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.

1. Introduction

Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H.

Recall that S is said to be a strictly pseudo contractive mapping if there exists a constant 0≤ρ<1 such that‖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〈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 mapping A:C→H is said to be α-inverse strongly monotone if〈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.

Let B 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〈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 operator B 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,‖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.

Let C 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
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
0<a≤λn≤b<2α,<c≤βn≤d<1,limn→∞(λn+1-λn)=0,limn→∞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: for λ>0 and μ>0, there holds the identityJλBx=JμB(μλx+(1-μλ)JλBx),x∈H.
We use the following notation:

xn⇀x stands for the weak convergence of {xn} to x;

xn→x stands for the strong convergence of {xn} to x.

We need the following lemmas for the next section.

Let C 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).

Let C 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
‖(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.

Let C 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.

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.

Assume that {an} is a sequence of nonnegative real numbers such that
an+1≤(1-γn)an+δnγn,
where {γn} is a sequence in (0,1) and {δn} is a sequence such that

∑n=1∞γn=∞,

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.

Let C 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
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
‖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
xt=κ(ρxt+(1-ρ)Sxt)+(1-κ)JλB((1-t)xt-λAxt).

In order to prove Theorem 3.1, we need the following propositions.

Proposition 3.3.

Under the assumptions of Theorem 3.1, the net {xt} defined by (3.1) and hence (3.3) is bounded.

Proof.

Let z∈F(S)∩(A+B)-10. It follows that
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
‖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
‖(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
‖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
‖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
‖xt-z‖≤‖z‖.
Therefore, {xt} is bounded.

Remark 3.4.

Since A 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 Theorem 3.1 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
‖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,
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
limt→0+‖Axt-Az‖=0.
Next, we show ∥xt-Sxt∥→0. By using the firm nonexpansivity of JλB, we have
‖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
‖(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
‖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,
‖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
‖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,
‖(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
limt→0+‖(1-t)xt-JλB((1-t)xt-λAxt)‖=0.
Therefore,
limt→0+‖xt-JλB((1-t)xt-λAxt)‖=0.
Hence,
limt→0+‖xt-Sxt‖=limt→0+‖xt-JλB((1-t)xt-λAxt)‖=0.

Finally, we prove Theorem 3.1.

Proof.

From (3.5) and (3.8), we have
‖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
‖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
supt∈(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
‖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
limn→∞‖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
(1-tn)xn-λAxn∈(I+λB)zn⟹1-tnλxn-Axn-znλ∈Bzn.
Since B is monotone, we have, for (u,v)∈B,
〈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
〈Ax̃+v,x̃-u〉≤1λ‖xn-zn‖‖zn-u‖+tnλ‖xn‖‖zn-u‖+‖Ax̃-Axn‖‖zn-u‖+〈Ax̃+v,x̃-zn〉.
Since
〈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
〈-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
‖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
‖x̃-z‖2≤-〈z,x̃-z〉,z∈F(S)∩(A+B)-10.
Equivalently,
‖x̃‖2≤〈x̃,z〉,z∈F(S)∩(A+B)-10.
This clearly implies that
‖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.

Let C 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
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.

Let C 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
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. Let C be a nonempty, closed, and convex subset of a Hilbert space, and let G:C×C→R be a bifunction satisfying the following conditions:

G(x,x)=0, for all x∈C,

G is monotone, that is, G(x,y)+G(y,x)≤0, for all x, y∈C,

for all x, y, z∈C, limsupt↓0G(tz+(1-t)x,y)≤G(x,y),

for all x∈C, G(x,·) is convex and lower semicontinuous.

Then, the mathematical model related to equilibrium problems (with respect to C) is to find x̃∈C such thatG(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.

Let C 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
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 that G:C×C→R satisfies (E1)–(E4). For r>0 and x∈H, define a mapping Tr:H→C as follows:
Tr(x)={z∈C:G(z,y)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈H. Then, the following hold:

Tr is single valued,

Tr is a firmly nonexpansive mapping, that is, for all x,y∈H,
‖Trx-Try‖2≤〈Trx-Try,x-y〉,

F(Tr)=EP(G),

EP(G) is closed and convex.

We call such Tr 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.

Let H 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
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,
Trx=(I+rAG)-1x.

From Lemma 4.3, Theorem 3.1, and Lemma 4.2, one has the following results.

Corollary 4.4.

Let C 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
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.

Let C 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
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.

Let C 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
xt=Tr((1-t)xt),t∈(0,1),
then the net {xt} converges strongly, as t→0+, to a point x̃=PEP(G)(0).

Acknowledgments

H.-J. Li was supported in part by Colleges and Universities Science and Technology Development Foundation (20110322) of Tianjin. Y.-C. Liou was supported in part by NSC 100-2221-E-230-012. C.-L. Li was supported in part by NSFC 71161001-G0105. Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.

NoorM. A.NoorK. I.Sensitivity analysis for quasi-variational inclusionsNoorM. A.Generalized set-valued variational inclusions and resolvent equationsYaoY.YaoJ. C.On modified iterative method for nonexpansive mappings and monotone mappingsChenR.SuY.XuH. K.Regularization and iteration methods for a class of monotone variational inequalitiesFangY. P.HuangN. J.H-monotone operator and resolvent operator technique for variational inclusionsTakahashiS.TakahashiW.ToyodaM.Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spacesYaoY.ShahzadN.New methods with perturbations for non-expansive mappings in Hilbert spacesRockafellarR. T.On the maximal monotonicity of subdifferential mappingsAslam NoorM.RassiasT. M.Projection methods for monotone variational inequalitiesNoorM. A.HuangZ.Some resolvent iterative methods for variational inclusions and nonexpansive mappingsYaoY.LiouY.-C.KangS. M.Two-stepprojection methods for a system of variational inequality problems in Banach spacesJournal of Global Optimization. In press10.1007/s10898-011-9804-0YaoY.ChenR.LiouY.-C.A unified implicit algorithm for solving the triple-hierarchical constrained optimization problemAdlyS.Perturbed algorithms and sensitivity analysis for a general class of variational inclusionsMarinoG.XuH.-K.Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spacesTakahashiW.ToyodaM.Weak convergence theorems for nonexpansive mappings and monotone mappingsSuzukiT.Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spacesXuH.-K.Iterative algorithms for nonlinear operatorsAoyamaK.KimuraY.TakahashiW.ToyodaM.On a strongly nonexpansive sequence in Hilbert spacesBlumE.OettliW.From optimization and variational inequalities to equilibrium problemsCombettesP. L.HirstoagaS. A.Equilibrium programming in Hilbert spacesYaoY.LiouY.-C.YaoJ.-C.An extragradient method for fixed point problems and variational inequality problemsYaoY.LiouY.-C.WuY.-J.An extragradient method for mixed equilibrium problems and fixed point problemsYaoY.LiouY. C.MarinoG.Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spacesCianciarusoF.MarinoG.MugliaL.YaoY.On a two-step algorithm for hierarchical fixed point problems and variational inequalities