Volume 2011, Article ID 284363,20pages doi:10.1155/2011/284363
Research Article
A General Iterative Approach to Variational
Inequality Problems and Optimization Problems
Jong Soo Jung
Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
Correspondence should be addressed to Jong Soo Jung,[email protected]
Received 4 October 2010; Accepted 14 November 2010
Academic Editor: Jen Chih Yao
Copyrightq2011 Jong Soo Jung. 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 a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.
1. Introduction
LetH be a real Hilbert space with inner product ·,·and induced norm · . LetCbe a nonempty closed convex subset ofHandS :C → Cbe self-mapping onC. We denote by FSthe set of fixed points ofSand byPC the metric projection ofHontoC.
LetAbe a nonlinear mapping ofCintoH. The variational inequality problem is to find au∈Csuch that
v−u, Au ≥0, ∀v∈C. 1.1
We denote the set of solutions of the variational inequality problem1.1by VIC, A. The variational inequality problem has been extensively studied in the literature; see1–5and the references therein.
introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapansive mappingSand the set of solutions of the problem1.1for an inverse-strong monotone mappingA:x1∈Cand
xn1αnx 1−αnSPCxn−λnAxn, n≥1, 1.2
where{αn} ⊂ 0,1 and{λn} ⊂ 0,2α. They proved that the sequence generated by1.2
strongly converges strongly toPFS∩VIC,Ax. In 2010, Jung10provided the following new
composite iterative scheme for the fixed point problem and the problem1.1:x1x∈Cand
ynαnfxn 1−αnSPCxn−λnAxn,
xn1
1−βn
ynβnSPC
yn−λnAyn
, n≥1, 1.3
wheref is a contraction with constantk ∈ 0,1,{αn},{βn} ∈ 0,1, and{λn} ⊂ 0,2α. He
proved that the sequence{xn}generated by1.3strongly converges strongly to a point in
FS∩VIC, A, which is the unique solution of a certain variational inequality.
On the other hand, the following optimization problem has been studied extensively by many authors:
min
x∈Ω
μ
2Bx, x 1 2x−u
2−hx, 1.4
where Ω ∞n1Cn, C1, C2, . . . are infinitely many closed convex subsets of H such that
∞
n1Cn/∅,u∈H,μ≥0 is a real number,Bis a strongly positive bounded linear operator on
Hi.e., there is a constantγ >0 such thatBx, x ≥γx2, for allx∈H, andhis a potential function forγf i.e.,hx γfxfor all x ∈ H. For this kind of optimization problems, see, for example, Deutsch and Yamada11, Jung10, and Xu12,13whenΩ Ni1Ciand
hx x, bfor a given pointbinH.
In 2007, related to a certain optimization problem, Marino and Xu14introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:
xn1αnγfxn I−αnBSxn, n≥0, 1.5
where {αn} ∈ 0,1 and γ > 0. They proved that the sequence {xn} generated by 1.5
converges strongly to the unique solution of the variational inequality
B−γfx∗, x−x∗≥0, x∈FS, 1.6
which is the optimality condition for the minimization problem
min
x∈FS
1
2Bx, x −hx, 1.7
In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem1.1for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al.6, Iiduka and Takahashi8, Jung10, and others.
2. Preliminaries and Lemmas
LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. We write xn xto indicate that the sequence{xn}converges weakly tox.xn → ximplies that{xn}
converges strongly tox.
First we recall that a mappingf :C → Cis acontractiononCif there exists a constant k ∈ 0,1such thatfx−fy ≤ kx−y,x, y ∈ C. A mapping T : C → Cis called
nonexpansiveifTx−Ty ≤ x−y, x, y∈C. We denote byFTthe set of fixed points ofT. For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such
that
x−PCx ≤x−y 2.1
for all y ∈ C. PC is called the metric projection of H onto C. It is well known that PC is
nonexpansive andPC satisfies
x−y, PCx−PC
y≥PCx−PC
y2 2.2
for everyx, y∈H. Moreover,PCxis characterized by the properties:
x−y2
≥ x−PCx2y−PCx2,
uPCx⇐⇒
x−u, u−y≥0, ∀x∈H, y∈C.
2.3
In the context of the variational inequality problem for a nonlinear mappingA, this implies that
u∈VIC, A⇐⇒uPCu−λAu, for anyλ >0. 2.4
It is also well known thatHsatisfies theOpial condition, that is, for any sequence{xn}with
xn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y 2.5
A mappingAofCintoH is calledinverse-strongly monotoneif there exists a positive real numberαsuch that
x−y, Ax−Ay≥αAx−Ay2 2.6
for allx, y ∈C; see4,7,17. For such a case,Ais calledα-inverse-strongly monotone. We know that ifAI−T, whereTis a nonexpansive mapping ofCinto itself andIis the identity mapping ofH, thenAis 1/2-inverse-strongly monotone and VIC, A FT. A mappingA ofCintoHis calledstrongly monotoneif there exists a positive real numberηsuch that
x−y, Ax−Ay≥ηx−y2 2.7
for allx, y∈C. In such a case, we sayAisη-strongly monotone. IfAisη-strongly monotone andκ-Lipschitz continuous, that is,Ax−Ay ≤ κx−yfor allx, y ∈ C, thenAisη/κ2
-inverse-strongly monotone. If Ais an α-inverse-strongly monotone mapping ofC intoH, then it is obvious thatAis 1/α-Lipschitz continuous. We also have that for allx, y ∈Cand λ >0,
I−λAx−I−λAy2
x−y−λAx−Ay2
x−y2−2λx−y, Ax−Ayλ2Ax−Ay2
≤x−y2λλ−2αAx−Ay2.
2.8
So, ifλ≤2α, thenI−λAis a nonexpansive mapping ofCintoH. The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda9.
Proposition 2.1. LetCbe a bounded closed convex subset of a real Hilbert space and let A be an
α-inverse-strongly monotone mapping ofCintoH. Then,VIC, Ais nonempty.
A set-valued mappingT :H → 2His calledmonotoneif for allx, y∈H,f ∈Tx, and
g∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2Hismaximalif the graphGT
ofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for every
y, g∈GTimpliesf ∈Tx. LetAbe an inverse-strongly monotone mapping ofCintoH and letNCvbe thenormal conetoCatv, that is,NCv{w∈H:v−u, w ≥0, for allu∈C},
and define
Tv
⎧ ⎨ ⎩
AvNCv, v∈C,
∅, v /∈C.
2.9
We need the following lemmas for the proof of our main results.
Lemma 2.2. In a real Hilbert spaceH, there holds the following inequality:
xy2≤ x22y, xy, 2.10
for allx, y∈H.
Lemma 2.3Xu12. Let{sn}be a sequence of nonnegative real numbers satisfying
sn1≤1−λnsnβnγn, n≥1, 2.11
where{λn}and{βn}satisfy the following conditions:
i{λn} ⊂0,1and
∞
n1λn∞or, equivalently,
∞
n11−λn 0;
iilim supn→ ∞βn/λn≤0or
∞
n1|βn|<∞;
iiiγn≥0 n≥1,
∞
n1γn<∞. Thenlimn→ ∞sn0.
Lemma 2.4Marino and Xu14. Assume that A is a strongly positive linear bounded operator on a Hilbert spaceHwith constantγ >0and0< ρ≤ B−1. ThenI−ρB ≤1−ργ.
The following lemma can be found in20,21 see alsoLemma 2.2in22.
Lemma 2.5. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letg : C →
R∪ {∞}be a proper lower semicontinunous differentiable convex function. Ifx∗is a solution to the minimization problem
gx∗ inf
x∈Cgx, 2.12 then
gx, x−x∗≥0, x∈C. 2.13
In particular, ifx∗solves the optimization problem
min
x∈C
μ
2Bx, x 1 2x−u
2−hx, 2.14
then
uγf−IμBx∗, x−x∗≤0, x∈C, 2.15
3. Main Results
In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.
Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceHsuch thatC ± C⊂ C. Let A be anα-inverse-strongly monotone mapping ofCintoHandSa nonexpansive mapping of
Cinto itself such thatFS∩VIC, A/∅. Letu∈Cand letBbe a strongly positive bounded linear operator onCwith constantγ ∈0,1andfa contraction ofCinto itself with constantk ∈0,1. Assume thatμ >0and0< γ <1μγ/k. Let{xn}be a sequence generated by
x1x∈C,
ynαn
uγfxn
I−αn
IμBSPCxn−λnAxn,
xn1
1−βn
ynβnSPC
yn−λnAyn
, n≥1,
IS
where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. Let{αn},{λn}, and{βn}satisfy the following conditions:
iαn → 0 n → ∞;
∞
n1αn∞;
iiβn⊂0, afor alln≥0and for somea∈0,1;
iiiλn∈c, dfor somec,dwith0< c < d <2α;
iv∞n1|αn1−αn|<∞,
∞
n1|βn1−βn|<∞,
∞
n1|λn1−λn|<∞.
Then{xn}converges strongly toq∈FS∩VIC, A, which is a solution of the optimization problem
min
x∈FS∩VIC,A μ
2Bx, x 1 2x−u
2−hx, OP1
wherehis a potential function forγf.
Proof. We note that from the control conditioni, we may assume, without loss of generality, thatαn≤1μB−1. Recall that ifBis bounded linear self-adjoint operator onH, then
Bsup{|Bu, u|:u∈H, u1}. 3.1
Observe that
I−αn
IμBu, u1−αn−αnμBu, u
≥1−αn−αnμB
≥0,
which is to say thatI−αnIμBis positive. It follows that
I−αn
IμBsupI−αn
IμBu, u:u∈H, u1
sup1−αn−αnμBu, u:u∈H, u1
≤1−αn
1μγ
<1−αn
1μγ.
3.3
Now we divide the proof into several steps.
Step 1. We show that{xn}is bounded. To this end, letzn PCxn−λnAxnandwnPCyn−
λnAynfor every n ≥ 1. Letp ∈ FS∩VIC, A. SinceI −λnAis nonexpansive andp
PCp−λnApfrom2.4, we have
zn−p≤xn−λnAxn−
p−λnAp
≤xn−p.
3.4
Similarly, we have
wn−p≤yn−p. 3.5
Now, setB IμB. Letp∈FS∩VIC, A. Then, fromISand3.4, we obtain
yn−pαnuαn
γfxn−Bp
I−αnB
Szn−p
≤1−1μγαnzn−pαnu
αnγfxn−f
pαnγf
p−Bp
≤1−1μγαnzn−pαnu
αnγkxn−pαnγf
p−Bp
1−1μγ−γkαnxn1−p
1μγ−γkαn
γfp−Bpu
1μγ−γk .
From3.5and3.6, it follows that
xn1−p1−βn
yn−p
βn
Swn−p
≤1−βnyn−pβnwn−p
≤1−βnyn−pβnyn−p
yn−p
≤max
⎧ ⎨
⎩xn−p,
γfp−Bpu
1μγ−γk
⎫ ⎬ ⎭.
3.7
By induction, it follows from3.7that
xn−p≤max⎧⎨
⎩x1−p,
γfp−Bpu
1μγ−γk
⎫ ⎬
⎭ n≥1. 3.8
Therefore, {xn} is bounded. So {yn}, {zn}, {wn},{fxn}, {Axn}, {Ayn}, and {BSzn} are
bounded. Moreover, sinceSzn−p ≤ xn−pandSwn−p ≤ yn−p,{Szn}and{Swn}
are also bounded. And by the conditioni, we have
yn−Sznαnuγfxn
−IμBSzn
αn
uγfxn
−BSzn−→0 asn−→ ∞.
3.9
Step 2. We show that limn→ ∞xn1−xn0 and limn→ ∞yn1−yn0. Indeed, sinceI−λnA
andPCare nonexpansive andznPCxn−λnAxn, we have
zn−zn−1 ≤ xn−λnAxn−xn−1−λn−1Axn−1
≤ xn−xn−1|λn−λn−1|Axn−1.
3.10
Similarly, we get
wn−wn−1 ≤yn−yn−1|λn−λn−1|Ayn−1. 3.11
Simple calculations show that
yn−yn−1αn
uγfxn
I−αnB
Szn−αn−1
uγfxn−1
−I−αn−1B
Szn−1
αn−αn−1
uγfxn−1−BSzn−1
αnγ
fxn−fxn−1
I−αnB
Szn−Szn−1.
So, we obtain
yn−yn−1≤ |αn−αn−1|
uγfxn−1BSzn−1
αnγkxn−xn−1
1−1μγαn
zn−zn−1
≤ |αn−αn−1|
uγfxn−1BSzn−1
αnγkxn−xn−1
1−1μγαn
xn−xn−1
|λn−λn−1|Axn−1.
3.13
Also observe that
xn1−xn
1−βn
yn−yn−1
βn−βn−1
Swn−1−yn−1
βnSwn−Swn−1.
3.14
By3.11,3.13, and3.14, we have
xn1−xn ≤
1−βnyn−yn−1βn−βn−1Sxn−1yn−1
βnwn−wn−1
≤1−βnyn−yn−1βnyn−yn−1βn|λn−λn−1|Ayn−1
βn−βn−1Swn−1yn−1
≤yn−yn−1|λn−λn−1|Ayn−1βn−βn−1Swn−1yn−1
≤1−1μγ−γkαn
xn−xn−1
|αn−αn−1|
uγfxn−1BSzn−1
|λn−λn−1|Ayn−1Axn−1
βn−βn−1Swn−1yn−1
≤1−1μγ−γkαn
xn−xn−1
M1|αn−αn−1|M2|λn−λn−1|M3βn−βn−1,
3.15
whereM1sup{uγfxnBTnzn:n≥1},M2sup{AynAxn:n≥1}, and
M3sup{Swnyn:n≥1}. From the conditionsiandiv, it is easy to see that
lim
n→ ∞
1μγ−γkαn 0,
∞
n1
1μγ−γkαn∞,
∞
n2
M1|αn−αn−1|M2|λn−λn−1|M3βn−βn−1<∞.
ApplyingLemma 2.3to3.15, we obtain
lim
n→ ∞xn1−xn0. 3.17
Moreover, by3.10and3.13, we also have
lim
n→ ∞zn1−zn0, nlim→ ∞yn1−yn0. 3.18
Step 3. We show that limn→ ∞xn−yn0 and limn→ ∞xn−Szn0. Indeed,
xn1−ynβnSwn−yn
≤βn
Swn−SznSzn−yn
≤awn−znSzn−yn
≤ayn−xnSzn−yn
≤ayn−xn1xn1−xnSzn−yn
3.19
which implies that
xn1−yn≤ a 1−a
xn1−xnSzn−yn. 3.20
Obviously, by3.9andStep 2, we havexn1−yn → 0 asn → ∞. This implies that
xn−yn≤ xn−xn1xn1−yn−→0 asn−→ ∞. 3.21
By3.9and3.21, we also have
Step 4. We show that limn→ ∞xn−zn0 and limn→ ∞yn−zn0. To this end, letpFS∩
VIC, A. SinceznPCxn−λnAxnandpPCp−λnp, we have
yn−p2 αn
uγfxn−Bp
I−αnB
Szn−p
2
≤αnuγfxn−BpI−αnBSzn−p
2
≤αnuγfxn−Bp
2
1−αn
1μγzn−p2
2αn
1−αn
1μγuγfxn−Bpzn−p
≤αnuγfxn−Bp
2
1−αn
1μγxn−p2λnλn−2αAxn−Ap2
2αn
1−αn
1μγγufxn−Bpzn−p
≤αnuγfxn−Bp
2
xn−p2
1−αn
1μγcd−2αAxn−Ap2
2αnuγfxn−Bpzn−p.
3.23
So we obtain
−1−αn
1μγcd−2αAxn−Ap2
≤αnγufxn−Bp
2
xn−pyn−pxn−p−yn−p
2αnγufxn−Bpzn−p
≤αnγufxn−Bp
2
xn−pyn−pxn−yn
2αnγufxn−Bpzn−p.
Sinceαn → 0 from the conditioniandxn−yn → 0 fromStep 3, we haveAxn−Ap →
0n → ∞. Moreover, from2.4we obtain
zn−p2PCxn−λnAxn−PC
p−λnAp2
≤xn−λnAxn−
p−λnAp
, zn−p
1
2
xn−λnAxn−
p−λnAp2zn−p2
−xn−λnAxn−
p−λnAp
−zn−p2
≤ 1
2
xn−p2zn−p2− xn−zn2
2λn
xn−zn, Axn−Ap
−λ2nAxn−Ap2
,
3.25
and so
zn−p2≤
xn−p2− xn−zn22λn
xn−zn, Axn−Ap
−λ2nAxn−Ap2. 3.26
Thus
yn−p2≤αnuγfxn−Bp
2
1−αn
1μγzn−p2
2αn
1−αn
1μγγufxn−Bpzn−p
≤αnuγfxn−Bp
2
xn−p2−
1−αn
1μγxn−zn2
21−αn
1μγλn
xn−zn, Axn−Ap
−1−αn
1μγλ2nAxn−Ap2
2αnuγfxn−Bzn−p.
Then, we have
1−αn
1μγxn−zn2
≤αnuγfxn−Bp
2
xn−pyn−pxn−p−yn−p
21−αn
1μγλn
xn−zn, Axn−Ap
−1−αn
1μγλ2nAxn−Ap2
2αnuγfxn−Bpzn−p
≤αnuγfxn−Bp
2
xn−pyn−pxn−yn
21−αn
1μγλn
xn−zn, Axn−Ap
−1−αn
1μγλ2nAxn−Ap2
2αnuγfxn−Bpzn−p.
3.28
Sinceαn → 0,xn−yn → 0 andAxn−Au → 0, we getxn−zn → 0. Also by3.21
yn−zn≤yn−xnxn−zn −→0 n−→ ∞. 3.29
Step 5. We show that limn→ ∞Szn−zn0. In fact, since
Szn−zn ≤Szn−ynyn−zn
αnuγfxn−BSznyn−zn,
3.30
from3.9and3.29, we have limn→ ∞Szn−zn0. Step 6. We show that
lim sup
n→ ∞
uγf−IμBq, yn−q
lim sup
n→ ∞
uγf−Bq, yn−q
≤0, 3.31
whereqis a solution of the optimization problemOP1. First we prove that
lim sup
n→ ∞
uγf−Bq, Szn−q
Since{zn}is bounded, we can choose a subsequence{zni}of{zn}such that
lim sup
n→ ∞
uγf−Bq, Szn−q
lim
i→ ∞
uγf−Bq, Szni−q
. 3.33
Without loss of generality, we may assume that{zni}converges weakly toz∈C.
Now we will show thatz∈FS∩VIC, A. First we show thatz∈FS. Assume that z /∈FS. Sincezni zandSz /z, by the Opial condition andStep 5, we obtain
lim inf
i→ ∞ zni −z<lim inf
i→ ∞ zni−Sz ≤lim inf
i→ ∞ zni−SzniSzni−Sz
lim inf
i→ ∞ Szni−Sz ≤lim inf
i→ ∞ zni−z,
3.34
which is a contradiction. Thus we havez∈FS. Next, let us show thatz∈VIC, A. Let
Tv
⎧ ⎨ ⎩
AvNCv, v∈C,
∅, v /∈C.
3.35
ThenT is maximal monotone. Letv, w∈GT. Sincew−Av∈NCvandzn∈C, we have
v−zn, w−Av ≥0. 3.36
On the other hand, fromzn PCxn−λnAxn, we havev−zn, zn−xn−λnAzn ≥ 0 and
hence
v−zn,zn−xn
λn
Therefore, we have
v−zni, w ≥ v−zni, Av
≥ v−zni, Av −
v−zni,
zni−xni λni
Axni
v−zni, Av−Axni−
zni −xni λni
v−zni, Av−Azniv−zni, Azni −Axni −
v−zni,
zni−xni λni
≥ v−zni, Azni−Axni −
v−zni,
zni−xni λni
.
3.38
Sincezn−xn → 0 inStep 4andAisα-inverse-strongly monotone, we havev−z, w ≥0
asi → ∞. SinceT is maximal monotone, we havez∈T−10 and hencez∈VIC, A.
Therefore,z∈FS∩VIC, A. Now fromLemma 2.5andStep 5, we obtain
lim sup
n→ ∞
uγf−Bq, Szn−q
lim
i→ ∞
uγf−Bq, Szni−q
lim
i→ ∞
uγf−Bq, zni−q
uγf−Bq, z−q
≤0.
3.39
By3.9and3.39, we conclude that
lim sup
n→ ∞
uγf−Bq, yn−q
≤lim sup
n→ ∞
uγf−Bq, yn−Szn
lim sup
n→ ∞
uγf−Bq, Szn−q
≤lim sup
n→ ∞
uγf−Bqyn−Sznlim sup n→ ∞
uγf−Bq, Szn−q
≤0.
Step 7. We show that limn→ ∞xn−q0 and limn→ ∞un−q 0, whereqis a solution of
the optimization problemOP1. Indeed fromISandLemma 2.2, we have
xn1−q2≤
yn−q2
αn
uγfxn−Bq
I−αnB
Szn−q
≤I−αnBSzn−q
2
2αn
uγfxn−Bq, yn−q
≤1−1μγαn
2
zn−q22αnγ
fxn−f
q, yn−q
2αn
uγfq−Bq, yn−q
≤1−1μγαn
2
xn−q22αnγkxn−qyn−q
2αn
uγf−Bq, yn−q
≤1−1μγαn
2x
n−q22αnγkxn−qyn−xnxn−q
2αn
uγf−Bq, yn−q
1−21μγ−γkαnxn−q2
α2
n
1μγ2xn−q22αnγkxn−qyn−xn
2αn
uγf−Bq, yn−q
,
3.41
that is,
xn1−q2 ≤
1−21μγ−γkαnxn−q2
α2n1μγ2M422αnγkyn−xnM4
2αn
uγf−Bq, yn−q
1−αnxn−q2βn,
3.42
whereM4sup{xn−q:n≥1},αn21μγ−γkαn, and
βnαn
αn
1μγ2M242γkyn−xnM42
uγf−Bq, yn−q
. 3.43
From i, yn −xn → 0 in Steps 3, and6, it is easily seen that αn → 0,
∞
n1αn ∞,
and lim supn→ ∞βn/αn ≤ 0. Hence, byLemma 2.3, we concludexn → qasn → ∞. This
completes the proof.
Corollary 3.2. Let H, C, S, B, f, u, γ, γ, k, and μ be as in Theorem 3.1. Let {xn} be a sequence generated by
x1x∈C,
ynαn
uγfxn
I−αn
IμBSxn,
xn1
1−βn
ynβnSyn, n≥1,
3.44
where {αn} and {βn} ⊂ 0,1. Let {αn} and {βn} satisfy the conditions (i), (ii), and (iv) in
Theorem 3.1. Then {xn} converges strongly to q ∈ FS, which is a solution of the optimization problem
min
x∈FS μ
2Bx, x 1 2x−u
2−hx, OP2
wherehis a potential function forγf.
Corollary 3.3. Let H, C, A, B, f, u, γ, γ, k, and μ be as in Theorem 3.1. Let {xn} be a sequence generated by
x1x∈C,
ynαn
uγfxn
I−αn
IμBPCxn−λnAxn,
xn1
1−βn
ynβnPC
yn−λnAyn
, n≥1,
3.45
where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. Let{αn},{λn}and{βn}satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then {xn}converges strongly to q ∈ VIC, A, which is a solution of the optimization problem
min
x∈V IC,A μ
2Bx, x 1 2x−u
2−hx, OP3
wherehis a potential function forγf.
Remark 3.4. 1 Theorem 3.1 and Corollary 3.3 improve and develop the corresponding results in Chen et al.6, Iiduka and Takahashi8, and Jung10.
2Even thoughβn 0 forn≥1, the iterative scheme3.44inCorollary 3.2is a new
one for fixed point problem of a nonexpansive mapping.
4. Applications
A mappingT :C → Cis calledstrictly pseudocontractiveif there existsαwith 0≤α <1 such that
Tx−Ty2≤
x−y2αI−Tx−I−Ty2 4.1
for everyx, y ∈ C. Ifk 0, thenT is nonexpansive. PutA I−T, whereT : C → Cis a strictly pseudo-contractive mapping with constantα. ThenAis1−α/2-inverse-strongly monotone; see2. Actually, we have, for allx, y∈C,
I−Ax−I−Ay2≤x−y2αAx−Ay2. 4.2
On the other hand, sinceHis a real Hilbert space, we have
I−Ax−I−Ay2
x−y2Ax−Ay2−2x−y, Ax−Ay. 4.3
Hence we have
x−y, Ax−Ay≥ 1−α
2 Ax−Ay
2
. 4.4
UsingTheorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.
Theorem 4.1. LetH,C,S,B,f,u,γ,γ,k, andμbe as inTheorem 3.1. LetTbe anα-strictly pseudo-contractive mapping ofCinto itself such thatFS∩FT/∅. Let{xn}be a sequence generated by
x1x∈C,
yn αn
uγfxn
I−αn
IμBS1−λnxnλnTxn,
xn1
1−βn
ynβnS
1−λnynλnTyn
, n≥1,
4.5
where {λn} ⊂ 0,1−α,{αn} ⊂ 0,1, and{βn} ⊂ 0,1. Let {αn},{λn}, and{βn} satisfy the conditions (i), (ii), (iii), and (iv) inTheorem 3.1. Then{xn}converges strongly toq∈FS∩FT, which is a solution of the optimization problem
min
x∈FS∩FT μ
2Bx, x 1 2x−u
2−hx, OP4
wherehis a potential function forγf.
Proof. PutAI−T. ThenAis1−α/2-inverse-strongly monotone. We haveFT VIC, A andPCxn−λnAxn 1−λnxnλnTxn. Thus, the desired result follows fromTheorem 3.1.
Theorem 4.2. LetHbe a real Hilbert space. Let Abe anα-inverse-strongly monotone mapping of
HintoHand S a nonexpansive mapping ofHinto itself such thatFS∩A−10/∅. Letu∈H, and
letBbe a strongly positive bounded linear operator on Hwith constant γ > 0and f : H → H
a contraction with constantk ∈0,1. Assume thatμ >0and0 < γ < 1μγ/k. Let{xn}be a sequence generated by
x1x∈H,
ynαn
uγfxn
I−αn
IμBSxn−λnAxn,
xn1
1−βn
ynβnS
yn−λnAyn
, n≥1,
4.6
where {λn} ⊂ 0,2α, {αn} ⊂ 0,1, and {βn} ⊂ 0,1. Let {αn}, {λn}, and {βn} satisfy the conditions (i), (ii), (iii), and (iv) inTheorem 3.1. Then{xn}converges strongly toq∈FS∩A−10, which is a solution of the optimization problem
min
x∈FS∩A−10 μ
2Bx, x 1 2x−u
2−hx, OP5
wherehis a potential function forγf.
Proof. We haveA−10 VIH, A. So, puttingP
H I, byTheorem 3.1, we obtain the desired
result.
Remark 4.3. 1Theorems4.1and4.2complement and develop the corresponding results in Chen et al.6and Jung10.
2In all our results, we can replace the condition∞n1|αn1−αn|<∞on the control
parameter{αn}by the conditionαn ∈0,1forn≥1, limn→ ∞αn/αn1 112,13or by the
perturbed control condition|αn1−αn|< oαn1 σn,
∞
n1σn<∞23.
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology2010-0017007.
References
1 F. E. Browder, “Nonlinear monotone operators and convex sets in Banach spaces,”Bulletin of the American Mathematical Society, vol. 71, pp. 780–785, 1965.
2 R. E. Bruck, Jr., “On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 61, no. 1, pp. 159–164, 1977.
3 J.-L. Lions and G. Stampacchia, “Variational inequalities,” Communications on Pure and Applied Mathematics, vol. 20, pp. 493–519, 1967.
4 F. Liu and M. Z. Nashed, “Regularization of nonlinear ill-posed variational inequalities and convergence rates,”Set-Valued Analysis, vol. 6, no. 4, pp. 313–344, 1998.
Feasibility and Optimization and Their Applications (Haifa, 2000), vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
6 J. Chen, L. Zhang, and T. Fan, “Viscosity approximation methods for nonexpansive mappings and monotone mappings,”Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1450–1461, 2007.
7 H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for monotone mappings,”Panamerican Mathematical Journal, vol. 14, no. 2, pp. 49–61, 2004.
8 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005.
9 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
10 J. S. Jung, “A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces,”Journal of Inequalities and Applications, vol. 2010, Article ID 251761, 16 pages, 2010.
11 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,”Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998.
12 H.-K. Xu, “Iterative algorithms for nonlinear operators,”Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
13 H.-K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
14 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
15 A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
16 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
17 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
18 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
19 R. T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.
20 J. T. Oden,Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.
21 Y. Yao, M. A. Noor, S. Zainab, and Y.-C. Liou, “Mixed equilibrium problems and optimization problems,”Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 319–329, 2009.
22 J. S. Jung, “Iterative algorithms with some control conditions for quadratic optimizations,”
Panamerican Mathematical Journal, vol. 16, no. 4, pp. 13–25, 2006.
