AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation51746010.1155/2012/517460517460Research ArticleAn Implicit Algorithm for Maximal Monotone Operators and Pseudocontractive MappingsLiHong-Jun1LiouYeong-Cheng2LiCun-Lin3Noor Muhammad Aslam4YaoYonghong1NoorKhalida Inayat1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387Chinatjpu.edu.cn2Department of Information ManagementCheng Shiu UniversityKaohsiung 833Taiwancsu.edu.tw3School of Management, North University for Nationalities, Yinchuan 750021China4Mathematics DepartmentCOMSATS Institute of Information Technology, IslamabadPakistanciit.edu.pk20122152012201221022012280220122012Copyright © 2012 Hong-Jun Li et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 0ρ<1 such thatSx-Sy2x-y2+ρ(I-S)x-(I-S)y2,x,yC. 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 toSx-Sy,x-yx-y2-1-ρ2(I-S)x-(I-S)y2,x,yC. We denote by F(S) the set of fixed points of S.

A mapping A:CH is said to be α-inverse strongly monotone ifAx-Ay,x-yαAx-Ay2, for some α>0 and for all x, yC. It is known that if A is an α-inverse strongly monotone, then Ax-Ay1/αx-y  for all x, yC.

Let B be a mapping of H into 2H. The effective domain of B is denoted by dom(B), that is, dom(B)={xH:Bx}. A multi valued mapping B is said to be a monotone operator on H iffx-y,u-v0, for all x, ydom(B), uBx, and vBy. 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={xH:0Bx}.

For a maximal monotone operator B on H and λ>0, we may define a single-valued operator JλB=(I+λB)-1:Hdom(B), which is called the resolvent of B for λ. It is known that the resolvent JλB is firmly nonexpansive, that is,JλBx-JλBy2JλBx-JλBy,x-y, for all x, yC 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 . Especially, Takahashi et al.  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=xC and let {xn}C, be a sequence generated by xn+1=βnxn+(1-βn)S(αnx+(1-αn)JλnB(xn-λnAxn)), for all n0, where {λn}(0,2α), {αn}(0,1) and {βn}(0,1) satisfy 0<aλnb<2α,<cβnd<1,limn(λn+1-λn)=0,limnαn=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),xH. We use the following notation:

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

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

We need the following lemmas for the next section.

Lemma 2.1 (see[<xref ref-type="bibr" rid="B14">14</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S:CH be a ρ-strict pseudo contraction. Define T:CH by Tx=αx+(1-α)Tx for each xC, then, as α[ρ,1), T is nonexpansive such that F(S)=F(T).

Lemma 2.2 (see[<xref ref-type="bibr" rid="B15">15</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping A:CH be α-inverse strongly monotone and λ>0 a constant, then one has (I-λA)x-(I-λA)y2x-y2+λ(λ-2α)Ax-Ay2,x,yC. In particular, if 0λ2α, then I-λA is nonexpansive.

Lemma 2.3 (see[<xref ref-type="bibr" rid="B14">14</xref>]).

Let C be a nonempty, closed and convex of a real Hilbert space H. Let T:CC be a λ-strictly pseudo-contractive mapping, then I-T is demi closed at 0, that is, if xnxC and xn-Txn0, then x=Tx.

Lemma 2.4 (see[<xref ref-type="bibr" rid="B16">16</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space X, and let {βn} be a sequence in [0,1] with 0<liminfnβnlimsupnβn<1. Suppose that xn+1=(1-βn)yn+βnxn for all n0 and limsupn(yn+1-yn-xn+1-xn)0, then limnyn-xn=0.

Lemma 2.5 (see[<xref ref-type="bibr" rid="B17">17</xref>]).

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=1γn=,

limsupnδn0 or n=1|δnγn|<, then limnan=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 aλb where [a,b](0,2α). Let κ(0,1) be a constant and S:CC 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 t0+, 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, yC, 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)κρ(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 zF(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)-z2=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+tz2=(1-t)(xt-z)-λ(Axt-Az)/(1-t)2+tz2=(1-t)(xt-z2-2λ1-tAxt-Az,xt-z+λ2(1-t)2Axt-Az2)+tz2(1-t)(xt-z2-2αλ1-tAxt-Az2+λ2(1-t)2Axt-Az2)+tz2=(1-t)(xt-z2+λ(1-t)2(λ-2(1-t)α)Axt-Az2)+tz2. By the assumption, we have λ-2(1-t)α0, for all t(0,1-λ/2α). Then, from (3.5) and (3.6), we obtain JλB((1-t)xt-λAxt)-z2(1-t)(xt-z2+λ(1-t)2(λ-2(1-t)α)Axt-Az2)+tz2(1-t)xt-z2+tz2. It follows from (3.3) and (3.7) that xt-z2κ(ρI+(1-ρ)S)xt-z2+(1-κ)JλB((1-t)xt-λAxt)-z2κxt-z2+(1-κ)JλB((1-t)xt-λAxt)-z2κxt-z2+(1-κ)[(1-t)xt-z2+tz2]. It follows that xt-zz. 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 limt0+xt-Sxt=0 and limt0+xt-JλB((1-t)xt-λAxt)=0.

Proof.

By (3.7) and (3.8), we obtain xt-z2[1-(1-κ)t]xt-z2+λ(1-κ)(1-t)(λ-2(1-t)α)Axt-Az2+(1-κ)tz2. So, λ(1-t)(2(1-t)α-λ)Axt-Az2tz2-txt-z20. Since liminft0+(λ/(1-t))(2(1-t)α-λ)>0, we obtain limt0+Axt-Az=0. Next, we show xt-Sxt0. By using the firm nonexpansivity of JλB, we have JλB((1-t)xt-λAxt)-z2=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)-z2-(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+tz2(1-t)xt-z2+tz2. It follows that JλB((1-t)xt-λAxt)-z212((1-t)xt-z2+tz2+JλB((1-t)xt-λAxt)-z2-(1-t)xt-JλB((1-t)xt-λAxt)-λ(Axt-Az)2). Thus, JλB((1-t)xt-λAxt)-z2(1-t)xt-z2+tz2-(1-t)xt-JλB((1-t)xt-λAxt)-λ(Axt-Az)2=(1-t)xt-z2+tz2-(1-t)xt-JλB((1-t)xt-λAxt)2+2λ(1-t)xt-JλB((1-t)xt-λAxt),Axt-Az-λ2Axt-Az2(1-t)xt-z2+tz2-(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-z2JλB((1-t)xt-λAxt)-z2(1-t)xt-z2+tz2-(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)2t(z2-xt-z2)+2λ(1-t)xt-JλB((1-t)xt-λAxt)Axt-Az. Since Axt-Az0 (by (3.12)), we deduce limt0+(1-t)xt-JλB((1-t)xt-λAxt)=0. Therefore, limt0+xt-JλB((1-t)xt-λAxt)=0. Hence, limt0+xt-Sxt=limt0+xt-JλB((1-t)xt-λAxt)=0.

Finally, we prove Theorem 3.1.

Proof.

From (3.5) and (3.8), we have xt-z2(1-t)((xt-λ1-tAxt)-(z-λ1-tAz))-tz2=(1-t)2(xt-λ1-tAxt)-(z-λ1-tAz)2-2t(1-t)z,(xt-λ1-tAxt)-(z-λ1-tAz)+t2z2(1-t)2xt-z2-2t(1-t)z,xt-λ1-t(Axt-Az)-z+t2z2=(1-2t)xt-z2+2t{-(1-t)z,xt-λ1-t(Axt-Az)-z+t2(z2+xt-z2)}. It follows that xt-z2-z,xt-λ1-t(Axt-Az)-z+t2(z2+xt-z2)+tzxt-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){z2+xt-z2+zxt-λ1-t(Axt-Az)-z}M. Next we show that {xt} is relatively norm compact as t0+. Assume that {tn}(0,1-λ/2α) is such that tn0+ as n. Put xn:=xtn. From (3.23), we have xn-z2-z,xn-λ1-tn(Axn-Az)-z+tnM,  zF(S)(A+B)-10. Since {xn} is bounded, without loss of generality, we may assume that xnx̃C. From (3.21), we have limnxn-Sxn=0. We can apply Lemma 2.3 to (3.26) to deduce x̃F(S). Further, we show that x̃ is also in (A+B)-10. Let vBu. Set zn=JλB((1-tn)xn-λAxn), for all n, then we have (1-tn)xn-λAxn(I+λB)zn1-tnλxn-Axn-znλBzn.   Since B is monotone, we have, for (u,v)B, 1-tnλxn-Axn-znλ-v,zn-u0(1-tn)xn-λAxn-zn-λv,zn-u0Axn+v,zn-u1λxn-zn,zn-u-tnλxn,zn-uAx̃+v,zn-u1λxn-zn,zn-u-tnλxn,zn-u+Ax̃-Axn,zn-uAx̃+v,zn-u1λxn-znzn-u+tnλxnzn-u+Ax̃-Axnzn-u. It follows that Ax̃+v,x̃-u1λxn-znzn-u+tnλxnzn-u+Ax̃-Axnzn-u+Ax̃+v,x̃-zn. Since xn-x̃,Axn-Ax̃αAxn-Ax̃2,AxnAz, and xnx̃, we have AxnAx̃. We also observe that tn0, xn-zn0 and znx̃. Then, from (3.29), we derive -Ax̃-v,x̃-u0. Since B is maximal monotone, we have -Ax̃Bx̃. This shows that 0(A+B)x̃. So, we have x̃F(S)(A+B)-10. Hence, xn-(λ/1-tn)  (Axn-Az)x̃ because of Axn-Az0. Therefore, we can substitute x̃ for z in (3.25) to get xn-x̃2-x̃,xn-λ1-tn(Axn-Ax̃)-x̃+tnM. Consequently, the weak convergence of {xn} to x̃ actually implies that xnx̃. This has proved the relative norm compactness of the net {xt} as t0+.

Now we return to (3.25) and take the limit as n to get   x̃-z2-z,x̃-z,  zF(S)(A+B)-10. Equivalently,   x̃2x̃,z,  zF(S)(A+B)-10. This clearly implies that x̃z,  zF(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 aλb where [a,b](0,2α). Let κ(0,1) be a constant and S:CC 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 t0+, 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 aλb 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 t0+, 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×CR be a bifunction satisfying the following conditions:

G(x,x)=0, for all xC,

G is monotone, that is, G(x,y)+G(y,x)0, for all x, yC,

for all x, y, zC, limsupt0G(tz+(1-t)x,y)G(x,y),

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

Then, the mathematical model related to equilibrium problems (with respect to C) is to find x̃C such thatG(x̃,y)0, for all yC. The set of such solutions x̃ is denoted by EP(G). The following lemma appears implicitly in Blum and Oettli .

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 xH, then there exists zC such that G(z,y)+1ry-z,z-x0,yC.

The following lemma was given in Combettes and Hirstoaga .

Lemma 4.2.

Assume that G:C×CR satisfies (E1)–(E4). For r>0 and xH, define a mapping Tr:HC as follows: Tr(x)={zC:G(z,y)+1ry-z,z-x0,  yC}, for all xH. Then, the following hold:

Tr is single valued,

Tr is a firmly nonexpansive mapping, that is, for all x,yH,   Trx-Try2Trx-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  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×CR satisfy (E1)–(E4). Let AG be a multivalued mapping of H into itself defined by AGx={{zH:G(x,y)y-x,z,yC},xC,,xC, then, EP(G)=AG-1(0), and AG is a maximal monotone operator with dom(AG)C. Further, for any xH 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×CR satisfying (E1)–(E4), and let Tr be the resolvent of G for r>0. Let κ(0,1) be a constant and S:CC 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 t0+, 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×CR satisfying (E1)–(E4), and let Tr be the resolvent of G for r>0. Let κ(0,1) be a constant and S:CC 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 t0+, 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×CR 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 t0+, 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.

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