈Bu. Set zn=JλB((1-tn)xn-λAxn), for all n, then we have
(3.27)(1-tn)xn-λAxn∈(I+λB)zn⟹1-tnλxn-Axn-znλ∈Bzn.
Since B is monotone, we have, for (u,v)∈B,
(3.28)〈1-tnλxn-Axn-znλ-v,zn-u〉≥0⟹〈(1-tn)xn-λAxn-zn-λv,zn-u〉≥0⟹〈Axn+v,zn-u〉≤1λ〈xn-zn,zn-u〉-tnλ〈xn,zn-u〉⟹〈Ax̃+v,zn-u〉≤1λ〈xn-zn,zn-u〉-tnλ〈xn,zn-u〉+〈Ax̃-Axn,zn-u〉⟹〈Ax̃+v,zn-u〉≤1λ‖xn-zn‖‖zn-u‖+tnλ‖xn‖‖zn-u‖+‖Ax̃-Axn‖‖zn-u‖.
It follows that
(3.29)〈Ax̃+v,x̃-u〉≤1λ‖xn-zn‖‖zn-u‖+tnλ‖xn‖‖zn-u‖+‖Ax̃-Axn‖‖zn-u‖+〈Ax̃+v,x̃-zn〉.
Since
(3.30)〈xn-x̃,Axn-Ax̃〉≥α‖Axn-Ax̃‖2,Axn→Az, and xn⇀x̃, we have Axn→Ax̃. We also observe that tn→0, ∥xn-zn∥→0 and zn⇀x̃. Then, from (3.29), we derive
(3.31)〈-Ax̃-v,x̃-u〉≥0.
Since B is maximal monotone, we have -Ax̃∈Bx̃. This shows that 0∈(A+B)x̃. So, we have x̃∈F(S)∩(A+B)-10. Hence, xn-(λ/1-tn)(Axn-Az)⇀x̃ because of ∥Axn-Az∥→0. Therefore, we can substitute x̃ for z in (3.25) to get
(3.32)‖xn-x̃‖2≤-〈x̃,xn-λ1-tn(Axn-Ax̃)-x̃〉+tnM.
Consequently, the weak convergence of {xn} to x̃ actually implies that xn→x̃. This has proved the relative norm compactness of the net {xt} as t→0+.
Now we return to (3.25) and take the limit as n→∞ to get
(3.33)‖x̃-z‖2≤-〈z,x̃-z〉,z∈F(S)∩(A+B)-10.
Equivalently,
(3.34)‖x̃‖2≤〈x̃,z〉,z∈F(S)∩(A+B)-10.
This clearly implies that
(3.35)‖x̃‖≤‖z‖,z∈F(S)∩(A+B)-10.
Therefore, x̃ is the minimum norm element in F(S)∩(A+B)-10. This completes the proof.Corollary 3.6.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H, and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let JλB=(I+λB)-1 be the resolvent of B for λ which satisfies a≤λ≤b where [a,b]⊂(0,2α). Let κ∈(0,1) be a constant and S:C→C a nonexpansive mapping such that F(S)∩(A+B)-10≠∅. For t∈(0,1-λ/2α), let {xt}⊂C be a net defined by
(3.36)xt=κ(1-ρ)1-κρSxt+1-κ1-κρJλB((1-t)xt-λAxt),
then the net {xt} converges strongly, as t→0+, to a point x̃=PF(S)∩(A+B)-10(0).Corollary 3.7.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H, and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let JλB=(I+λB)-1 be the resolvent of B for λ>0 such that (A+B)-10≠∅. Let λ be a constant satisfying a≤λ≤b where [a,b]⊂(0,2α). For t∈(0,1-λ/2α), let {xt}⊂C be a net generated by
(3.37)xt=JλB((1-t)xt-λAxt),
then the net {xt} converges strongly, as t→0+, to a point x̃=P(A+B)-10(0).


## 4. Applications

Next, we consider the problem for finding the minimum norm solution of a mathematical model related to equilibrium problems. LetC be a nonempty, closed, and convex subset of a Hilbert space, and let G:C×C→R be a bifunction satisfying the following conditions:(E1)
G(x,x)=0, for all x∈C,(E2)
G is monotone, that is, G(x,y)+G(y,x)≤0, for all x, y∈C,(E3)
for allx, y, z∈C, limsupt↓0G(tz+(1-t)x,y)≤G(x,y),(E4)
for allx∈C, G(x,·) is convex and lower semicontinuous.Then, the mathematical model related to equilibrium problems (with respect toC) is to find x̃∈C such that(4.1)G(x̃,y)≥0,
for all y∈C. The set of such solutions x̃ is denoted by EP(G). The following lemma appears implicitly in Blum and Oettli [19].Lemma 4.1.
LetC be a nonempty, closed, and convex subset of H, and let G be a bifunction of C×C into R satisfying (E1)–(E4). Let r>0 and x∈H, then there exists z∈C such that
(4.2)G(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.The following lemma was given in Combettes and Hirstoaga [20].Lemma 4.2.
Assume thatG:C×C→R satisfies (E1)–(E4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
(4.3)Tr(x)={z∈C:G(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then, the following hold:(1)
Tr is single valued,(2)
Tr is a firmly nonexpansive mapping, that is, for all x,y∈H,
(4.4)‖Trx-Try‖2≤〈Trx-Try,x-y〉,(3)
F(Tr)=EP(G),(4)
EP(G) is closed and convex.
We call suchTr the resolvent of G for r>0. Using Lemmas 4.1 and 4.2, we have the following lemma. See [18] for a more general result.Lemma 4.3.
LetH be a Hilbert space, and let C be a nonempty, closed, and convex subset of H. Let G:C×C→R satisfy (E1)–(E4). Let AG be a multivalued mapping of H into itself defined by
(4.5)AGx={{z∈H:G(x,y)≥〈y-x,z〉,∀y∈C},x∈C,∅,x∉C,
then, EP(G)=AG-1(0), and AG is a maximal monotone operator with dom(AG)⊂C. Further, for any x∈H and r>0, the resolvent Tr of G coincides with the resolvent of AG, that is,
(4.6)Trx=(I+rAG)-1x.
From Lemma4.3, Theorem 3.1, and Lemma 4.2, one has the following results.Corollary 4.4.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let G be a bifunction from C×C→R satisfying (E1)–(E4), and let Tr be the resolvent of G for r>0. Let κ∈(0,1) be a constant and S:C→C a ρ-strict pseudocontraction with ρ∈[0,1) such that F(S)∩EP(G)≠∅. For t∈(0,1), let {xt}⊂C be a net defined by
(4.7)xt=κ(1-ρ)1-κρSxt+1-κ1-κρTr((1-t)xt),
then the net {xt} converges strongly, as t→0+, to a point x̃=PF(S)∩EP(G)(0).Corollary 4.5.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let G be a bifunction from C×C→R satisfying (E1)–(E4), and let Tr be the resolvent of G for r>0. Let κ∈(0,1) be a constant and S:C→C be a nonexpansive mapping such that F(S)∩EP(G)≠∅. For t∈(0,1), let {xt}⊂C be a net defined by
(4.8)xt=κ(1-ρ)1-κρSxt+1-κ1-κρTr((1-t)xt),
then the net {xt} converges strongly, as t→0+, to a point x̃=PF(S)∩EP(G)(0).Corollary 4.6.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let G be a bifunction from C×C→R satisfying (E1)–(E4), and let Tr be the resolvent of G for r>0. Suppose EP(G)≠∅. For t∈(0,1), let {xt}⊂C be a net generated by
(4.9)xt=Tr((1-t)xt),t∈(0,1),
then the net {xt} converges strongly, as t→0+, to a point x̃=PEP(G)(0).



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*Source: 517460-2012-05-21.xml*