Volume 2009, Article ID 520301, 16 pagesdoi:10.1155/2009/520301 Research Article A New Approximation Method for Solving Variational Inequalities and Fixed Points of Nonexpansive Mappings
Trang 1Volume 2009, Article ID 520301, 16 pages
doi:10.1155/2009/520301
Research Article
A New Approximation Method for Solving
Variational Inequalities and Fixed Points of
Nonexpansive Mappings
Chakkrid Klin-eam and Suthep Suantai
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
Correspondence should be addressed to Suthep Suantai,scmti005@chiangmai.ac.th
Received 3 June 2009; Revised 31 August 2009; Accepted 1 November 2009
Recommended by Vy Khoi Le
A new approximation method for solving variational inequalities and fixed points of nonexpansive mappings is introduced and studied We prove strong convergence theorem of the new iterative scheme to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for the inverse-strongly monotone mapping which solves some variational inequalities Moreover, we apply our main result to obtain strong convergence
to a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping in a Hilbert space
Copyrightq 2009 C Klin-eam and S Suantai 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
1 Introduction
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems Convex minimization problems have a great impact and influence
in the development of almost all branches of pure and applied sciences A typical problem is
to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping
on a real Hilbert spaceH:
θx 12Ax, x −x, y ∀x ∈ FS, 1.1
whereA is a linear bounded operator, FS is the fixed point set of a nonexpansive mapping
S, and y is a given point in H.
LetH be a real Hilbert space and C be a nonempty closed convex subset of H.
Trang 2Recall that a mappingS : C → C is called nonexpansive if Sx − Sy ≤ x − y for all x, y ∈ C.
The set of all fixed points ofS is denoted by FS, that is, FS {x ∈ C : x Sx} A
linear bounded operatorA is strongly positive if there is a constant γ > 0 with the property
Ax, x ≥ γx2 for allx ∈ H A self-mapping f : C → C is a contraction on C if there is a
constantα ∈ 0, 1 such that fx − fy ≤ αx − y for all x, y ∈ C We use Π C to denote the collection of all contractions onC Note that each f ∈ Π Chas a unique fixed point inC.
A mappingB of C into H is called monotone if Bx − By, x − y ≥ 0 for all x, y ∈ C The
variational inequality problem is to findx ∈ C such that
Bx, y − x≥ 0 ∀y ∈ C. 1.2
The set of solutions of the variational inequality is denoted byV IC, B A mapping B
ofC to H is called inverse-strongly monotone if there exists a positive real number β such that
x − y, Bx − By≥ βBx − By2 ∀x, y ∈ C. 1.3
For such a case, B is β-inverse-strongly monotone If B is a β-inverse-strongly monotone
mapping ofC to H, then it is obvious that B is 1/β-Lipschitz continuous.
In 2000, Moudafi1 introduced the viscosity approximation method for nonexpansive mapping and proved that if H is a real Hilbert space, the sequence {x n} defined by the iterative method below, with the initial guessx0∈ C is chosen arbitrarily:
where{α n } ⊂ 0, 1 satisfies certain conditions, converges strongly to a fixed point of S say
x ∈ C which is the unique solution of the following variational inequality:
I − fx, x − x≥ 0 ∀x ∈ FS. 1.5
In 2004, Xu2 extended the results of Moudafi 1 to a Banach space In 2006, Marino and Xu3 introduced a general iterative method for nonexpansive mapping They defined the sequence{x n} by the following algorithm:
x0∈ C, x n1 α n γfx n I − α n ASx n , n ≥ 0, 1.6
where{α n } ⊂ 0, 1 and A is a strongly positive linear bounded operator, and they proved
that ifC H and the sequence {α n } satisfies appropriate conditions, then the sequence {x n} generated by1.6 converges strongly to a fixed point of S say x ∈ H which is the unique
solution of the following variational inequality:
A − γfx, x − x≥ 0 ∀x ∈ FS, 1.7 which is the optimality condition for minimization problem minx∈C 1/2Ax, x − hx,
whereh is a potential function for γf i.e., h x γf for all x ∈ H.
Trang 3For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, Iiduka and Takahashi4 introduced following iterative process:
x0∈ C, x n1 α n u 1 − α n SP C x n − λ n Bx n , n ≥ 0, 1.8
where P C is the projection ofH onto C, u ∈ C, {α n } ⊂ 0, 1 and {λ n } ⊂ a, b for some
a, b with 0 < a < b < 2β They proved that under certain appropriate conditions imposed
on {α n } and {λ n }, the sequence {x n} generated by 1.8 converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mappingsay x ∈ C which solves
the variational inequality
In 2007, Chen et al.5 introduced the following iterative process: x0∈ C,
x n1 α n fx n 1 − α n SP C x n − λ n Bx n , n ≥ 0, 1.10
where {α n } ⊂ 0, 1 and {λ n } ⊂ a, b for some a, b with 0 < a < b < 2β They proved
that under certain appropriate conditions imposed on {α n } and {λ n }, the sequence {x n} generated by 1.10 converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mappingsay x ∈ C which solves the variational inequality
I − fx, x − x≥ 0 ∀x ∈ FS ∩ V IC, B. 1.11
In this paper, we modify the iterative methods 1.6 and 1.10 by purposing the following general iterative method:
x0 ∈ C, x n1 P C
α n γfx n I − α n ASP C x n − λ n Bx n, n ≥ 0, 1.12
where P C is the projection of H onto C, f is a contraction, A is a strongly positive linear
bounded operator,B is a β-inverse strongly monotone mapping, {α n } ⊂ 0, 1 and {λ n} ⊂
a, b for some a, b with 0 < a < b < 2β.
We note that whenA I and γ 1, the iterative scheme 1.12 reduces to the iterative scheme1.10
The purpose of this paper is twofold First, we show that under some control conditions the sequence{x n} defined by 1.12 strongly converges to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for the inverse-strongly monotone mapping B in a real Hilbert space which
solves some variational inequalities Secondly, by using the first results, we obtain a strong convergence theorem for a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping Moreover, we consider the problem of finding a common element of the set of fixed points of nonexpansive mapping and the set of zeros of inverse-strongly monotone mapping
Trang 42 Preliminaries
LetH be real Hilbert space with inner product ·, ·, C a nonempty closed convex subset of
H Recall that the metric nearest point projection P Cfrom a real Hilbert spaceH to a closed
convex subsetC of H is defined as follows: given x ∈ H, P C x is the only point in C with the
propertyx − P C x inf{x − y : y ∈ C} In what followsLemma 2.1can be found in any standard functional analysis book
Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H Given x ∈ H and y ∈ C, then
i y P C x if and only if the inequality x − y, y − z ≥ 0 for all z ∈ C,
ii P C is nonexpansive,
iii x − y, P C x − P C y ≥ P C x − P C y2 for all x, y ∈ H,
iv x − P C x, P C x − y ≥ 0 for all x ∈ H and y ∈ C.
UsingLemma 2.1, one can show that the variational inequality1.2 is equivalent to a fixed point problem
Lemma 2.2 The point u ∈ C is a solution of the variational inequality 1.2 if and only if u satisfies
the relation u P C u − λBu for all λ > 0.
We writex n x to indicate that the sequence {x n } converges weakly to x and write
x n → x to indicate that {x n } converges strongly to x It is well known that H satisfies the
Opial’s condition6, that is, for any sequence {x n } with x n x, the inequality
lim inf
n → ∞ x n − x < lim inf
holds for everyy ∈ H with x / y.
A set-valued mappingT : H → 2 H is called monotone if for all x, y ∈ H, u ∈ Tx, and
v ∈ Ty imply x − y, u − v ≥ 0 A monotone mapping T : H → 2 H is maximal if the graph
GT of T is not properly contained in the graph of any other monotone mapping It is known
that a monotone mappingT is maximal if and only if for x, u ∈ H × H, x − y, u − v ≥ 0 for
everyy, v ∈ GT implies u ∈ Tx Let B be an inverse-strongly monotone mapping of C to
H and let N C v be normal cone to C at v ∈ C, that is, N C v {w ∈ H : v −u, w ≥ 0, ∀u ∈ C},
and define
Tv
⎧
⎨
⎩
Bv N C v, if v ∈ C,
ThenT is a maximal monotone and 0 ∈ Tv if and only if v ∈ V IC, B 7 In the sequel, the following lemmas are needed to prove our main results
Lemma 2.3 see 8 Assume {a n } is a sequence of nonnegative real numbers such that a n1 ≤
1 − γ n a n δ n , n ≥ 0, where {γ n } ⊂ 0, 1 and {δ n } is a sequence in R such that
n1 γ n ∞,
ii lim supn → ∞ δ n /γ n ≤ 0 or ∞
n1 |δ n | < ∞.
Then lim n → ∞ a n 0.
Trang 5Lemma 2.4 see 9 Let C be a closed convex subset of a real Hilbert space H and let T : C → C
be a nonexpansive mapping such that FT / ∅ If a sequence {x n } in C is such that x n z and
x n − Tx n → 0, then z Tz.
Lemma 2.5 see 3 Assume A is a strongly positive linear bounded operator on a Hilbert space H
with coefficient γ > 0 and 0 < ρ ≤ A−1, then I − ρA ≤ 1 − ργ.
3 Main Results
In this section, we prove a strong convergence theorem for nonexpansive mapping and inverse strongly monotone mapping
Theorem 3.1 Let H be a real Hilbert space, let C be a closed convex subset of H, and let B : C → H
be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator
of H into itself with coefficient γ > 0 such that A 1 and let f : C → C be a contraction with coefficient α 0 < α < 1 Assume that 0 < γ < γ/α Let S be a nonexpansive mapping of C into itself such that Ω FS ∩ V IC, B / ∅ Suppose {x n } is the sequence generated by the following
algorithm: x0∈ C,
x n1 P C
α n γfx n I − α n ASP C x n − λ n Bx n 3.1
for all n 0, 1, 2, , where {α n } ⊂ 0, 1 and {λ n } ⊂ 0, 2β If {α n } and {λ n } are chosen so that
λ n ∈ a, b for some a, b with 0 < a < b < 2β,
C1: lim
n1
α n ∞,
C3: ∞
n1
|α n1 − α n | < ∞, C4: ∞
n1
|λ n1 − λ n | < ∞,
3.2
then {x n } converges strongly to q ∈ Ω, where q PΩγf I − Aq which solves the following
variational inequality:
γf − Aq, p − q≤ 0 ∀p ∈ Ω. 3.3
Proof First, we show the mapping I − λ n B is nonexpansive Indeed, since B is a β-strongly
monotone mapping and 0< λ n < 2β, we have that for all x, y ∈ C,
I − λ n Bx − I − λ n By2x − y − λ n
Bx − By2
x − y2− 2λ n
x − y, Bx − By λ2
n Bx − By2
≤x − y2 λ nλ n − 2βBx − By2
≤x − y2,
3.4
Trang 6which implies that the mappingI − λ n B is nonexpansive Next, we show that the sequence
{x n } is bounded Put y n P C x n − λ n x n for all n ≥ 0 Let u ∈ Ω, we have
y n − u P C x n − λ n Bx n − P C u − λ n Bu
≤ x n − λ n Bx n − u − λ n Bu
≤ I − λ n Bx n − I − λ n Bu
≤ x n − u.
3.5
Then, we have
x n1 − u P C
α n γfx n I − α n ASy n
− P C u
≤α n
γfx n − Au I − α n ASy n − u
≤ α n γfx n − Au 1 − α n γ y n − u
≤ α n γfx n − γfu α n γfu − Au 1 − α n γ y n − u
≤ αγα n x n − u α n γfu − Au 1 − α n γx n − u
1−γ − γαα nx n − u α n γfu − Au
1−γ − γαα nx n − u γ − γαα n γfu − Au
γ − γα
≤ max
x n − u, γfu − Au
γ − γα .
3.6
It follows from induction that
x n − u ≤ max
x0− u, γfu − Au
γ − γα , n ≥ 0. 3.7
Therefore, {x n } is bounded, so are {y n },{Sy n },{Bx n }, and {fx n } Since I − λ n B is
nonexpansive andy n P C x n − λ n Bx n, we also have
y n1 − y n ≤ x n1 − λ n1 Bx n1 − x n − λ n Bx n
≤ x n1 − λ n1 Bx n1 − x n − λ n1 Bx n |λ n − λ n1 |Bx n
≤ I − λ n1 Bx n1 − I − λ n1 Bx n |λ n − λ n1 |Bx n
≤ x n1 − x n |λ n − λ n1 |Bx n .
3.8
Trang 7So we obtain
x n1 − x n P C
α n γfx n I − α n ASy n
−P Cα n−1 γfx n−1 I − α n−1 ASy n−1
≤I − α n ASy n − Sy n−1
− α n − α n−1 ASy n−1
γα n
fx n − fx n−1 γα n − α n−1 fx n−1
≤1− α n γ y n − y n−1 |α n − α n−1|ASyn−1
γαα n x n − x n−1 γ|α n − α n−1|fxn−1
≤1− α n γx n − x n−1 |λ n−1 − λ n |Bx n−1 |α n − α n−1|ASyn−1
γαα n x n − x n−1 γ|α n − α n−1|fxn−1
≤1− α n γx n − x n−1 |λ n−1 − λ n |Bx n−1 |α n − α n−1|ASyn−1
γαα n x n − x n−1 γ|α n − α n−1|fxn−1
1−γ − γαα nx n − x n−1 L|λ n−1 − λ n | M|α n − α n−1 |,
3.9
where L sup{Bx n−1 : n ∈ N}, M max{sup n∈N ASy n−1 , sup n∈N γfx n−1} Since
∞
n1 |α n − α n−1 | < ∞ and ∞
n1 |λ n−1 − λ n | < ∞, byLemma 2.3, we havex n1 − x n → 0 For
u ∈ Ω and u P C u − λ n Bu, we have
x n1 − u2P C
α n γfx n I − α n ASy n
− P C u2
≤α n
γfx n − Au I − α n ASy n − u2
≤α n γfx n − Au I − α n A Sy n − u2
≤α n γfx n − Au 1 − α n γ y n − u2
≤ α n γfx n − Au21− α n γ y n − u2
2α n1− α n γ γfx n − Auy n − u
≤ α n γfx n − Au21− α n γI − λ n Bx n − I − λ n Bu2
2α n1− α n γ γfx n − Auy n − u
≤ α n γfx n − Au21− α n γx n − u2 λ n
λ n − 2βBx n − Bu2
2α n1− α n γ γfx n − Auy n − u
≤ α n γfx n − Au2
x n − u21− α n γab − 2βBx n − Bu2
2α n
1− α n γ γfx n − Auy n − u.
3.10
Trang 8So, we obtain
−1− α n γab − 2βBx n − Bu2
≤ α n γfx n − Au2
x n − u x n1 − ux n − u − x n1 n
≤ α n γfx n − Au2
n x n − x n1 x n − u x n1 − u,
3.11
where n 2α n 1 − α n γγfx n − Auy n − u Since α n → 0 and x n1 − x n → 0, we obtain thatBx n − Bu → 0 as n → ∞ Further, byLemma 2.1iii, we have
y n − u2
P C x n − λ n Bx n − P C u − λ n Bu2
≤x n − λ n Bx n − u − λ n Bu, y n − u
1 2
x n − λ n Bx n − u − λ n Bu2y n − u2
−xn − λ n Bx n − u − λ n Bu −y n − u2
≤ 1 2
x n − u2y n − u2−x n − y n
− λ n Bx n − Bu2
1 2
x n − u2y n − u2−x n − y n2
1 2
2λ nx n − y n , Bx n − Bu− λ2
n Bx n − Bu2
.
3.12
So, we obtain that
y n − u2
≤ x n − u2−x n − y n2 2λ n
x n − y n , Bx n − Bu− λ2
n Bx n − Bu2. 3.13
So, we have
x n1 − u2 P C
α n γfx n I − α n ASy n
− P C u2
≤α n
γfx n − Au I − α n ASy n − u2
≤α n γfx n − Au I − α n A Sy n − u2
≤α n γfx n − Au21− α n γ y n − u2
≤ α n γfx n − Au21− α n γ y n − u2 2α n1− α n γ γfx n − Auy n − u
≤ α n γfx n − Au21− α n γx n − u2−1− α n γ x n − y n2
21− α n γλ nx n − y n , Bx n − Bu−1− α n γλ2
n Bx n − Bu2
2α n1− α n γ γfx n − Auy n − u
Trang 9≤ α n γfx n − Au2
x n − u2−1− α n γ x n − y n2
21− α n γλ n x n − y n , Bx n − Bu −1− α n γλ2
n Bx n − Bu2
2α n
1− α n γ γfx n − Auy n − u,
3.14 which implies
1− α n γ x n − y n2≤ α n γfx n − Au2
x n − u x n1 − ux n − x n1
21− α n γλ nx n − y n , Bx n − Bu−1− α n γλ2
n Bx n − Bu2
2α n
1− α n γ γfx n − Auy n − u.
3.15
Sinceα n → 0, x n1 − x n → 0, and Bx n − Bu → 0, we obtain x n − y n → 0 as n → ∞.
Next, we have
x n1 − Sy n P C
α n γfx n I − α n ASy n
− P CSy n
≤α n γfx n I − α n ASy n − Sy n
Sinceα n → 0 and {fx n }, {ASy n } are bounded, we have x n1 − Sy n → 0 as n → ∞ Since
x n − Sy n ≤ x n − x n1 x n1 − Sy n , 3.17
it implies thatx n − Sy n → 0 as n → ∞ Since
x n − Sx n ≤x n − Sy n Sy n − Sx n
we obtain thatx n − Sx n → 0 as n → ∞ Moreover, from
y n − Sy n ≤ y n − x n x n − Sy n , 3.19
it follows thaty n − Sy n → 0 as n → ∞.
Trang 10Observe thatPΩγf I − A is a contraction Indeed, byLemma 2.5, we have that
I − A ≤ 1 − γ and since 0 < γ < γ/α, we have
PΩ
γf I − Ax − PΩ
γf I − Ay ≤ γf I − Ax − γf I − Ay
≤ γfx − fy I − Ax − y
≤ γαx − y 1 − γx − y
1−γ − γα x − y.
3.20
Then Banach’s contraction mapping principle guarantees thatPΩγf I − A has a unique
fixed point, say q ∈ H That is, q PΩγf I − Aq ByLemma 2.1i, we obtain that
γf − Aq, p − q ≤ 0 for all p ∈ Ω Choose a subsequence {y n k } of {y n} such that
lim sup
n → ∞
γf − Aq, Sy n − q lim
k → ∞
γf − Aq, Sy n k − q. 3.21
As{y n k } is bounded, there exists a subsequence {y n kj } of {y n k} which converges weakly to
p We may assume without loss of generality that y n k p Since y n − Sy n → 0, we obtain
Sy n k p Since x n − Sx n → 0, x n − y n → 0 and byLemma 2.4, we havep ∈ FS Next,
we show thatp ∈ V IC, B Let
Tv
⎧
⎨
⎩
Bv N C v, if v ∈ C,
whereN C v is normal cone to C at v ∈ C, that is, N C v {w ∈ H : v − u, w ≥ 0, ∀u ∈ C}.
ThenT is a maximal monotone Let v, w ∈ GT Since w − Bv ∈ N C v and y n ∈ C, we have
v − y n , w − Bv ≥ 0 On the other hand, byLemma 2.1iv and from y n P C x n − λ n Bx n, we have
v − y n , y n − x n − λ n Bx n≥ 0, 3.23 and hencev − y n , y n − x n /λ n Bx n ≥ 0 Therefore, we have
v − y n k , w≥v − y n k , Bv
≥v − y n k , Bv−
v − y n k , y n k − x n k
λ n Bx n k
v − y n k , Bv − Bx n k−y n k − x n k
λ n
v − y n k , Bv − By n kv − y n k , By n k − Bx n k−
v − y n k , y n k − x n k
λ n
≥v − y n k , By n k − Bx n k
−
v − y n k , y n k − x n k
λ n
.
3.24
... Trang 9≤ α n γfx n − Au2
x n... γfx n − Auy n − u.
3.10
Trang 8So, we obtain
−1−... − λ n1 |Bx n .
3.8
Trang 7So we obtain
x n1