Volume 2010, Article ID 278973,11pages doi:10.1155/2010/278973
Research Article
On the Convergence for an Iterative Method for
Quasivariational Inclusions
Yu Li
1and Changqun Wu
21Research Institute of Management Science and Engineering, Henan University, Kaifeng 475004, China 2School of Business and Administration, Henan University, Kaifeng 475004, China
Correspondence should be addressed to Changqun Wu,[email protected]
Received 27 September 2009; Accepted 13 December 2009
Academic Editor: Tomonari Suzuki
Copyrightq2010 Y. Li and C. Wu. 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.
We introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·,·and the norm · . Let S : H → H be a nonlinear mapping. In this paper, we useFSto denote the fixed point set ofS.
Recall the following definitions.
1The mappingSis said to becontractivewith the coefficientα∈0,1if
Sx−Sy≤αx−y, ∀x, y∈H. 1.1
2The mappingSis said to benonexpansiveif
3The mappingSis said to bestrictly pseudocontractivewith the coefficientk∈0,1if
Sx−Sy2≤x−y2kI−Sx−I−Sy2, ∀x, y∈H. 1.3
4The mappingSis said to bepseudocontractiveif
Sx−Sy2≤x−y2I−Sx−I−Sy2, ∀x, y∈H. 1.4
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example,1–6and the references therein.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mappingSon a real Hilbert spaceH:
min x∈FS
1
2Ax, x −hx, 1.5
whereAis a linear bounded and strongly positive operator andhis a potential function for
γfi.e.,hx γfxforx∈H.
Recently, Marino and Xu2studied the following iterative scheme:
x0∈H, xn1 I−αnASxnαnγfxn, n≥0. 1.6
They proved that the sequence{xn}generated in the above iterative scheme converges strongly to the unique solution of the variational inequality:
A−γfx∗, x−x∗≥0, x∈FS, 1.7
which is the optimality condition for the minimization problem1.5.
Next, letB:H → Hbe a nonlinear mapping. Recall the following definitions. 1The mappingBis said to bemonotoneif for eachx, y∈H, we have
Bx−By, x−y≥0. 1.8
2Bis said to beμ-strongly monotoneif
Bx−By, x−y≥μx−y2, ∀x, y∈H. 1.9
3The mapping Bis said to beμ-inverse-strongly monotoneif there exists a constant
μ >0 such that
4The mappingBis said to berelaxedδ-cocoerciveif there exists a constantδ >0 such that
Bx−By, x−y≥−δBx−By2, ∀x, y∈H. 1.11
5The mappingBis said to berelaxedδ, r-cocoerciveif there exist two constantsδ, r > 0 such that
Bx−By, x−y≥−δBx−By2rx−y2, ∀x, y∈H. 1.12
6Recall also that a set-valued mapping M : H → 2His called monotoneif for all
x, y ∈H,f ∈Mxandg ∈Myimplyx−y, f−g ≥0.The monotone mapping
M:H → 2His maximalif the graph ofGMofT is not properly contained in the graph of any other monotone mapping.
The so-called quasi-variational inclusion problem is to find au∈Hfor a given element
f∈Hsuch that
f∈BuMu, 1.13
whereB:H → HandM:H → 2Hare two nonlinear mappings. See, for example,7–12.
A special case of the problem1.13is to find an elementu∈Hsuch that
0∈BuMu. 1.14
In this paper, we use V IH, B, M to denote the solution of the problem 1.14. A
number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.
Next, we consider two special cases of the problem1.14.
AIf M ∂φ : H → 2H, where φ : H → R∪ {∞} is a proper convex lower semicontinuous function and ∂φ is the subdifferential of φ, then the variational inclusion problem1.14is equivalent to findingu∈Hsuch that
Bu, v−uφv−φu≥0, ∀v∈H, 1.15
which is said to be themixed quasi-variational inequality. See, for example,7,8for more details.
BIfφis the indicator function ofC,then the variational inclusion problem1.14is equivalent to the classical variational inequality problem, denoted byV IC, B, to findu∈Csuch that
For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality 1.16, Iiduka and Takahashi 13
proved the following theorem.
Theorem IT. LetC be a closed convex subset of a real Hilbert space H. Let B be anα -inverse-strongly monotone mapping ofCintoHand letSbe a nonexpansive mapping ofCinto itself such thatFS∩V IC, B/∅. Suppose thatx1x∈Cand{xn}is given by
xn1αnx 1−αnSPCxn−λnBxn 1.17
for everyn1,2, . . . ,where{αn}is a sequence in0,1and{λn}is a sequence ina, b. If{αn}and
{λn}are chosen so that{λn} ∈a, bfor somea, bwith0< a < b <2α,
lim n→ ∞αn0,
∞
n1
αn∞,
∞
n1
|αn1−αn|<∞,
∞
n1
|λn1−λn|<∞, 1.18
then{xn}converges strongly toPFS∩V IC,Bx.
Recently, Zhang et al.11 considered the problem1.14. To be more precise, they
proved the following theorem.
Theorem ZLC. Let H be a real Hilbert space, B : H → H an α-inverse-strongly monotone mapping,M:H → 2Ha maximal monotone mapping, andS:H → Ha nonexpansive mapping. Suppose that the setFS∩V IH, B, M/∅, whereV IH, B, Mis the set of solutions of variational inclusion1.14. Suppose thatx0x∈Hand{xn}is the sequence defined by
xn1αnx0 1−αnSyn,
ynJM,λxn−λBxn, n≥0,
1.19
whereλ∈0,2αand{αn}is a sequence in0,1satisfying the following conditions: alimn→ ∞αn0,∞n1αn∞;
b∞n0|αn1−αn|<∞.
Then{xn}converges strongly toPFS∩V IH,B,Mx0.
In this paper, motivated by the research work going on in this direction, see, for instance, 2, 3, 7–21, we introduce an iterative method for finding a common element
of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem1.14with multivalued maximal monotone mapping and relaxedδ, r-cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.
In order to prove our main results, we need the following conceptions and lemmas.
Definition 1.1see11. LetM : H → 2H be a multivalued maximal monotone mapping. Then the single-valued mappingJM,λ :H → Hdefined byJM,λu IλM−1u,for all
Lemma 1.2see4. Assume that{αn}is a sequence of nonnegative real numbers such thatαn1≤ 1−γnαnδn,where{γn}is a sequence in0,1and{δn}is a sequence such that
a∞n1γn∞;
blim supn→ ∞δn/γn≤0or∞n1|δn|<∞.
Thenlimn→ ∞αn0.
Lemma 1.3see22. Let{xn}and{yn}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Suppose thatxn1 1−βnyn
βnxnfor alln≥0andlim supn→ ∞yn1−yn − xn1−xn≤0.Thenlimn→ ∞yn−xn0.
Lemma 1.4see11. u∈His a solution of variational inclusion1.14if and only ifuJM,λu−
λBu,for allλ >0, that is,
V IH, B, M FJM,λI−λB, ∀λ >0. 1.20
Lemma 1.5 see 11. The resolvent operator JM,λ associated with M is single-valued and nonexpansive for allλ >0.
Lemma 1.6see23. LetCbe a closed convex subset of a strictly convex Banach spaceE. LetSand
T be two nonexpansive mappings onC. Suppose thatFT∩FSis nonempty. Then a mappingR onCdefined byRxaSx 1−aTx, wherea∈0,1, forx∈Cis well defined and nonexpansive andFR FT∩FSholds.
Lemma 1.7see24. LetHbe a real Hilbert space, letCbe a nonempty closed convex subset ofH, and letS:C → Cbe a nonexpansive mapping. ThenI−Sis demiclosed at zero.
Lemma 1.8see25. LetC be a nonempty closed convex subset of a real Hilbert spaceH and
T :C → Cak-strict pseudocontraction. DefineS:C → HbySxαx 1−αTxfor eachx∈C. Then, asα∈k,1,Sis nonexpansive such thatFS FT.
2. Main Results
Theorem 2.1. LetHbe a real Hilbert space andM:H → 2Ha maximal monotone mapping. Let
B : H → H be a relaxedδ, r-cocoercive andν-Lipschitz continuous mapping, andSak-strict pseudocontraction with a fixed point. Define a mappingSk : H → HbySkx kx 1−kSx. Letfbe a contraction ofHinto itself with the contractive coefficientα0< α <1,andAa strongly positive linear bounded self-joint operator with the coefficientγ >0. Assume that0 < γ < γ/αand Ω FS∩V IH, B, M/∅. Letx1∈Hand{xn}be a sequence generated by
ynJM,λxn−λBxn,
xn1αnγfxn βnxn 1−βnI−αnA μSkxn1−μyn, ∀n≥1,
2.1
C10< a≤βn≤b <1, for alln≥1, C2limn→ ∞αn0and∞n1αn∞,
then{xn}converges strongly toz∈Ω,which solves uniquely the following variational inequality:
A−γfz, z−x∗≤0, ∀x∗∈Ω. 2.2
Equivalently, one hasPΩI−Aγfzz.
Proof. The uniqueness of the solution of the variational inequality2.2is a consequence of the strong monotonicity ofA−γf. Suppose thatz1 ∈Ωandz2∈Ωboth are solutions to2.2; thenA−γfz1, z1−z2 ≤0 andA−γfz2, z2−z1 ≤0.Adding up the two inequalities, we see that
A−γfz1−
A−γfz2, z1−z2
≤0. 2.3
The strong monotonicity of A −γf see 2, Lemma 2.3 implies that z1 z2 and the uniqueness is proved. Below we usezto denote the unique solution of2.2.
Next, we show that the mappingI−λBis nonexpansive. Indeed, for allx, y∈H, one see from the conditionλ∈0,2r−γμ2/μ2that
I−λBx−I−λBy2
x−y−λBx−By2
x−y2−
2λBx−By, x−yλ2Bx−By2
≤x−y2−
2λ−δBx−By2rx−y2λ2ν2x−y2
1λ2ν2−2λr2λδν2x−y2
≤x−y2,
2.4
which implies that the mappingI−λBis nonexpansive. Takingx∗∈Ω,we havex∗JM,λx∗−
λBx∗.It follows fromLemma 1.5that
yn−x∗JM,λxn−λBxn−JM,λx∗−λBx∗ ≤ xn−x∗. 2.5
Note that from the conditionsC1andC2, we may assume, without loss of generality, that
αn ≤ 1−βnA−1 for all n ≥ 1. SinceAis a strongly positive linear bounded self-adjoint operator, we haveA sup{|Ax, x|:x∈ H,x 1}. Now forx∈ Hwithx 1, we see that
that is,1−βnI−αnAis positive. It follows that
1−βnI−αnAsup1−βnI−αnAx, x:x∈C,x1 sup1−βn−αnAx, x:x∈C,x1
≤1−βn−αnγ.
2.7
SettnμSkxn 1−μyn. FromLemma 1.8, we see thatSkis nonexpansive. It follows from 2.5that
tn−x∗ ≤μSkxn−x∗1−μyn−x∗≤ xn−x∗. 2.8
From2.7and2.8, we arrive at
xn1−x∗
αnγfxn βnxn 1−βnI−αnAtn−x∗
≤αnγfxn−Ax∗βnxn−x∗1−βnI−αnAtn−x∗
≤αnγfxn−Ax∗βnxn−x∗1−βn−αnγtn−x∗
≤αnγfxn−γfx∗αnγfx∗−Ax∗βnxn−x∗ 1−βn−αnγxn−x∗
≤ααnγxn−x∗αnγfx∗−Ax∗βnxn−x∗ 1−βn−αnγxn−x∗
1−αnγ−αγxn−x∗αnγfx∗−Ax∗.
2.9
By simple inductions, one obtains thatxn−x∗ ≤ max{x1−x∗,γfx∗−Ax∗/γ−αγ}, which gives that the sequence{xn}is bounded, so are{yn}and{tn}.
On the other hand, we see from the nonexpansivity of the mappingsJM,λthat
yn1−ynJM,λxn1−λBxn1−JM,λxn−λBxn ≤ xn1−xn. 2.10
It follows that
tn1−tnμSkxn1
1−μyn1− μSkxn1−μyn
≤μSkxn1−Skxn1−μyn1−yn
≤ xn1−xn.
2.11
Setting
xn1
we see that
en1−en
αn1γfxn1 1−βn1
I−αn1A
tn
1−βn1 −
αnγfxn 1−βnI−αnAtn 1−βn
αn1 1−βn1
γfxn1−Atntn1−
αn
1−βn γfxn−Atn
−tn.
2.13
It follows that
en1−en ≤ αn1 1−βn1
γfxn1−Atn αn 1−βn
γfxn−Atntn1−tn, 2.14
which combines with2.11yields that
en1−en − xn1−xn ≤ 1α−nβ1 n1
γfxn1−Atn1α−nβ n
γfxn−Atn. 2.15
It follows from the conditionsC1andC2that lim supn→ ∞en1−en − xn1−xn≤0. Hence, fromLemma 1.3, one obtains limn→ ∞en−xn0.From2.12, one hasxn1−xn 1−βnen−xn.Thanks to the conditionC1, we see that
lim
n→ ∞xn1−xn0. 2.16
On the other hand, we have
xn1−xnαnγfxn βnxn 1−βnI−αnAtn−xn
αnγfxn−Atn1−βntn−xn.
2.17
It follows that
1−βntn−xn ≤ xn1−xnαnγfxn−Atn. 2.18
From the conditionsC1andC2and2.16, we see that
lim
n→ ∞tn−xn0. 2.19
Next, we prove that lim supn→ ∞γf−Az, xn−z ≤0,wherezPΩI−A−γfz.
To see this, we choose a subsequence{xni}of{xn}such that
lim sup n→ ∞
γf−Az, xn−zilim
→ ∞
Since{xni}is bounded, there exists a subsequence{xnij}of {xni} which converges weakly to w. Without loss of generality, we can assume that xni w.Next, we show that w ∈
FS∩V IH, M, B. Define a mappingDby
DxμSkx1−μJM,λI−λB, ∀x∈H. 2.21
In view ofLemma 1.6, we see thatDis nonexpansive such that
FD FSk∩FJM,λI−λB FS∩V IH, B, M. 2.22
From2.19, we obtain limn→ ∞Dxni−xni 0.It follows fromLemma 1.7thatw ∈FD. That is,w∈FS∩V IH, M, B.Thanks to2.20, we arrive at
lim sup n→ ∞
γf−Az, xn−z lim i→ ∞
γf−Az, xni−zγf−Az, w−z≤0. 2.23
Finally, we show thatxn → z,asn → ∞.Indeed, we have
xn1−z2
αnγfxn βnxn 1−βnI−αnAtn−z, xn1−z αnγfxn−Az, xn1−zβnxn−z, xn1−z
1−βnI−αnAtn−z, xn1−z
≤αnγfxn−fz, xn1−zαnγfz−Az, xn1−z βnxn−zxn1−z
1−βn−αnγxn−zxn1−z
≤ γα
2 αn
xn−z2xn1−z2
αnγfz−Az, xn1−z
1−αnγxn−zxn1−z
≤ γα
2 αn
xn−z2xn1−z2
αnγfz−Az, xn1−z
1−αnγ 2
xn−z2xn1−z2
1−αn
γ−αγ
2 xn−z 21
2xn1−z 2α
nγfz−Az, xn1−z,
2.24
which implies that
xn1−z2≤ 1−αnγ−αγxn−z22αnγfz−Az, xn1−z
. 2.25
Lettingγ 1 andA I, the identity mapping, we can obtain fromTheorem 2.1the following result immediately.
Corollary 2.2. LetHbe a real Hilbert space andM:H → 2H a maximal monotone mapping. Let
B : H → H be a relaxedδ, r-cocoercive andν-Lipschitz continuous mapping, andSak-strict pseudocontraction with a fixed point. Define a mappingSk : H → HbySkx kx 1−kSx. Letf be a contraction ofH into itself with the contractive coefficientα0 < α < 1.Assume that Ω FS∩V IH, B, M/∅. Letx1∈Hand{xn}be a sequence generated by
ynJM,λxn−λBxn,
xn1αnfxn βnxn1−βn−αn μSkxn1−μyn, ∀n≥1,
2.26
where{αn}and{βn}are sequences in0,1. Assume thatλ ∈ 0,2r−δν2/ν2,r > δν2. If the control consequences{αn}and{βn}satisfy the following restrictions:
C10< a≤βn≤b <1, for alln≥1, C2limn→ ∞αn0and∞n1αn∞, then{xn}converges strongly toz∈Ω.
Remark 2.3. Corollary 2.2improvesTheorem 2.1of Zhang et al.11in the following sense: 1from nonexpansive mappings to strict pseudocontractions;
2the analysis technique used in this paper is different from11’s: the proof is also
more concise than11’s;
3the restriction imposed on the parameter{αn}is relaxed.
Acknowledgments
The authors are extremely grateful to the referee for useful suggestions that improved the contents of the paper. This work was supported by Important Science and Technology Research Project of Henan province, China092102210134.
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