Volume 2011, Article ID 620284,17pages doi:10.1155/2011/620284
Research Article
Iterative Methods for Variational Inequalities
over the Intersection of the Fixed Points Set of
a Nonexpansive Semigroup in Banach Spaces
Issa Mohamadi
Department of Mathematics, Islamic Azad University, Sanandaj Branch, Sanandaj 418, Kurdistan, Iran
Correspondence should be addressed to Issa Mohamadi,[email protected]
Received 8 November 2010; Accepted 19 November 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2011 Issa Mohamadi. 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.
This paper presents a framework of iterative methods for finding specific common fixed points of a nonexpansive self-mappings semigroup in a Banach space. We prove, with appropriate conditions, the strong convergence to the solution of some variational inequalities.
1. Introduction
LetCbe a nonempty closed convex subset of a Hilbert spaceH, and let F : C → H be a nonlinear map. The classical variational inequality which is denoted by VIF, Cis formulated as findingx∗∈Csuch that
Fx∗, x−x∗ ≥0, 1.1
for allx∈C. We recall thatFis calledη-strongly monotone, if for eachx, y∈C, we have
Fx−Fy, x−y≥ηx−y2, 1.2
for a constantη >0, and alsoκ-Lipschitzian if for eachx, y∈C, we have
for a constant κ > 0. Existence and uniqueness of solutions are important problems of the VIF, C. It is known that ifFis a strongly monotone and Lipschitzian mapping onC, then VIF, Chas a unique solution. An important problem is how to find a solution of VIF, C. It is known that
x∗∈VIF, C⇐⇒x∗PCx∗−λFx∗, 1.4
where λ > 0 is an arbitrarily fixed constant and PC is the projection of H onto C. This alternative equivalence has been used to study the existence theory of the solution and to develop several iterative type algorithms for solving variational inequalities. But the fixed point formulation in1.4involves the projectionPC, which may not be easy to compute, due to the complexity of the convex setC. So, projection methods and their variant forms can be implemented for solving variational inequalities.
In order to reduce the complexity probably caused by the projectionPC, Yamada1 see also2introduced a hybrid steepest-descent method for solving VIF, C. His idea is stated now. Assume thatCis the fixed point set of a nonexpansive mappingT :H → H. Recall thatT is nonexpansive if
Tx−Ty≤x−y, ∀x, y∈H. 1.5
Assume thatF isη-strongly monotone andκ-Lipschitzian onC. Take a fixed number μ ∈
0,2η/κ2and a sequence{λ
n}in0,1satisfying the following conditions: C1limn→ ∞λn0,
C2∞n1λn∞,
C3limn→ ∞λn−λn1/λ2n10.
Starting with an arbitrary initial guess x0 ∈ H, generate a sequence {xn} by the following algorithm:
xn1:Txn−λn1μFTxn, n≥0. 1.6
Yamada1proved that the sequence{xn}converges strongly to a unique solution of VIF, C. Xu and Kim3further considered and studied the hybrid steepest-descent algorithm1.6. Their major contribution is that the strong convergence of 1.6 holds with conditionC3 being replaced by the following condition:
C3limn→ ∞λn−λn1/λn10.
It is clear that condition C3 is strictly weaker than conditionC3, coupled with conditionsC1andC2. Moreover,C3includes the important and natural choice{1/n}
for{λn}whereasC3does not. For more related results, see4,5.
LetXbe a Banach space we recall that a nonexpansive semigroup is a family{Tt :
t >0}of self-mappings ofXsatisfies the following conditions:
iT0xxforx∈X,
iiilimt→0Ttxxforx∈X,
ivfor eacht >0, Ttis nonexpansive. that is,
Ttx−Tty≤x−y, ∀x, y∈X. 1.7
The problem is to find some fixed point inC t>0FixTt. For this, so many algorithms have been developed and under some restrictions partial answers have been obtained6–11. Assume thatF :X → Xis a strongly monotone and Lipschitzian mapping and{Tt:
t >0}is a nonexpansive semigroup of self-mappings onX. For an appropriateμand starting from an arbitrary initial pointx0∈X, we devise the following implicit, explicit, and modified
iterations:
xn:λnxn 1−λnTtnxn−λnμFxn, 1.8
xn1:λnxn 1−λnTtnxn−λnμFxn, 1.9
xn1:λnyn 1−λnTtnxn,
yn:
1−μn
xnμnTtn−Fxn,
1.10
forn ≥ 1. With some appropriate assumptions, we prove the strong convergence of 1.8,
1.9, and1.10to the unique solution of the variational inequalityFx∗, Jx−x∗ ≥0 inC, whereJis the single-valued normalized duality mapping fromXinto 2X∗.
Our main purpose is to improve some of the conditions and results in the mentioned papers, especially those of Song and Xu11.
2. Preliminaries
LetS:{x ∈X :x 1}be the unit sphere of the Banach spaceX. The spaceXis said to haveGateaux differentiable normorXis said to besmooth, if the limit
lim t→0
xty− x
t , 2.1
exists for eachx, y∈S, andXis said to have auniformly Gateaux differentiablenorm if for each
y∈S, the limit2.1converges uniformly forx∈S. Further,Xis said to beuniformly smooth if the limit2.1exists uniformly forx, y∈S×S.
We denoteJthe normalized duality mapping fromXto 2X∗defined by
Jx f∗:x, f∗x2f∗2, ∀x∈X. 2.2
Recall that a Banach spaceX is said to bestrictly convexifx y 1 andx /y
impliesxy/2<1. In a strictly convex Banach spaceX, we have that ifλx 1−λy1 forλ∈0,1andx, y∈X, thenxy.
Now, we recall the concept of uniformly asymptotically regular semigroup. A continuous operator semigroup {Tt : t > 0} on X is said to be uniformly asymptotically regularonXif for allh >0 and any bounded subsetDofX, we have
lim
t→ ∞supx∈DThTtx−Ttx0. 2.3
The nonexpansive semigroup{σtx 1/t t
0Tsxds : t > 0}is an example of uniformly
asymptotically regular operator semigroup11.
Letμbe a continuous linear functional onl∞satisfyingμ1μ1. Then, we know thatμis ameanonNif and only if
inf{an:n∈N} ≤μa≤sup{an:n∈N}, 2.4
for everya a1, a2, . . .∈l∞. Sometimes, we useμnaninstead ofμa. A meanμonNis calleda Banach limitifμnan μnan1. We know that ifμis a Banach limit, then
lim inf
n→ ∞ an≤μa≤lim supn→ ∞ an, 2.5
for everya a1, a2, . . .∈l∞. Thus, ifan → casn → ∞, then we have
μnan μa c. 2.6
A discussion on these and related concepts can be found in12.
We make use of the following well-known results throughout the paper.
Lemma 2.1see12, Lemma 4.5.4. LetDbe a nonempty closed convex subset of a Banach space
Xwith a uniformly Gateaux differentiable norm, and let{yn}be a bounded sequence inX. Ifz0∈D,
then
μnyn−z02min
y∈D μn
yn−y2, 2.7
if and only if
μn
y−z0, J
yn−z0
≤0, 2.8
for ally∈D.
Lemma 2.2see13. Let{sn},{cn} ⊂R,{an} ⊂0,1, and let{bn} ⊂Rbe sequences such that
for alln≥0. Assume also thatn≥0|cn|<∞. Then, the following results hold:
iifbn≤βan(whereβ≥0), then{sn}is bounded, iiif we have
n≥0
an∞, lim sup n→ ∞
bn
an ≤
0, 2.10
thenlimn→ ∞sn0.
Lemma 2.3. LetXbe a real normed linear space, and letJbe the normalized duality mapping onX. Then, for anyx, y∈X, andj∈J, the following inequality holds:
xy2≤ x22y, jxy. 2.11
In order to reduce any possible complexity in writing, we setCt>0FixTtfor a nonexpansive semigroup{Tt:t >0}and
Dxn
x∈X:gx inf y∈Xg
y, 2.12
wheregx μnxn−x2, for allx∈X, and{xn}is a bounded sequence inX.
3. Implicit Iterative Method
Recall that ifJis the single-valued normalized duality mapping from a Banach spaceXinto 2X∗, a nonlinear operatorF :X → Xis calledη-strongly monotone if for everyx, y∈X, the following inequality holds:
Fx−Fy, Jx−y≥ηx−y2, 3.1
for a constantη >0.
The following lemma will be be used to show the convergence of1.8and1.9.
Lemma 3.1. LetXbe a Banach space, and letJbe the single-valued normalized duality mapping from
Xinto2X∗. Assume also thatF:X → Xisη-strongly monotone andκ-Lipschitzian onX. Then,
ψx Ix−μFx 3.2
Proof. By using Lemma2.3, we have
ψx−ψy2≤I−μFx−I−μFy2
x−yμFy−Fx2
≤x−y22μFy−Fx, Jx−yμFy−Fx
≤x−y22μFy−Fx, Jx−y2μ2Fy−Fx, JFy−Fx
≤x−y2−2μFx−Fy, Jx−y2μ2Fy−FxJFy−Fx
≤x−y2−2μηx−y22μ2Fy−Fx2
≤x−y2−2μηx−y22μ2κ2x−y2
≤1−2μη2μ2κ2x−y2,
3.3
Thus, we obtain
ψx−ψy≤
1−2μη−μκ2x−y, 3.4
and forμ∈0, η/κ2, we have
1−2μη−μκ2 ∈0,1. That is,ψ is a contraction, and the
proof is complete.
In the following theorem, which is the main result in this section, we establish the strong convergence of the sequence defined by1.8.
Theorem 3.2. LetX be a real Banach space with a uniformly Gateaux differentiable norm, and let
{Tt:t >0}be a nonexpansive semigroup fromXinto itself. Let also{xn}defined by1.8satisfies the following condition:
C∩Dxn/∅. 3.5
Assume that F : X → X is η-strongly monotone andκ-Lipschitzian. Assume also that{tn} is a sequence of positive numbers thatlimn→ ∞tn ∞and {λn} ⊂ 0,1. Ifμ ∈ 0, η/κ2, then{xn} converges strongly to some fixed pointx∗ ∈C, which is the unique solution inCto the variational inequality VI∗F, C, that is
Proof. We divide the proof into several steps.
Step 1. We first prove the uniqueness of the solution to VI∗F, C; for this, we supposex∗, y∗∈ Care two solutions of VI∗F, C. Thus, we have
Fx∗, Jx∗−y∗≤0,
Fy∗, Jy∗−x∗≤0. 3.7
By adding up the last two inequalities, we obtain
ηx∗−y∗2≤Fx∗−Fy∗, Jx∗−y∗≤0, 3.8
and so,x∗y∗.
Step 2. We claim that{xn}is bounded. In fact, taking a fixedx∗∈C, we have
xn−x∗λnxn 1−λnTtnxn−λnμFxn−x∗
≤λnxn−μFxn−x∗ 1−λnTtnxn−x∗
≤λnI−μF
xn−
I−μFx∗λnI−μF
x∗−x∗ 1−λnxn−x∗
≤λn
1−2μη−μκ2x
n−x∗ 1−λnxn−x∗λnμFx∗
≤
1−λn
1−
1−2μη−μκ2xn−x∗λnμFx∗.
3.9
Takingγ 1−
1−2μη−μκ2and by using induction, we obtain
xn−x∗ ≤max
x0−x∗,
μ γFx
∗, 3.10
therefore,{xn−x∗}is bounded and so is{xn}.
Step 3. The sequence {xn} is sequentially compact. To prove this, we assume that the set
Dxncontains somex∗such thatTtx∗ x∗for an arbitraryt >0. So, by using Lemma2.1, we can obtain
μnx−x∗, Jxn−x∗ ≤0, ∀x∈X. 3.11
On the other hand, for anyq∈C, we have xn−q2
λn
xn−q
1−λn
Ttnxn−q
−μλnFxn, J
xn−q
≤λnI−μF
xn−
I−μFqxn−qλn
−μFq, Jxn−q
1−λn
Ttnxn−Ttnq, J
xn−q
≤1−λnγxn−q2λn
−μFq, Jxn−q
.
Thus,
xn−q2≤ 1
γ
I−μFq−q, Jxn−q
. 3.13
Also, we have xn−q2≤
λn
I−μFxn−q, J
xn−q
1−λn
Ttnxn−q, J
xn−q
≤λn
I−μFxn−q, J
xn−q
1−λnxn−q2.
3.14
It follows that
xn−q2≤
I−μFxn−q, J
xn−q
. 3.15
Combining3.11and3.13together, we get
μnxn−x∗2≤
μn
γ
I−μFx∗−x∗, Jxn−x∗
≤0. 3.16
This yieldsμnxn−x∗ 0. Hence, there exists a subsequence of{xn} such as{xnk} that converges strongly tox∗; that is,{xn}is sequentially compact.
Step 4. We claim thatx∗is the solution of VI∗F, C. Since{xn}is bounded, for any fixed point
x∈C, there exist a constantL >0 such thatxn−x ≤L. Therefore, we obtain
xn−x2λn
I−μFxn−
I−μFx∗, Jxn−x
λn
−μFx∗, Jxn−x
1−λnTtnxn−Ttnx, Jxn−xλnx∗−x, Jxn−x
≤2−γλnLxn−x∗λn
−μFx∗, Jxn−x
xn−x2.
3.17
Hence,
Fx∗, Jxn−x ≤
L2−γ μ xn−x
∗. 3.18
Note that the duality mapping J is single valued X is smooth, and norm topology to weak∗ uniformly continuous on bounded sets of Banach spaceX with uniformly Gateaux differentiable norm. Therefore,
Fx∗, Jxnk−x −→ Fx∗, Jx∗−x, 3.19
and by taking limit asnk → ∞in two sides of3.18, we obtain
Fx∗, Jx∗−x ≤0, ∀x∈C. 3.20
Step 5. xn → x∗ in norm. Indeed, we show that each cluster point of the sequence{xn}is equal tox∗. Assume thaty∗is another strong limit point of{xn}inC. Thanks to3.15, we have the following two inequalities:
x∗−y∗2≤I−μFx∗−y∗, Jx∗−y∗,
y∗−x∗2≤
I−μFy∗−x∗, Jy∗−x∗.
3.21
Therefore,
2x∗−y∗2 ≤I−μFx∗−I−μFy∗x∗−y∗, Jx∗−y∗≤2−γx∗−y∗2.
3.22
It yields that x∗−y∗2 0, which proves the uniqness of x∗. Thus, {xn} itself converges strongly tox∗. This completes the proof.
4. Explicit Iterative Method
In this section, we will present our result of the strong convergence of1.9, but first, we need to prove, with different approach, the following lemma.
Lemma 4.1. LetX,{Tt : t > 0}, F,{λn},{tn}, μ, and{xn}be as those in Theorem3.2. Ifx∗ limn→ ∞xn, and there exists a bounded sequence{yn}such that
lim
n→ ∞Ttyn−yn0, ∀t >0. 4.1
Then,
lim sup n→ ∞
Fx∗, Jx∗−yn
≤0. 4.2
Proof. By the uniqueness ofx∗and with no loss of generality, we can chooseλnsuch that
λn−→0,
Ttyn−yn
λn −→0,
4.3
asn → ∞. Letx∗λnbe the fixed point of the contraction
Then,
x∗λn−ynλnI−μF
x∗λn −yn
1−λn
Ttnx∗λn−yn
. 4.5
Now, by using Lemma2.3, we have
x∗λn−yn
2
1−λn2Ttnx∗λn−yn
2
2λn
I−μFx∗λn−yn, J
x∗λn−yn
≤1−λn2Ttnx∗λn−TtnynTtnyn−yn
2
2λnx∗λn−yn
2
2λn
−μFx∗λn, Jxλn∗ −yn
≤1λn2x∗λn−yn
2
Ttnyn−yn2xλn∗ −ynTtnyn−yn
2λn
−μFx∗λn, Jxλn∗ −yn
.
4.6
Therefore,
μFx∗λn, Jx∗λn−yn
≤ λn 2
xλn∗ −yn
2
Ttnyn−yn
2λn
2x∗λn −ynTtnyn−yn. 4.7
Because{yn},{Ttnyn}and{x∗λn}are bounded, from4.3and4.7, we conclude that
lim sup n→ ∞
μFx∗λn, Jx∗λn −yn
≤0. 4.8
Moreover, we have
μFxλn∗ , Jx∗λn−yn
x∗−I−μFx∗λn, Jx∗−yn
x∗−I−μFx∗λn, Jx∗λn−yn
−Jx∗−yn
x∗λn−x∗, Jx∗λn−yn
.
4.9
By Theorem3.2,x∗λn → x∗, asn → ∞. So, using the boundedness of{yn}, we get
x∗λn −x∗, Jx∗λn−yn
−→0, n−→ ∞. 4.10
On the other hand, noticing that the sequence{x∗λn−yn}is bounded and the duality mappingJis single-valued and norm to weak∗uniformly continuous on bounded subsets of
X, we conclude that
x∗−I−μFx∗λn, Jx∗λn −yn
−Jx∗−yn
Therefore, from4.8and4.9, we obtain
lim sup n→ ∞
Fx∗, Jx∗−yn
≤0. 4.12
This completes the proof.
Next, we prove the strong convergence of explicit iteration scheme1.9.
Theorem 4.2. LetX be a real Banach space with a uniformly Gateaux differentiable norm, and let
{Tt:t >0}be a nonexpansive semigroup fromXinto itself. Let also{xn}defined by1.9satisfies the following conditions:
iC∩Dxn/∅,
iilimn→ ∞xn−Ttxn0for allt >0.
Assume thatF : X → X isη-strongly monotone andκ-Lipschitzian, and that {tn}is a sequence of positive numbers thatlimn→ ∞tn ∞. Assume also that the sequence{λn}in0,1satisfies the control condition
∞
n1
λn∞. 4.13
If μ ∈ 0, η/κ2, then{xn} converges strongly to some fixed point x∗ ∈ C, which is the unique solution inCfor the following variational inequality:
Fx∗, Jx−x∗ ≥0, ∀x∈C. 4.14
Proof. Existence and uniqueness of the solution of VI∗F, Cis attained from Theorem3.2. Now, we claim that{xn}is bounded. Indeed, taking a fixedx∗∈C, we have
xn1−x∗λnxn 1−λnTtnxn−λnμFxn−x∗
≤λnxn−μFxn−x∗ 1−λnTtnxn−x∗
≤λnI−μF
xn−
I−μFx∗λnI−μF
x∗−x∗ 1−λnxn−x∗
≤λn
1−2μη−μκ2x
n−x∗ 1−λnxn−x∗λnμFx∗
≤1−λn
1−
1−2μη−μκ2xn−x∗λnμFx∗.
4.15
Takingan λn1−
1−2μη−μκ2,bn λnμFx∗,cn 0, and using Lemma2.2,
Next, we prove that{xn}converges strongly to the unique solutionx∗of VI∗F, C. By definition of the algorithm and takingγ 1−1−2μη−μκ2, we have
xn1−x∗2 λn
I−μFxn−x∗, Jxn1−x∗
1−λnTtnxn−x∗, Jxn1−x∗
≤λn
I−μFxn−
I−μFx∗, Jxn1−x∗
λn
−μFx∗, Jxn1−x∗
1−λnTtnxn−x∗xn1−x∗
≤μλn−Fx∗, Jxn1−x∗λn
1−2μη−μκ2x
n−x∗xn1−x∗
1−λnxn−x∗xn1−x∗
≤μλnFx∗, Jx∗−xn1λn
1−2μη−μκ2xn−x∗
2x
n1−x∗2
2
1−λn
xn−x∗2xn1−x∗2
2
≤1−λnγ
xn−x∗22μλnFx∗, Jx∗−xn1.
4.16
Takinganλnγ,bn2μλnFx∗, Jx∗−xn1, andcn 0 and using Lemma4.1together with Lemma2.2lead to limn→ ∞xn1−x∗20, that is,xn → x∗in norm. This completes the proof.
Corollary 4.3. Let X be a real reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm. Let also{Tt:t >0}be a nonexpansive semigroup fromXinto itself such that
C t>0FixTt/∅. Assume that{xn}defined by1.9satisfies conditioniiin Theorem4.2, then conditioniholds.
Proof. Clearly, gx μnxn−x2 is a convex and continuous function. Because X is a reflexive Banach space, according to 12, Theorem 1.3.11, Dxn is nonempty. Also, by convexity and continuity of g, the set Dxn is a closed convex subset of X. Since limn→ ∞Ttxn−xn0 for everyx∈Dxnandt >0, we have
gTtx μnxn−Ttx2μnxn−TtxnTtxn−Tx2≤μnxn−x2gx. 4.17
So, Ttx ∈ D, and therefore TtDxn ⊂ Dxn. Suppose that u ∈ C. Because every nonempty closed convex subset of a strictly convex and reflexive Banach space X is a Chebyshev set, according to14, Corollary 5.1.19, there exists a unique x∗ ∈ Dxn such that
u−x∗ inf
x∈Dxnu−x. 4.18
On the other hand,Ttuufor allt >0,Ttx∗∈DxnandTtis nonexpansive, so we get
that is,Ttx∗ x∗, by uniqueness ofx∗ ∈Dxn. Thus,x∗ ∈C∩Dxn. This completes the proof.
Corollary 4.4. LetX be a real Banach space, and let{Tt : t > 0}be a nonexpansive uniformly asymptotically regular semigroup fromX into itself. If{xn} is defined by1.9, where λn satisfies C1, then conditioniiin Theorem4.2holds.
Proof. From1.9,C1, and the boundedness of{xn}, we conclude that
xn1−Ttnxnλnxn−Ttnxn−μFxn−→0, 4.20
asn → ∞. On the other hand, the semigroup{Tt : t > 0} is uniformly asymptotically regular, limn→ ∞tn∞, andS{xn}is a bounded subset inX, so for allt >0, we have
lim
n→ ∞TtTtnxn−Ttnxn ≤nlim→ ∞supx∈STtTtnx−Ttnx0. 4.21
Hence,
xn1−Ttxn1 ≤ xn1−TtnxnTtnxn−TtTtnxnTtTtnxn−Ttxn1
≤2xn1−TtnxnTtTtnxn−Ttnxn.
4.22
So, from4.20,4.21, and4.22, we get
lim
n→ ∞xn−Ttxn0, ∀t >0, 4.23
and it completes the proof.
Remark 4.5. According to Corollaries4.3and4.4, our assumptions are weaker than those of Song and Xu11. Also, noticing that for a contractionf :X → X, the mapping1/μI−f
is strongly monotone and Lipschitzian. So, by replacing1/μI−fbyFin1.8and1.9, the following schemes are, respectively, obtained:
xn:λnfxn 1−λnTtnxn,
xn1:λnfxn 1−λnTtnxn.
4.24
Remark 4.6. In the same way and with the same conditions mentioned in Theorem4.2, it’s easy to see that the sequence{xn}defined by
xn1:Ttnxn−λn1μFTtnxn, n≥0 4.25
5. Modified Iterative Method
In this section, we show that the modified sequence{xn}defined by 1.10also converges strongly to the solution of variational equality VI∗F, C, but first, we need to prove the following lemma.
Lemma 5.1. LetX be a Banach space. Assume thatF : X → X isη-strongly monotone andκ -Lipschitzian nonlinear operator andT :X → X a nonexpansive mapping. Ifμ ∈0, η/σ2, where
σκ2, then
ϕx Ix−μFI−Tx 5.1
is a contraction onX.
Proof. Considering the inequality
xy2
≤ x22y, Jxy, 5.2
from Lemma2.3in a Banach spaceX, whereJ : X → 2X∗is the normalized single-valued duality, we have
ϕx−ϕy2I−μFI−Tx−I−μFI−Ty2
x−yμFI−Ty−FI−Tx2
≤x−y22μFI−Ty−FI−Tx, Jx−y
μFI−Ty−FI−Tx
≤x−y22μFI−Ty−FI−Tx, Jx−y2μ2FI−Ty
−FI−Tx, JFI−Ty−FI−Tx
≤x−y2−2μFI−Tx−FI−Ty, Jx−y
2μ2FI−Ty−FI−TxJFI−Ty−FI−Tx
≤x−y2−2μFx−Fy, Jx−y−2μI−Tx−I−Ty, Jx−y
2μ2FI−Ty−FI−TxJFI−Ty−FI−Tx.
5.3
Noticing that
we obtain
ϕx−ϕy2≤
x−y2−2μηx−y22μ2FI−Ty−FI−Tx2
≤x−y2−2μηx−y22μ2κ22x−y2
≤1−2μη2μ2σ2x−y2.
5.5
Thus, we have
ϕx−ϕy≤1−2μ
η−μσ2x−y. 5.6
Note that forμ∈0, η/σ2, we conclude
1−2μη−μσ2∈0,1. That is,ϕis a contraction
and the proof is complete.
Theorem 5.2. LetXbe a real Banach space with a uniformly Gateaux differentiable norm and{Tt:
t > 0} a nonexpansive semigroup fromX into itself. Let also {xn} defined by1.10 satisfies the following conditions:
iC∩Dxn/∅,
iilimn→ ∞xn−Ttxn0for allt >0.
Assume thatF :X → Xisη-strongly monotone andκ-Lipschitzian and{tn}a sequence of positive numbers thatlimn→ ∞tn∞. Assume also that the sequences{μn} ⊂0, η/1σ2, whereσκ2, and{λn}in0,1satisfy the following control conditions:
C1∞n1λn∞,
C2{μn}does not take 0 as it’s limit point.
Then,{xn}converges strongly to some fixed pointx∗ ∈C, which is the unique solution inC for the variational inequality VI∗F, C.
Proof. Existence and uniqueness of the solution of VI∗F, Cis obtained from Theorem3.2. We claim that{xn}is bounded. Indeed, taking a fixedx∗∈C, we have
xn1−x∗λnyn 1−λnTtnxn−x∗
≤λnI−μnFI−Ttn
xn−x∗ 1−λnTtnxn−x∗
≤λnI−μnFI−Ttn
xn−
I−μnFI−Ttn
x∗
λnI−μnFI−Ttn
x∗−x∗ 1−λnxn−x∗
≤λn
1−2μn
η−μnσ2
xn−x∗ 1−λnxn−x∗λnμnFx∗
≤
1−λn
1−
1−2μn
η−μnσ2
xn−x∗λnμnFx∗.
Noticing that
μ2n<1−
1−2μn
η−μnσ2
, 5.8
and by assumption that there exists >0 so thatμn> , for alln∈N. Thus, we get
xn1−x∗ ≤
1−λnμ2n
xn−x∗ 1
λnμn
2Fx∗, 5.9
and from Lemma2.2, we conclude that{xn}is bounded.
By Theorem 3.2, there exists a unique solution x∗ to VI∗F, C. We prove that {xn} converges strongly tox∗
xn1−x∗2λn
I−μnFI−Ttn
xn−x∗, Jxn1−x∗
1−λnTtnxn−x∗, Jxn1−x∗
≤λn
I−μnFI−Ttn
xn−
I−μnFI−Ttn
x∗, Jxn1−x∗
λn
−μnFI−Ttnx∗, Jxn1−x∗
1−λnTtnxn−x∗xn1−x∗
≤μnλn−Fx∗, Jxn1−x∗λn
1−2μn
η−μnσ2
xn−x∗xn1−x∗
1−λnxn−x∗xn1−x∗
≤μnλnFx∗, Jx∗−xn1λn
1−2μn
η−μnσ2
xn−x∗2xn1−x∗2
2
1−λn
xn−x∗2xn1−x∗2
2
≤
1−λn
1−
1−2μn
η−μnσ2
xn−x∗22λnμnFx∗, Jx∗−xn1
≤1−λnμ2n
xn−x∗2 2λnμ2n
Fx
∗, Jx∗−x n1.
5.10
Takinganλnμ2n,bn 2λnμ2n/Fx∗, Jx∗−xn1, andcn 0 and using Lemma4.1 together with Lemma2.2imply limn→ ∞xn1−x∗20, that is,
lim
n→ ∞xn−x
∗0.
5.11
This completes the proof.
Remark 5.3. In Theorem5.2, ifη >1, thenμn <1−
Remark 5.4. We can easily see that under some restrictions all the strongly monotone and Lipschitzian nonlinear operators used in this paper are replaceable by strongly accretive and strictly pseudocontractive onessee15.
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