Volume 2010, Article ID 547828,9pages doi:10.1155/2010/547828
Research Article
Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces
Yuan Qing,
1Xiaolong Qin,
1Haiyun Zhou,
2and Shin Min Kang
31Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea
Correspondence should be addressed to Shin Min Kang,[email protected] Received 16 July 2010; Revised 30 November 2010; Accepted 20 December 2010 Academic Editor: Ljubomir B. Ciric
Copyrightq2010 Yuan Qing 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.
Let H be a Hilbert space and C a nonempty closed convex subset of H. Let A : C → H be a maximal monotone mapping and f : C → C a bounded demicontinuous strong pseudocontraction. Let{xt}be the unique solution to the equationfx xtAx. Then{xt}is bounded if and only if{xt}converges strongly to a zero point of A ast → ∞which is the unique solution inA−10, whereA−10denotes the zero set ofA, to the following variational inequality fp−p, y−p ≤0, for ally∈A−10.
1. Introduction and Preliminaries
Throughout this work, we always assume thatHis a real Hilbert space, whose inner product and norm are denoted by·,·and · , respectively. Let Cbe a nonempty closed convex subset ofHandAa nonlinear mapping. We useDAandRAto denote the domain and the range of the mappingA. → anddenote strong and weak convergence, respectively.
Recall the following well-known definitions.
1A mappingA:C → His said to be monotone if Ax−Ay, x−y
≥0, ∀x, y∈C. 1.1
2The single-valued mappingA : C → His maximal if the graphGAofAis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingAis maximal if and only if forx, Ax∈H×H,x−y, Ax−g ≥0 for everyy, g∈GAimpliesgAy.
3A : C → H is said to be pseudomonotone if for any sequence {xn} in C which converges weakly to an elementxinCwith lim supn→ ∞Axn, xn−x ≤0 we have
lim inf
n→ ∞
Axn, xn−y
≥
Ax, x−y
, ∀y∈C. 1.2
4A :C → His said to be bounded if it carries bounded sets into bounded sets; it is coercive ifAx, x/x → ∞asx → ∞.
5LetX, Ybe linear normed spaces.T :DT⊂X → Yis said to be demicontinuous if, for any{xn} ⊂DTwe haveTxn Tx0asxn → x0.
6LetTbe a mapping of a linear normed spaceXinto its dual spaceX∗.Tis said to be hemicontinuous if it is continuous from each line segment inXto the weak topology inX∗.
7The mapping f with the domain Df and the range Rf in H is said to be pseudocontractive if
fx−f y
, x−y
≤x−y2, ∀x, y∈D f
. 1.3
8The mappingfwith the domainDfand the rangeRfinHis said to be strongly pseudocontractive if there exists a constantβ∈0,1such that
fx−f y
, x−y
≤βx−y2, ∀x, y∈D f
. 1.4
Remark 1.1. For the maximal monotone operatorA, we can defined the resolvent of Aby Jt ItA−1, t >0. It is well know thatJt:H → DAis nonexpansive.
Remark 1.2. It is well-known that ifTis demicontinuous, thenTis hemicontinuous, however, the converse, in general, may not be true. In reflexive Banach spaces, for monotone mappings defined on the whole Banach space, demicontinuity is equivalent to hemicontinuity.
To find zeroes of maximal monotone operators is the central and important topics in nonlinear functional analysis. We observe thatp is a zero of the monotone mapping Aif and only if it is a fixed point of the pseudocontractive mappingT : I−A. Consequently, considerable research works, especially, for the past 40 years or more, have been devoted to the existence and convergence of zero points for monotone mappings or fixed points of pseudocontractions, see, for instance,1–23.
In 1965, Browder1proved the existence result of fixed point for demicontinuous pseudocontractions in Hilbert spaces. To be more precise, he proved the following theorem.
Theorem B1. LetHbe a Hilbert space,Ca nonempty bounded and closed convex subset ofHand T :C → Ca demicontinuous pseduo-contraction. ThenT has a fixed point inC.
In 1968, Browder4proved the existence results of zero points for maximal monotone mappings in reflexive Banach spaces. To be more precise, he proved the following theorem.
Theorem B2. LetX be a reflexive Banach space, T1 : DT1 ⊆ X → 2X∗ a maximal monotone mapping andT2a bounded, pseudomonotone and coercive mapping. Then, for anyh∈X∗, there exists u∈Xsuch thath∈T1T2u, orRT1T2is all ofX∗.
For the existence of continuous paths for continuous pseudocontractions in Banach spaces, Morales and Jung15proved the following theorem.
Theorem MJ. LetEbe a Banach space. Suppose thatCis a nonempty closed convex subset ofEand T :C → Eis a continuous pseudocontraction satisfying the weakly inward condition. Then for each z ∈ C, there exists a unique continuous patht → yt ∈ C,t ∈ 0,1, which satisfies the following equationyt 1−tztTyt.
In 2002, Lan and Wu14partially improved the result of Morales and Jung15from continuous pseudocontractions to demicontinuous pseudocontractions in the framework of Hilbert spaces. To be more precise, they proved the following theorem.
Theorem LW. Let K be a bounded closed convex set in H. Assume that T : K → H is a demicontinuous weakly inward pseudocontractive map. ThenT has a fixed point inK. Moreover;
for everyx0∈K,{xt}defined byxt 1−tTxttx0converges to a fixed point ofT.
In this work, motivated by Browder 3, Lan and Wu 14, Morales and Jung 15, Song and Chen19, and Zhou22,23, we consider the existence of convergence of paths for maximal monotone mappings in the framework of real Hilbert spaces.
2. Main Results
Lemma 2.1. LetCbe a nonempty closed convex subset of a Hilbert spaceH and T : C → H a demicontinuous monotone mapping. ThenTis pseudomonotone.
Proof. For any sequence{xn} ⊂Cwhich converges weakly to an elementxinCsuch that lim sup
n→ ∞ Txn, xn−x ≤0, 2.1
we see from the monotonicity ofTthat
Tx, xn−x ≤ Txn, xn−x. 2.2
Combining2.1with2.2, we obtain that lim sup
n→ ∞ Tx, xn−x0. 2.3 By takingz, g∈GraphT, we arrive at
Txn, xn−zTxn, xn−xTxn, x−z, 2.4
which yields that
lim inf
n→ ∞ Txn, xn−zlim inf
n→ ∞ Txn, x−z. 2.5
Noticing that
g, xn−z
≤ Txn, xn−z, 2.6 we have
g, x−z
≤lim inf
n→ ∞ Txn, x−z. 2.7
Letzt 1−txty, for ally∈Candt∈0,1. By takingzt zandgt g in2.7, we see that
gt, x−y
≤lim inf
n→ ∞
Txn, x−y
. 2.8
Noting thatzt → x,t → 0,gtTzt, andT :C → His demicontinuous, we havegtTzt Txast → 0, and hence
lim inf
n→ ∞
Txn, xn−y
lim inf
n→ ∞
Txn, x−y
≥
Tx, x−y
. 2.9
This completes the proof.
Lemma 2.2. LetCbe a nonempty closed convex subset of a Hilbert spaceH,A:C → Ha maximal monotone mapping, andT :C → Ha bounded, demicontinuous, and strongly monotone mapping.
ThenAT has a unique zero inC.
Proof. By usingLemma 2.1and Theorem B2, we can obtain the desired conclusion easily.
Lemma 2.3. LetCbe a nonempty closed convex subset of a Hilbert spaceH,A:C → Ha maximal monotone mapping, andf :C → Ha bounded, demicontinuous strong pseudocontraction with the coefficientβ∈0,1. Fort >0, consider the equation
0TxtAx, 2.10
whereT I−f. Then, One has the following.
iEquation2.10has a unique solutionxt∈Cfor everyt >0.
iiIf{xt}is bounded, thenAxt → 0 ast → ∞.
iiiIfA−10/∅, then{xt}is bounded and satisfies xt−fxt, xt−p
≤0, ∀p∈A−10, 2.11 whereA−10denotes the zero set ofA.
Proof. iFromLemma 2.2, one can obtain the desired conclusion easily.
iiWe usext ∈Cto denote the unique solution of2.10. That is, 0 TxttAxt. It follows that 0 I−fxttAxt. Notice that
Axt fxt−xt
t . 2.12
From the boundedness offand{xt}, one has limt→ ∞Axt0.
iiiForp∈A−10, one obtains that xt−p2
xt−p, xt−p fxt−p, xt−p −
Axt, xt−p
≤ fxt−f p
, xt−pf p
−p, xt−p −
Axt, xt−p
≤βxt−p2 f
p
−p, xt−p .
2.13
It follows that
xt−p2≤ 1 1−β
f p
−p, xt−p
. 2.14
That is,xt−p ≤1/1−βfp−p, for allt >0. This shows that{xt}is bounded. Noticing thatxt−fxt −tAxt, one arrives at
xt−fxt, xt−p −t
Axt, xt−p
≤0, ∀t >0. 2.15
This completes the proof.
Lemma 2.4. Let C be a nonempty closed convex subset of a Hilbert space H and A a maximal monotone mapping. Then C ⊆ IAC. If one definesg : C → C bygx IA−1x, for allx∈C, theng:C → Cis a nonexpansive mapping withFg A−10andx−gx ≤ Ax, whereFgdenotes the set of fixed points ofg.
Proof. Noticing that A is maximal monotone, one has RI A H. It follows that C ⊆ IAC. For anyx, y∈C, one sees that
gx−g
yIA−1x−IA−1y≤x−y, 2.16 which yields thatgis nonexpansive mapping. Notice that
xgx⇐⇒IAxx⇐⇒Ax0. 2.17
That is,Fg A−10. On the other hand, for anyx∈C, we have x−gxgg−1x−gx
≤g−1x−x
IAx−x Ax.
2.18
This completes the proof.
SetS 0,1. LetBSdenote the Banach space of all bounded real value functions onS with the supremum norm,X a subspace of BS, and μan element in X∗, where X∗ denotes the dual space ofX. Denote byμfthe value ofμatf∈X. Ifes 1, for allx∈S, sometimesμewill be denoted byμ1. WhenXcontains constants, a linear functionalμon Xis called a mean onXifμ1 μ1. We also know that ifXcontains constants, then the following are equivalent.
1μ1 μ1.
2infs∈Sfs≤μf≤sups∈Sfs, for allf∈X.
To prove our main results, we also need the following lemma.
Lemma 2.5see20, Lemma 4.5.4. LetCbe a nonempty and closed convex subset of a Banach spaceE. Suppose that norm ofEis uniformly Gˆateaux differentiable. Let{xt}be a bounded set inX andz∈C. Letμtbe a mean onX. Then
μtxt−z2min
y∈Cxt−y 2.19
if and only if
μt
y−z, xt−z
≤0, y∈C. 2.20
Now, we are in a position to prove the main results of this work.
Theorem 2.6. Let H be a Hilbert space and Ca nonempty closed convex subset of H. Let A : C → H be a maximal monotone mapping and f : C → Ca bounded demicontinuous strong pseudocontraction. Let{xt}be as inLemma 2.3. Then{xt}is bounded if and only if{xt}converges strongly to a zero pointpofAast → ∞which is the unique solution inA−10to the following variational inequality:
f p
−p, y−p
≤0, ∀y∈A−10. 2.21
Proof. The part ⇐ is obvious and we only prove ⇒. From Lemma 2.3, one sees that Axt → 0 as t → ∞. It follows fromLemma 2.4 that xt−gxt → 0 as t → ∞.
Definehx μtxt−x,x∈C, whereμtis a Banach limit. Thenhis a convex and continuous function withhx → ∞asx → ∞. Put
K
x∈C:hx min
y∈C h y
. 2.22
From the convexity and continuity ofh, we can get the convexity and continuity of the set K. Sincehis continuous andH is a Hilbert space, we see that hattains its infimum over K; see20for more details. ThenK is nonempty bounded and closed convex subset ofC.
Indeed,K contains one point only. Setgx IA−1x, whereg : K → K. Notice that g is nonexpansive. Since every nonempty bounded and closed convex subset has the fixed point property for nonexpansive self-mapping in the framework of Hilbert spaces, thenghas a fixed pointpinK, that is,gp p. It follows fromLemma 2.4thatAp 0. On the other hand, one hasμtxt−pminy∈Chy. In view ofLemma 2.5, we obtain that
μt
y−p, xt−p
≤0, ∀y∈C. 2.23
By takingyfpin2.23, we arrive at μt
f p
−p, xt−p
≤0, ∀y∈C. 2.24
Combining2.14with2.23yields thatμtxt−p2 0. Hence, there exists a subnet{xtα}of {xt}such that{xtα} → p. FromiiiofLemma 2.3, one has
xtα−fxtα, xtα−y
≤0, ∀y∈A−10. 2.25
Taking limit in2.25, one gets that p−f
p
, p−y ≤0, ∀y∈A−10. 2.26
If there exists another subset{xtβ}of{xt}such that{xtβ} → q, thenqis also a zero ofA. It follows from2.26that
p−f p
, p−q
≤0. 2.27
By usingiiiofLemma 2.3again, one arrives at xtβ−f
xtβ
, xtβ−p
≤0. 2.28
Taking limit in2.28, we obtain that q−f
q , q−p
≤0. 2.29
Adding2.27and2.29, we have p−qf
q
−f p
, p−q
≤0, 2.30
which yields that
p−q2≤ f
p
−f q
, p−q
≤βp−q2. 2.31 It follows thatp q. That is, {xt} converges strongly top ∈ A−10, which is the unique solution to the following variational inequality:
f p
−p, y−p
≤0, ∀y∈A−10. 2.32
Remark 2.7. FromTheorem 2.6, we can obtain the following interesting fixed point theorem.
The composition of bounded, demicontinuous, and strong pseudocontractions with the metric projection has a unique fixed point. That is,pPfp.
Acknowledgment
The third author was supported by the National Natural Science Foundation of ChinaGrant no. 10771050.
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