Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 748918, 17 pages
doi:10.1155/2011/748918
Research Article
An Implicit Hierarchical Fixed-Point Approach to
General Variational Inequalities in Hilbert Spaces
L. C. Zeng,1, 2 Ching-Feng Wen,3 and J. C. Yao3
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3
Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
2
Correspondence should be addressed to Ching-Feng Wen, cfwen@kmu.edu.tw
Received 13 December 2010; Accepted 5 March 2011
Academic Editor: Jong Kim
Copyright q 2011 L. C. Zeng 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 C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be
a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C
be nonexpansive mappings with FixT /
∅ where FixT denotes the fixed-point set of T , and
f : C → H be
a
ρ-contraction
with
coefficient
ρ ∈ 0, 1. Let 0 < μ < 2η/κ2 and 0 < γ ≤ τ,
where τ 1 − 1 − μ2η − μκ2 . For each s, t ∈ 0, 1, let xs,t be a unique solution of the fixed-point
equation xs,t PC sγf xs,t I − sμFtV 1 − tT xs,t . We derive the following conclusions on
the behavior of the net {xs,t } along the curve t ts: i if ts Os, as s → 0, then xs,ts → z∞
strongly, which is the unique solution of the variational inequality of finding z∞ ∈ FixT such
that μF − γf lI − V z∞ , x − z∞ ≥ 0, for all x ∈ FixT and ii if ts/s → ∞, as s → 0,
then xs,ts → x∞ strongly, which is the unique solution of some hierarchical variational inequality
problem.
1. Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper,
we write xn x to indicate that the sequence {xn } converges weakly to x. xn → x implies
that {xn } converges strongly to x. Let T : C → C be a nonexpansive mapping; namely,
Tx − Ty
≤ x − y
for all x, y ∈ C. The set of fixed-points of T is denoted by the set
FixT : {x ∈ C : Tx x}. It is well known that if FixT /
∅, then FixT is closed and
convex. Given nonexpansive mapping V : C → C, consider the variational inequality for
short, VI of finding hierarchically a fixed-point x∗ ∈ FixT of T with respect to V such that
I − V x∗ , y − x∗ ≥ 0,
∀y ∈ FixT.
1.1
2
Fixed Point Theory and Applications
Equivalently, x∗ PFixT V x∗ , that is, x∗ is a fixed-point of the nonexpansive mapping PFixT V ,
where PK denotes the metric projection from H onto a nonempty closed convex subset K
of H. Let S denote the solution set of the VI 1.1, and assume throughout the rest of this
paper that S / ∅. It is easy to see that S FixPFixT V . The VI 1.1 covers several topics
investigated in the literature; see, for example, 1–7. Related iterative methods for solving
fixed-point problems, variational inequalities, and optimization problems can also be found
in 8–20.
Let f : C → C be a ρ-contraction and define, for s, t ∈ 0, 1, two mappings Wt and fs,t
by
Wt tV 1 − tT,
fs,t sf 1 − sWt .
1.2
It is easy to verify that Wt is nonexpansive and fs,t is a 1 − 1 − ρs-contraction.
Let xs,t be the unique fixed-point of fs,t , that is, the unique solution of the fixed-point
equation
xs,t sfxs,t 1 − sWt xs,t .
1.3
Moudafi and Maingé 21 initiated the investigation of the iterated behavior of the net
{xs,t } as s → 0 firstly and t → 0 secondly. They made the following assumptions:
A1 for each t ∈ 0, 1, the fixed-point set FixWt of Wt is nonempty and the set
{FixWt : 0 < t < 1} FixWt 1.4
t∈0,1
is bounded;
A2 ∅ / S ⊂ · − lim inft → 0 FixWt : {z : ∃zt ∈ FixWt such that zt → z}.
Moudafi and Maingé 21 see also 22 proved that, for each fixed t ∈ 0, 1, as s → 0,
xs,t → xt ; moreover, as t → 0, xt x∞ which is the unique solution of the variational
inequality of finding x∞ ∈ S, such that
I − f x∞ , x − x∞ ≥ 0,
∀x ∈ S.
1.5
The following theorem, due to Xu 23, improves Moudafi and Maingé’s result
since it shows that {xt } actually strongly converges to x∞ . Moreover, it does not need the
boundedness assumption of the set t∈0,1 FixWt .
Theorem 1.1 see 23, Theorem 3.2. Let the above assumption (A2) hold. Assume also that, for
each t ∈ 0, 1, FixWt is nonempty (but not necessarily bounded), then the strong lims → 0 xs,t : xt
exists for each t ∈ 0, 1. Moreover, the strong limt → 0 xt : x∞ exists and solves the VI 1.5. Hence,
for each null sequence {sn } in 0, 1, there is another null sequence {tn } in 0, 1 such that xsn ,tn ,
converges strongly to x∞ , as n → ∞.
In 21, 23, the authors stated the problem of the convergence of {xs,t } when s, t →
0, 0 jointly. Very recently, Cianciaruso et al. 24 further investigated the behavior of the net
{xs,t } along the curve t ts, and their results point to a negative answer to this problem.
Fixed Point Theory and Applications
3
Theorem 1.2 see 24, Theorem 2.1. Let H be a real Hilbert space, and let C be a nonempty closed
convex subset of H. Let V, T : C → C be nonexpansive mappings with FixT /
∅. Let f : C → C
be a ρ-contraction with ρ ∈ 0, 1. Assume that ts Os, as s → 0, and let l lim sups → 0 ts /s,
then the net {xs,ts }s∈0,1 , defined by
xs,ts sfxs,ts 1 − sWts xs,ts ,
1.6
strongly converges to z∞ ∈ FixT which is the unique solution of the variational inequality of finding
z∞ ∈ FixT, such that
I − f lI − V z∞ , x − z∞ ≥ 0,
∀x ∈ FixT.
1.7
Theorem 1.3 see 24, Theorem 3.1. Let H be a real Hilbert space and let C be a nonempty closed
convex subset of H. Assume that V, T : C → C are nonexpansive mappings with FixT / ∅ and
f : C → C is a ρ-contraction with ρ ∈ 0, 1. Assume the condition (A2) holds. Let ts ts satisfy
lims → 0 ts /s ∞. Then the net {xs,ts }s∈0,1 defined by
xs,ts sfxs,ts 1 − sWts xs,ts ,
1.8
strongly converges to x∞ ∈ S which is the unique solution of the VI 1.5.
On the other hand, let F : H → H be a κ-Lipschitzian and η-strongly monotone
operator with constants κ, η > 0, and let T : H → H be nonexpansive such that FixT /
∅.
In 2001, Yamada 6 introduced the so-called hybrid steepest descent method for solving the
variational inequality problem: finding x ∈ FixT such that
x − x
≥ 0,
F x,
∀x ∈ FixT.
1.9
This method generates a sequence {xn } via the following iterative scheme:
xn1 Txn − λn1 μFTxn ,
∀n ≥ 0,
1.10
where 0 < μ < 2η/κ2 , the initial guess x0 ∈ H is arbitrary, and the sequence {λn } in 0, 1
satisfies the conditions
λn −→ 0,
∞
n0
λn ∞,
∞
|λn1 − λn | < ∞.
1.11
n0
A key fact in Yamada’s argument is that, for small enough λ > 0, the mapping
T λ x : Tx − λμFTx,
∀x ∈ H
is a contraction, due to the κ-Lipschitz continuity and η-strong monotonicity of F.
1.12
4
Fixed Point Theory and Applications
In this paper, let C be a nonempty closed convex subset of a real Hilbert space H.
Assume that F : C → H is a κ-Lipschitzian and η-strongly monotone operator with constants
κ, η > 0, f : C → H is a ρ-contraction with coefficient ρ ∈ 0, 1 and T, V : C → C are
nonexpansive mappings with FixT / ∅. Let 0 < μ < 2η/κ2 , and 0 < γ ≤ τ, where τ 1 −
1 − μ2η − μκ2 . Consider the hierarchical variational inequality problem for short, HVIP:
VI a: finding z∗ ∈ FixT such that I − V z∗ , z − z∗ ≥ 0, for all z ∈ FixT;
VI b: finding x∗ ∈ S such that μF − γ fx∗ , x − x∗ ≥ 0, for all x ∈ S.
Here, S denotes the nonempty solution set of the VI a.
Motivated and inspired by the above-hybrid steepest descent method and hierarchical
fixed-point approximation method, we define, for each s, t ∈ 0, 1, two mappings Wt and fs,t
by
Wt tV 1 − tT,
fs,t PC sγ f I − sμF Wt .
1.13
It is easy to see that Wt is a nonexpansive self-mapping on C. Moreover, utilizing Lemma 2.5
in Section 2, we can see that fs,t is a 1 − τ − γ ρs-contraction. Indeed, observe that
fs,t x − fs,t y PC sγ fx I − sμF Wt x − PC sγ f y I − sμF Wt y ≤ sγ fx I − sμF Wt x − sγ f y I − sμF Wt y ≤ sγ fx − f y I − sμF Wt x − I − sμF Wt y
≤ sγ ρx − y 1 − sτx − y
1 − τ − γ ρ s x − y .
1.14
Let xs,t be the unique fixed-point of fs,t in C, that is, the unique solution of the fixed-point
equation
xs,t PC sγ fxs,t I − sμF Wt xs,t .
1.15
We investigate the behavior of the net {xs,t } generated by 1.15 along the curve
t ts and our results give a negative answer to the problem put forth in 21, 23. Specifically,
we derive the following conclusions:
i if ts Os, as s → 0, then xs,ts → z∞ ∈ FixT, which is the unique solution of
the variational inequality of finding z∞ ∈ FixT such that
μF − γ f lI − V z∞ , x − z∞ ≥ 0,
∀x ∈ FixT,
1.16
ii if ts/s → ∞, as s → 0, then xs,ts → x∞ ∈ S, which is the unique solution of the
VI b.
In particular, if we put μ 1, F I, and γ τ 1, and let f be a contractive selfmapping on C with coefficient ρ ∈ 0, 1, then our results reduce to the above Theorems 1.2
Fixed Point Theory and Applications
5
and 1.3, respectively. There is no doubt that our results cover Theorems 1.2 and 1.3 as special
cases, respectively. In the meantime, our results also extend and improve Xu’s Theorem 3.2
23.
2. Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Recall that
the metric or nearest point projection from H onto C is the mapping PC : H → C which
assigns to each point x ∈ H the unique point PC x ∈ C satisfying the property
x − PC x
inf x − y : dx, C.
2.1
y∈C
Lemma 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and
z∈C
i that z PC x if and only if there holds the relation
x − z, y − z ≤ 0,
∀y ∈ C,
2.2
ii that z PC x if and only if there holds the relation:
2 2
x − z
2 ≤ x − y − y − z ,
∀y ∈ C,
2.3
iii there holds the relation
2
PC x − PC y, x − y ≥ PC x − PC y ,
∀x, y ∈ H.
2.4
Consequently, PC is nonexpansive and monotone.
Lemma 2.2 see 25, Demiclosedness principle. Let C be a nonempty closed convex subset of a
real Hilbert space H and let T : C → C be a nonexpansive mapping with FixT / ∅. If {xn } is a
sequence in C weakly converging to x and if {I − Txn } converges strongly to y, then I − Tx y;
in particular, if y 0, then x ∈ FixT.
The following lemmas are not difficult to prove.
Lemma 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H, f : C → H a
ρ-contraction with coefficient ρ ∈ 0, 1, and F : C → H a κ-Lipschitzian and η-strongly monotone
operator with constants κ, η > 0, then for 0 ≤ γ ρ < μη,
2
x − y, μF − γ f x − μF − γ f y ≥ μη − γ ρ x − y ,
that is, μF − γ f is strongly monotone with constant μη − γ ρ.
∀x, y ∈ C,
2.5
6
Fixed Point Theory and Applications
Lemma 2.4. There holds the following inequality in a real Hilbert space H:
x y
2 ≤ x
2 2 y, x y ,
∀x, y ∈ H.
2.6
The following lemma plays a key role in proving strong convergence of our implicit
hybrid method.
Lemma 2.5 see 5, Lemma 3.1. Let λ be a number in 0, 1 and let μ > 0. Let F : C → H be an
operator on C such that, for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone.
Associating with a nonexpansive mapping T : C → C, define the mapping T λ : C → H by
T λ x : Tx − λμFTx,
∀x ∈ C,
2.7
then T λ is a contraction provided μ < 2η/κ2 , that is,
λ
T x − T λ y ≤ 1 − λτx − y,
where τ 1 −
∀x, y ∈ C,
2.8
1 − μ2η − μκ2 ∈ 0, 1.
Remark 2.6. Put F I, where I is the identity operator of H. Then κ η 1 and hence
μ < 2η/κ2 2. Also, put μ 1, then it is easy to see that
τ 1−
1 − μ 2η − μκ2 1.
2.9
In particular, whenever λ > 0, we have T λ x : Tx − λμFTx 1 − λTx.
3. On Convergence of {xs,t }s,t∈0,1
In this section we study the convergence of the net {xs,t } along the curve t ts : ts , where
ts Os, as s → 0.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H
be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C
be nonexpansive mappings with FixT /
∅, and f : C → H bea ρ-contraction with coefficient
ρ ∈ 0, 1. Let 0 < μ < 2η/κ2 and 0 < γ ≤ τ, where τ 1 − 1 − μ2η − μκ2 . Assume that
ts Os, as s → 0, and let l lim sups → 0 ts /s, then the net {xs,ts }s∈0,1 defined by
xs,ts PC sγ fxs,ts I − sμF Wts xs,ts ,
3.1
where Wts ts V 1 − ts T, strongly converges to a fixed-point z∞ of T which is the unique solution
of the variational inequality of finding z∞ ∈ FixT such that
μF − γ f lI − V z∞ , x − z∞ ≥ 0,
∀x ∈ FixT.
3.2
Fixed Point Theory and Applications
7
Proof. First, let us show that the VI 3.2 has a unique solution. Indeed, it is sufficient to show
that the operator μF − γ f lI − V is strongly monotone. Observe that
μη ≥ τ ⇐⇒ μη ≥ 1 − 1 − μ 2η − μκ2
⇐⇒ 1 − μ 2η − μκ2 ≥ 1 − μη
⇐⇒ 1 − 2μη μ2 κ2 ≥ 1 − 2μη μ2 η2
⇐⇒ κ2 ≥ η2
⇐⇒ κ ≥ η,
μF − γ f lI − V x − μF − γ f lI − V y, x − y
μF − γ f x − μF − γ f y, x − y l I − V x − I − V y, x − y
≥ μF − γ f x − μF − γ f y, x − y
≥ μη − γ ρ x − y
2 , ∀x, y ∈ C.
3.3
Since
0 ≤ γ ρ < γ ≤ τ ≤ μη,
3.4
it follows that μF − γ f lI − V is strongly monotone with constant μη − γ ρ > 0. So the
variational inequality 3.2 has only one solution. Below we use z∞ ∈ FixT to denote the
unique solution of VI 3.2.
The remainder of the proof is divided into two steps.
The first step is to prove that the net {xs,ts }s∈0,1 is bounded. Indeed, set
ys,ts sγ fxs,ts I − sμF Wts xs,ts ,
3.5
where Wts ts V 1 − ts T. Take a fixed p ∈ FixT arbitrarily, then from 3.1 we obtain that
xs,ts PC ys,ts and
ys,ts − p sγ fxs,ts I − sμF Wts xs,ts − p
s γ fxs,ts − μFWts p I − sμF Wts xs,ts − I − sμF Wts p Wts p − p
sγ fxs,ts − f p s γ f p − μFWts p I − sμF Wts xs,ts − I − sμF Wts p
ts V − Ip.
3.6
Since PC is the metric projection from H onto C, utilizing Lemma 2.1, we have
PC ys,ts − ys,ts , PC ys,ts − p ≤ 0.
3.7
8
Fixed Point Theory and Applications
Thus, utilizing Lemma 2.5, we get
xs,ts − p2 PC ys,ts − ys,ts , PC ys,ts − p ys,ts − p, xs,ts − p
≤ ys,ts − p, xs,ts − p
sγ fxs,ts − f p , xs,ts − p s γ f p − μFWts p, xs,ts − p
I − sμF Wts xs,ts − I − sμF Wts p, xs,ts − p ts V − Ip, xs,ts − p
≤ sγ fxs,ts − f p xs,ts − p
s γ f p − μFWts p, xs,ts − p
I − sμF Wts xs,ts − I − sμF Wts p
xs,ts − p
ts V − Ip, xs,ts − p
2
≤ sγ ρ
xs,ts − p
2 s γ f p − μFWts p, xs,ts − p 1 − sτxs,ts − p
ts V − Ip, xs,ts − p
2
1 − s τ − γ ρ xs,ts − p s γ f p − μFWts p, xs,ts − p
ts V − Ip, xs,ts − p ,
3.8
which hence implies that
xs,t − p2 ≤
s
ts 1
γ f p − μFWts p, xs,ts − p V − Ip, xs,ts − p .
τ − γρ
s
3.9
In particular,
xs,ts − p
≤
1
ts
γ f p − μFWts p
V − Ip
.
τ − γρ
s
3.10
Note that
Wts p − p
ts V − Ip
≤ V − Ip
.
3.11
Wts p
≤ p
V − Ip
.
3.12
Hence, we have
Since ts Os, as s → 0, 3.10 implies the boundedness of {xs,ts }, and the first step is
proved.
Fixed Point Theory and Applications
9
The second step is to prove that the net xs,ts → z∞ ∈ FixT, as s → 0, where z∞ is the
unique solution of the VI 3.2. Indeed, observe that
xs,ts − Txs,ts ≤ sγ fxs,ts I − sμF Wts xs,ts − Txs,ts ≤ sγ fxs,ts sμ
FWts xs,ts Wts xs,ts − Txs,ts 3.13
sγ fxs,ts sμ
FWts xs,ts ts V xs,ts − Txs,ts .
From 3.11, it follows that
FWts xs,ts − Fp
FWts xs,ts − FWts p FWts p − Fp
≤ κ xs,ts − p
Wts p − p
≤ κ xs,ts − p
V − Ip
.
3.14
Since {xs,ts } is bounded when s → 0, 3.14 implies the boundedness of {FWts xs,ts }.
Consequently, noticing that {xs,ts } and {FWts xs,ts } are bounded when s → 0 hence ts → 0,
we conclude from 3.13 that
xs,ts − Txs,ts −→ 0.
3.15
We now claim that {xs,ts }s∈0,1 is relatively compact as s → 0 in the norm topology. To
see this, assume {sn } is a null sequence in 0, 1. Without loss of generality, we may assume
that xsn ,tsn x which implies from 3.15 and Lemma 2.2 that x ∈ FixT. It is clear that
→ F x as n → ∞. This implies that as n → ∞,
FWtsn x Ftsn V x 1 − tsn x
γ fx
− μFWtsn x,
xsn ,tsn − x xsn ,tsn − x γ fx
− μF x,
xsn ,tsn − x μF x − μFWtsn x,
−→ 0.
≤ γ fx
− μF x,
xsn ,tsn − x μ
F x − FWtsn x
x
sn ,tsn − x
3.16
We thus immediately get from 3.9 that xsn ,tsn → x.
We next further claim that x z∞ , the unique solution of the VI 3.2, which then
completes the proof. Indeed, observe that
1
1
μF − γ f xs,ts PC ys,ts − ys,ts − I − Wts xs,ts μFxs,ts − FWts xs,ts .
s
s
3.17
10
Fixed Point Theory and Applications
Hence, utilizing Lemma 2.1, we deduce from the monotonicity of μF − γ f and I − Wts that for
any fixed p ∈ FixT,
μF − γ f p, xs,ts − p ≤ μF − γ f xs,ts , xs,ts − p
1
1
PC ys,ts − ys,ts , PC ys,ts − p − I − Wts xs,ts , xs,ts − p
s
s
μ Fxs,ts − FWts xs,ts , xs,ts − p
≤−
1
I − Wts xs,ts , xs,ts − p μ Fxs,ts − FWts xs,ts , xs,ts − p
s
1
1
I − Wts xs,ts − I − Wts p, xs,ts − p − I − Wts p, xs,ts − p
s
s
μ Fxs,ts − FWts xs,ts , xs,ts − p
−
1
I − Wts p, xs,ts − p μ Fxs,ts − FWts xs,ts , xs,ts − p
s
ts V − Ip, xs,ts − p μ Fxs,ts − FWts xs,ts , xs,ts − p .
s
≤−
3.18
we have
Now, since xsn ,tsn → x,
Fxsn ,tsn − FWtsn xsn ,tsn Fxsn ,tsn − F tsn V xsn ,tsn 1 − tsn Txsn ,tsn −→ F x − F x 0.
3.19
So, putting s sn and t tsn in 3.18 and letting n → ∞, we immediately conclude that
μF − γ f p, x − p ≤ l V − Ip, x − p ,
∀p ∈ FixT,
3.20
that is,
μF − γ f lI − V p, x − p ≤ 0,
∀p ∈ FixT.
3.21
Upon replacing the p in the last inequality with x αq − x
∈ FixT, where α ∈ 0, 1 and
q ∈ FixT, we get
μF − γ f lI − V x α q − x , x − q ≤ 0.
3.22
Letting α → 0, we obtain the VI
x − q ≤ 0,
μF − γ f lI − V x,
∀q ∈ FixT.
3.23
We immediately see that x satisfies the VI 3.2 and therefore we must have x z∞ since z∞
is the unique solution of 3.2.
Fixed Point Theory and Applications
11
Remark 3.2. i If ts os i.e., l 0, then the above argument shows that the net {xs,ts }
actually converges in norm to the unique solution of the variational inequality of finding
x∞ ∈ FixT such that
μF − γ f x∞ , p − x∞ ≥ 0,
∀p ∈ FixT,
3.24
which is also the unique fixed-point of the contraction PFixT I − μF γ f, x∞ PFixT I −
μF γ fx∞ . In particular, if μ 1, F I, γ τ 1 and f is a ρ-contractive self-mapping on
C, then this is Theorem 3.2 in Xu 23.
ii The net {xs,t }s,t∈0,1 does not converge, in general, as s, t → 0, 0 jointly, to the
unique solution x∞ ∈ S of the VI b in Section 1. As a matter of fact, if {xs,t }s,t∈0,1 converges
to x∞ jointly as s, t → 0, 0, then by 3.24 we would have the relation and the VI b
x∞ PS I − μF γ f x∞ PFixT I − μF γ f x∞
3.25
for all ρ-contraction f. In particular, if μ 1, F I and γ τ 1, then x∞ PS fx∞ PFixT fx∞ for all ρ-contraction f. This implies that S FixT which is not true, in general.
iii Consider the case of l > 0. If x∞ , the unique solution of 3.24, belongs to S, then,
/ S, the following example shows that there are, in general, no links
clearly, x∞ z∞ . If x∞ ∈
among z∞ , S and x∞ . Take
C 0, 1,
μ 1,
fx x
,
2
F I,
γ τ 1,
V x 1 − x,
T I,
l 1,
3.26
then FixT 0, 1. Moreover, the unique solution x∞ of the variational inequality of finding
x∞ ∈ 0, 1 such that
μF − γ f x∞ , z − x∞ ≥ 0, ∀z ∈ 0, 1,
i.e., I − f x∞ , z − x∞ ≥ 0, ∀z ∈ 0, 1
3.27
is x∞ 0; the unique solution z∞ of the variational inequality of finding z∞ ∈ 0, 1 such that
μF − γ f lI − V z∞ , z − z∞ ≥ 0, ∀z ∈ 0, 1,
i.e.,
I − f I − V z∞ , z − z∞ ≥ 0, ∀z ∈ 0, 1
3.28
is z∞ 2/5, and the set S of solutions to the variational inequality of finding x ∈ 0, 1 such
that
I − V x, z − x ≥ 0,
is the singleton {1/2}.
∀z ∈ 0, 1,
3.29
12
Fixed Point Theory and Applications
Remark 3.3. Compared with Theorem 2.1 of Cianciaruso et al. 24, our Theorem 3.1 improves
and extends their Theorem 2.1 24 in the following aspects.
i The self contraction f : C → C in 24, Theorem 2.1 is extended to the case of
possibly nonself contraction f : C → H on a nonempty closed convex subset C
of H.
ii The convex combination of self contraction f and nonexpansive mapping Wts in
the implicit scheme 2.1 of Theorem 2.1 24 is extended to the linear combination
of possibly nonself contraction f and hybrid steepest descent method involving
Wt s .
iii In order to guarantee that the net {xs,ts } generated by the implicit scheme still lies
in C, the implicit scheme 2.1 in 24, Theorem 2.1 is extended to develop our new
implicit scheme 3.1 by virtue of the projection method.
iv The new technique of argument is applied to deriving our Theorem 3.1. For
instance, the characteristic properties Lemma 2.4 of the metric projection play
a key role in proving the strong convergence of the net {xs,ts }s∈0,1 in our
Theorem 3.1.
v If we put μ 1, F I, and γ τ 1, and let f be a contractive self-mapping on C
with coefficient ρ ∈ 0, 1, then our Theorem 3.1 reduces to Theorem 2.1 24. Thus,
our Theorem 3.1 covers Theorem 2.1 24 as a special case.
4. The Case of l ∞
In this section we examine the convergence of the net {xs,ts }s∈0,1 along the curve where
ts /s → ∞, as s → 0. We will prove that the net converges strongly to a point x∞ ∈ S which
is the unique solution of the VI b in Section 1.
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that
F : C → H is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T :
C → C are nonexpansive mappings with FixT / ∅, and f : C → H is a ρ-contraction with
coefficient ρ ∈ 0, 1. Assume
there
holds
the
condition
(A2) in Section 1. Let 0 < μ < 2η/κ2 and
0 < γ ≤ τ, where τ 1 −
{xs,ts }s∈0,1 defined by
1 − μ2η − μκ2 . Let ts ts satisfy lims → 0 ts /s ∞, then the net
xs,ts PC sγ fxs,ts I − sμF Wts xs,ts ,
4.1
where Wts ts V 1 − ts T, strongly converges to x∞ ∈ S which is the unique solution of the VI (b).
Proof. The proof is divided into three steps, the first of which is to prove the boundedness of
{xs,ts }s∈0,1 . Indeed, let z ∈ S. By condition A2, there exists ps ∈ FixWs such that ps → z
as s → 0. Now, set
ys,ts sγ fxs,ts I − sμF Wts xs,ts ,
4.2
Fixed Point Theory and Applications
13
where Wts ts V 1 − ts T, then from 4.1 we obtain that xs,ts PC ys,ts and
ys,ts − pts sγ fxs,ts I − sμF Wts xs,ts − pts
sγ fxs,ts − f pts s γ f − μF pts I − sμF Wts xs,ts − I − sμF Wts pts .
4.3
Since PC is the metric projection from H onto C, utilizing Lemma 2.1, we have
PC ys,ts − ys,ts , PC ys,ts − pts ≤ 0.
4.4
Thus, utilizing Lemma 2.5, we get
xs,ts − pts 2 PC ys,ts − ys,ts , PC ys,ts − pts ys,ts − pts , xs,ts − pts
≤ ys,ts − pts , xs,ts − pts
sγ fxs,ts − f pts , xs,ts − pts s γ f − μF pts , xs,ts − pts
I − sμF Wts xs,ts − I − sμF Wts pts , xs,ts − pts
≤ sγ fxs,ts − f pts xs,ts − pts s γ f − μF pts , xs,ts − pts
I − sμF Wts xs,ts − I − sμF Wts pts xs,ts − pts 2
2
≤ sγ ρxs,ts − pts s γ f − μF pts , xs,ts − pts 1 − sτxs,ts − pts 2
1 − s τ − γ ρ xs,ts − pts s γ f − μF pts , xs,ts − pts .
4.5
It follows that
xs,t − pt 2 ≤
s
s
1 γ f − μF pts , xs,ts − pts .
τ − γρ
4.6
This implies immediately that
xs,t − pt ≤
s
s
1 γ f − μF pt .
s
τ − γρ
4.7
From 4.7, the boundedness of {xs,ts }s∈0,1 follows since {ps } is bounded.
The second step is to prove that the set of weak cluster points of {xs,ts }s∈0,1 , ωw xs,ts ,
is a subset of S; moreover, ωw xs,ts ωS xs,ts . First observe that the boundedness of {xs,ts },
3.15, and Lemma 2.2 imply that ωw xs,ts ⊂ FixT.
Now, let w ∈ ωw xs,ts and assume that xn : xsn ,tsn w, where sn → 0. For
convenience, we write tn tsn for all n; thus, tn /sn → ∞ as n → ∞. Now, take a fixed
x ∈ FixT arbitrarily and set
yn sn γ fxn I − sn μF Wtn xn ,
4.8
14
Fixed Point Theory and Applications
where Wtn tn V 1 − tn T, then from 4.1 we obtain that xn PC yn and
− μFWtn x I − sn μF Wtn xn
yn − x sn γ fxn − fx
s γ fx
− I − sn μF Wtn x tn V − Ix.
4.9
Since PC is the metric projection from H onto C, utilizing Lemma 2.1, we have
PC yn − yn , PC yn − x
≤ 0.
4.10
Thus, utilizing Lemma 2.5, we obtain that for a constant M ≥ supn {
γ f −μFWtn x
x
n − x
},
xn − x
2 PC yn − yn , PC yn − x
yn − x,
xn − x
xn − x
≤ yn − x,
sn γ fxn − fx,
sn γ f − μFWtn x,
xn − x
xn − x
I − sn μF Wtn xn − I − sn μF Wtn x,
xn − x tn V − Ix,
xn − x
≤ sn γ fxn − fx
x
sn γ f − μFWtn x
x
n − x
n − x
tn V − Ix,
I − sn μF Wtn xn − I − sn μF Wtn x
x
xn − x
n − x
≤ sn γ ρ
xn − x
2 sn M 1 − sn τ
xn − x
2 tn V − Ix,
xn − x
1 − sn τ − γ ρ xn − x
2 tn V − Ix,
sn M.
xn − x
4.11
It follows that
xn − x
≤
I − V x,
sn M
−→ 0,
tn
4.12
as sn /tn → 0. But xn w, and we derive
w − x
≤ 0,
I − V x,
∀x ∈ FixT.
4.13
Upon replacing the x in 4.13 with w αx − w ∈ FixT, where α ∈ 0, 1 and x ∈ FixT,
we get
≤ 0.
I − V w αx − w, w − x
4.14
Letting α → 0, we obtain the VI
≤ 0,
I − V w, w − x
Therefore, w ∈ S.
∀x ∈ FixT.
4.15
Fixed Point Theory and Applications
15
Next using condition A2 again, we have a sequence ptn ∈ FixWtn such that ptn →
w. Then in relation 4.6, we replace z and pts with w and ptn , respectively, to derive
xn − pt 2 ≤
n
1 γ f − μF ptn , xn − ptn .
τ − γρ
4.16
Now since γ f − μFptn → γ f − μFw and xn − ptn 0, taking the limit in 4.16, we
immediately get xn → w. Hence w ∈ ωS xs,ts .
The third and final step is to prove that the net {xs,ts } converges in norm to x∞ PS I − μf γ fx∞ . It suffices to prove that each norm limit point w ∈ ωS xs,ts solves the VI
b in Section 1. We still use the same subsequence {xn } of the net {xs,ts } such that xn → w
as shown in the second step. On the other hand, for every p ∈ S, by condition A2, we have,
for each n, ptn ∈ FixWtn , such that ptn → p as n → ∞. Observe that
1
1
PC yn − yn − I − Wtn xn μFxn − FWtn xn ,
μF − γ f xn sn
sn
4.17
where yn sn γ fxn I − sn μFWtn xn and xn PC yn . Utilizing Lemmas 2.1 and 2.5, we
deduce from the monotonicity of I − Wtn that
μF − γ f xn , xn − ptn
1
1
PC yn − yn , PC yn − ptn −
I − Wtn xn , xn − ptn μ Fxn − FWtn xn , xn − ptn
sn
sn
1
1
PC yn − yn , PC yn − ptn −
I − Wtn xn − I − Wtn ptn , xn − ptn
sn
sn
μ Fxn − FWtn xn , xn − ptn
≤ μ Fxn − FWtn xn , xn − ptn .
4.18
Note that Fxn − FWtn xn Fxn − Ftn V xn 1 − tn Txn → Fw − Fw 0 as n → ∞. Passing
to the limit as n → ∞ in the last inequality, we conclude that
μF − γ f w, w − p ≤ 0,
∀p ∈ S.
4.19
This implies that w satisfies the VI b in Section 1. Hence w x∞ , as required.
Remark 4.2. Compared with Theorem 3.1 of Cianciaruso et al. 24, our Theorem 4.1 improves
and extends their Theorem 3.1 24 in the following aspects.
i The self contraction f : C → C in 24, Theorem 3.1 is extended to the case of
possibly nonself contraction f : C → H on a nonempty closed convex subset C
of H.
16
Fixed Point Theory and Applications
ii The convex combination of self contraction f and nonexpansive mapping Wts in
the implicit scheme 3.1 of Theorem 3.1 24 is extended to the linear combination
of possibly nonself contraction f and hybrid steepest descent method involving
Wt s .
iii In order to guarantee that the net {xs,ts } generated by the implicit scheme still lies
in C, the implicit scheme 3.1 in 24, Theorem 3.1 is extended to develop our new
implicit scheme 4.1 by virtue of the projection method.
iv The new technique of argument is applied to deriving our Theorem 4.1. For
instance, the characteristic properties Lemma 2.4 of the metric projection play
a key role in proving the strong convergence of the net {xs,ts }s∈0,1 in our
Theorem 4.1.
v If we put μ 1, F I, and γ τ 1, and let f be a contractive self-mapping on C
with coefficient ρ ∈ 0, 1, then our Theorem 4.1 reduces to Theorem 3.1 24. Thus,
our Theorem 4.1 covers Theorem 3.1 24 as a special case.
Acknowledgments
This research was partially supported by the National Science Foundation of China
11071169, Innovation Program of Shanghai Municipal Education Commission 09ZZ133,
Leading Academic Discipline Project of Shanghai Normal University DZL707, Science and
Technology Commission of Shanghai Municipality Grant 075105118, and Shanghai Leading
Academic Discipline Project S30405. This research was partially supported by Grant NSC
99-2115-M-110-004-MY3. This research was partially supported by the Grant NSC 99-2115M-037-001.
References
1 A. Cabot, “Proximal point algorithm controlled by a slowly vanishing term: applications to
hierarchical minimization,” SIAM Journal on Optimization, vol. 15, no. 2, pp. 555–572, 2005.
2 Z.-Q. Luo, J.-S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge
University Press, Cambridge, 1996.
3 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.
4 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical
Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
5 H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational
inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
6 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the
intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in
Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8,
pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
7 Y. Yao and Y.-C. Liou, “Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical
fixed point problems,” Inverse Problems, vol. 24, no. 1, p. 8, 2008.
8 L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “The viscosity approximation method for asymptotically
nonexpansive mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol.
69, no. 4, pp. 1402–1412, 2008.
9 L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points
of a finite family of nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol.
64, no. 11, pp. 2507–2515, 2006.
Fixed Point Theory and Applications
17
10 L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive
mappings and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 5, pp.
1464–1473, 2007.
11 L. C. Zeng, N. C. Wong, and J. C. Yao, “Convergence analysis of modified hybrid steepest-descent
methods with variable parameters for variational inequalities,” Journal of Optimization Theory and
Applications, vol. 132, no. 1, pp. 51–69, 2007.
12 L.-C. Ceng and J.-C. Yao, “Relaxed viscosity approximation methods for fixed point problems and
variational inequality problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10,
pp. 3299–3309, 2008.
13 L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Mann-type steepest-descent and modified hybrid steepestdescent methods for variational inequalities in Banach spaces,” Numerical Functional Analysis and
Optimization, vol. 29, no. 9-10, pp. 987–1033, 2008.
14 L. C. Ceng, S. Huang, and A. Petruşel, “Weak convergence theorem by a modified extragradient
method for nonexpansive mappings and monotone mappings,” Taiwanese Journal of Mathematics, vol.
13, no. 1, pp. 225–238, 2009.
15 L. C. Ceng, A. Petruşel, C. Lee, and M. M. Wong, “Two extragradient approximation methods for
variational inequalities and fixed point problems of strict pseudo-contractions,” Taiwanese Journal of
Mathematics, vol. 13, no. 2, pp. 607–632, 2009.
16 L.-C. Ceng, S. Huang, and Y.-C. Liou, “Hybrid proximal point algorithms for solving constrained
minimization problems in Banach spaces,” Taiwanese Journal of Mathematics, vol. 13, no. 2, pp. 805–
820, 2009.
17 L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general iterative method with strongly positive operators
for general variational inequalities,” Computers & Mathematics with Applications, vol. 59, no. 4, pp.
1441–1452, 2010.
18 L. C. Ceng and N. C. Wong, “Viscosity approximation methods for equilibrium problems and fixed
point problems of nonlinear semigroups,” Taiwanese Journal of Mathematics, vol. 13, no. 5, pp. 1497–
1513, 2009.
19 L.-C. Ceng and S. Huang, “Modified extragradient methods for strict pseudo-contractions and
monotone mappings,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1197–1211, 2009.
20 L. C. Ceng, D. R. Sahu, and J. C. Yao, “Implicit iterative algorithms for asymptotically nonexpansive
mappings in the intermediate sense and Lipschitz-continuous monotone mappings,” Journal of
Computational and Applied Mathematics, vol. 233, no. 11, pp. 2902–2915, 2010.
21 A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point
problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006.
22 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical
Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
23 H.-K. Xu, “Viscosity method for hierarchical fixed point approach to variational inequalities,”
Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463–478, 2010.
24 F. Cianciaruso, V. Colao, L. Muglia, and H.-K. Xu, “On an implicit hierarchical fixed point approach
to variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 80, no. 1, pp. 117–124,
2009.
25 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, 1990.