Volume 2008, Article ID 583082, 19 pagesdoi:10.1155/2008/583082 Research Article Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpans
Trang 1Volume 2008, Article ID 583082, 19 pages
doi:10.1155/2008/583082
Research Article
Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces
Somyot Plubtieng and Kasamsuk Ungchittrakool
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Somyot Plubtieng,somyotp@nu.ac.th
Received 2 July 2008; Accepted 23 December 2008
Recommended by Hichem Ben-El-Mechaiekh
The convex feasibility problemCFP of finding a point in the nonempty intersectionN
i1 C iis
considered, where N 1 is an integer and the C i’s are assumed to be convex closed subsets of a
Banach space E By using hybrid iterative methods, we prove theorems on the strong convergence
to a common fixed point for a finite family of relatively nonexpansive mappings Then, we apply our results for solving convex feasibility problems in Banach spaces
Copyrightq 2008 S Plubtieng and K Ungchittrakool 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
i1
space E This problem is a frequently appearing problem in diverse areas of mathematical
Hilbert spaces which captures applications in various disciplines such as image restoration
tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint
Trang 2So projection methods have dominated in the iterative approaches toCFP in Hilbert spaces;
the convex feasibility problem by convex combinations of sunny nonexpansive retractions
is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach
onto C is not generally nonexpansive Our purpose in the present paper is to obtain an
analogous result for a finite family of relatively nonexpansive mappings in Banach spaces
convergence theorems to approximate a fixed point of a single relatively nonexpansive
y n J−1
α n Jx n1− α n
JTx n
,
H n z ∈ C : φz, y n
φz, x n
,
W n z ∈ C :x n − z, Jx − Jx n
0,
x n1 ΠH n ∩W n x, n 0, 1, 2, ,
1.2
by
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1
β1n Jx n β2n JTx n β n3JSx n
,
H nz ∈ C : φz, y n
φz, x n
,
W nz ∈ C :x n − z, Jx − Jx n
0,
x n1 P H n ∩W n x, n 0, 1, 2, ,
1.3
F : FS ∩ FT.
We note that the block iterative method is a method which often used by many authors
a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming In this paper, we introduce the following iterative
Trang 3process by using the shrinking method proposed, whose studied by Takahashi et al.26,
F :N i1 FT i / ∅ Define {x n} in the two following ways:
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1 β1n Jx n N
i1
β i1 n JT i x n
,
C n1z ∈ C n : φ
z, y n
φz, x n
,
x n1 ΠC n1 x0, n 0, 1, 2, ,
1.4
and
x0∈ C,
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1 β1n Jx n N
i1
β i1 n JT i x n
,
H nz ∈ C : φz, y n
φz, x n
,
W nz ∈ C :x n − z, Jx0− Jx n
0,
x n1 ΠH n ∩W n x0, n 0, 1, 2, ,
1.5
where{α n }, {β i n } ⊂ 0, 1, N1 i1 β i n 1 satisfy some appropriate conditions
spaces Moreover, we apply our results to the problem of finding a common zero of a finite family of maximal monotone operators and equilibrium problems
Throughout the paper, we will use the notations:
i → for strong convergence and for weak convergence;
2 Preliminaries
Jx x∗∈ E∗:
x, x∗
x2x∗2
2.1
Trang 4A Banach space E is said to be strictly convex if x y/2 < 1 for all x, y ∈ E with
U {x ∈ E : x 1} be the unit sphere of E Then the Banach space E is said to be smooth
provided that
lim
t → 0
x ty − x
the duality mapping J is single valued It is also known that if E is uniformly smooth, then
J is uniformly norm-to-norm continuous on each bounded subset of E Some properties of
defined by
V
x, x∗
x2− 2x, x∗
for all x ∈ E and x∗ ∈ E∗ In other words, V x, x∗ φx, J−1x∗ for all x ∈ E and x∗ ∈ E∗
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive
This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section Some of them are known; others are not hard to derive
Trang 5Lemma 2.1 see 27,30,32 If E is a strictly convex and smooth Banach space, then for x, y ∈ E,
φy, x 0 if and only if x y.
Proof It is su fficient to show that if φy, x 0 then x y From 1, we have x y.
Lemma 2.2 Kamimura and Takahashi 17 Let E be a uniformly convex and smooth Banach
space and let {y n }, {z n } be two sequences of E If φy n , z n → 0 and either {y n } or {z n } is bounded,
then y n − z n → 0.
Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let T be a mapping from C into itself, and let FT be the set of all fixed points
mapping if the following conditions are satisfied:
R1 FT is nonempty;
R2 φu, Tx φu, x for all u ∈ FT and x ∈ C;
R3 FT FT.
Lemma 2.3 Alber 16, Alber and Reich 31, Kamimura and Takahashi 17 Let C be a
nonempty closed convex subset of a smooth Banach space E, let x ∈ E, and let x0∈ C Then, x0 ΠC x
if and only if x0− y, Jx − Jx0
Lemma 2.4 Alber 16, Alber and Reich 31, Kamimura and Takahashi 17 Let E be a
reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let x ∈ E Then φy, Π C x φΠ C x, x φy, x for all y ∈ C.
Lemma 2.5 Let E be a uniformly convex Banach space and let B r 0 {x ∈ E : x r} be a closed
ball of E Then there exists a continuous strictly increasing convex function g : 0, ∞ → 0, ∞ with g0 0 such that
N
i1
ω i x i
2
i1
ω i x i2− ω j ω k g x j − x k , for any j,k ∈ {1,2, ,N}, 2.5
where {x i}N
i1 ⊂ B r 0 and {ω i}N i1 ⊂ 0, 1 with N
i1 ω i 1.
Proof It sufficient to show that
N
i1
ω i x i
2
i1
Trang 6It is obvious that2.6 holds for N 1, 2 see 34 for more details Next, we assume that
N
i1
ω i x i
2
ω N x N
1− ω N N−1
i1
ω i
1− ω N x i
2
ω N x N21− ω N
N−1
i1
ω i
1− ω N x i
2
ω N x N21− ω N N−1
i1
ω i
1− ω N x i2−ω1ω2
1− ω N2g x1− x2
i1
ω i x i2−ω1ω2
1− ω N gx1− x2
i1
ω i x i2− ω1ω2g x1− x2.
2.7 This completes the proof
Lemma 2.6 Let C be a closed convex subset of a smooth Banach space E and let x, y ∈ E Then the
set K : {v ∈ C : φv, y φv, x} is closed and convex.
Proof As a matter of fact, the defining inequality in K is equivalent to the inequality
v, 2Jx − Jy x2− y2. 2.8 This inequality is affine in v and hence the set K is closed and convex
3 Main result
In this section, we prove strong convergence theorems for finding a common fixed point of
a finite family of relatively nonexpansive mappings in Banach spaces by using the hybrid method in mathematical programming
Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty closed convex subset of E Let {T i}N i1 be a finite family of relatively nonexpansive mappings from C into itself such that F :N i1 FT i is nonempty and let x0∈ E For C1 C and x1 ΠC1x0, define a sequence {x n } of C as follows:
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1 β1n Jx n N
i1
β i1 n JT i x n
,
C n1z ∈ C n : φ
z, y n
φz, x n
,
x n1 ΠC x0, n 0, 1, 2, ,
3.1
Trang 7where {α n }, {β n i } ⊂ 0, 1 satisfy the following conditions:
i 0 α n < 1 for all n ∈ N ∪ {0} and lim sup n → ∞ α n < 1,
ii 0 β i n 1 for all i 1, 2, , N 1, N1
i1 β i n 1 for all n ∈ N ∪ {0} If either
a lim infn → ∞ β1n β i1 n > 0 for all i 1, 2, , N or
b limn → ∞ β1n 0 and lim inf n → ∞ β k1 n β l1 n > 0 for all i / j, k, l 1, 2, , N Then the sequence {x n } converges strongly to Π F x0, whereΠF is the generalized projection from E onto F.
Proof We first show by induction that F ⊂ C n for all n ∈ N F ⊂ C1is obvious Suppose that
F ⊂ C k for some k ∈ N Then, we have, for u ∈ F ⊂ C k,
φ
u, y k
φu, J−1
α k Jx k1− α k
Jz k
Vu, α k Jx k1− α k
Jz k
α k V
u, Jx k
1− α k
V
u, Jz k
α k φ
u, x k
1− α k
φ
u, z k
,
φ
u, z k
V u, β1k Jx k N
i1
β i1 k JT i x k
β1k V
u, Jx k
i1
β i1 k V
u, JT i x k
φu, x k
.
3.2
It follow that
φ
u, y k
φu, x k
3.3
x n ΠC n x0, we have
φ
x n , x0
φu, x0
− φu, x n
φu, x0
φ
x n , x0
φu, x0
φ
x n , x0
min
y∈C φ
y, x0
φx n1 , x0
Trang 8Therefore{φx n , x0} is nondecreasing So there exists the limit of φx n , x0 ByLemma 2.4,
we have
φ
x n1 , x n
φx n1 , Π C n x0
φx n1 , x0
− φΠC n x0, x0
φx n1 , x0
− φx n , x0
.
3.7
φ
x n1 , y n
φx n1 , x n
lim
n → ∞ x n1 − y n lim
n → ∞ x n1 − x n 0. 3.9
Since J is uniformly norm-to-norm continuous on bounded sets, we have
lim
n → ∞ Jx n1 − Jy n limn → ∞ Jx n1 − Jx n 0. 3.10
SinceJx n1 − Jy n Jx n1 − α n Jx n − 1 − α n Jz n 1 − α n Jx n1 − Jz n − α n Jx n − Jx n1
Jx n1 − Jz n 11− α
n Jx n1 − Jy n α n Jx n − Jx n1
uniformly norm-to-norm continuous on bounded sets, it follows that
lim
n → ∞ x n1 − z n limn → ∞ J−1
Jx n1
− J−1
Fromx n − z n x n − x n1 x n1 − z n, we have limn → ∞ x n − z n 0
Trang 9Next, we show thatx n − T i x n → 0 for all i 1, 2, , N Since {x n} is bounded and
φp, T i x n φp, x n for all i 1, 2, , N, where p ∈ F We also obtain that {Jx n } and {JT i x n}
are bounded for all i 1, 2, , N Then there exists r > 0 such that {Jx n }, {JT i x n } ⊂ B r0 for
φ
p, z n
p, β n1Jx n N
i1
β i1 n JT i x n
β1n Jx n N
i1
β i1 n JT i x n
2
p2− 2β1n
p, Jx n
i1
β i1 n
p, JT i x n
β1n x n2 N
i1
β i1 n T i x n2
− β1n β i1 n g Jx n − JT i x n
β1n
p2− 2p, Jx n
x n2
i1
β i1 n
p2 2p, JT i x n
T i x n2
− β1n β i1 n g Jx n − JT i x n
β1n φ
p, x n
i1
β n i1 φ
p, T i x n
− β n1β i1 n g Jx n − JT i x n
φp, x n
− β n1β i1 n g Jx n − JT i x n
3.13
and hence
β1n β i1 n g Jx n − JT i x n φp,x n
− φp, z n
2p, z n − x n
x n z n x n − z n
2pz n − x n x n z n x n − z n
−→ 0,
3.14
Lemma 2.5 Bya, we have limn → ∞ gJx n − JT i x n 0 and then limn → ∞ Jx n − JT i x n 0
we obtain
lim
n → ∞ x n − T i x n lim
n → ∞ J−1
Jx n
− J−1
Trang 10for all i 1, 2, , N If b holds, we get
φ
p, z n
p, β n1Jx n N
i1
β i1 n JT i x n
β1n Jx n N
i1
β i1 n JT i x n
2
p2− 2β1n
p, Jx n
i1
β i1 n
p, JT i x n
β1n x n2 N
i1
β i1 n T i x n2
− β k1 n β l1 n g JT k x n − JT l x n
β1n
p2− 2p, Jx n
x n2
i1
β i1 n
p2 2p, JT i x n
T i x n2
− β k1 n β l1 n g JT k x n − JT l x n
β1n φ
p, x n
i1
β n i1 φ
p, T i x n
− β n k1 β n l1 g JT k x n − JT l x n
φp, x n
− β n k1 β l1 n g JT k x n − JT l x n
3.16 and hence
β k1 n β l1 n g JT k x n − JT l x n φp,x n
− φp, z n
2p, z n − x n
x n z n x n − z n
2pz n − x n x n z n x n − z n
−→ 0.
3.17
Then by the same argument above, we have limn → ∞ T k x n − T l x n 0 for all k, l 1, 2, , N.
Next, we observe that
φT k x n , z n V T k x n , β n1Jx n N
i1
β i1 n JT i x n
β1n V
T k x n , Jx n
i1
β i1 n V
T k x n , JT i x n
β1n φ
T k x n , x n
i1
β i1 n φ
T k x n , T i x n
−→ 0.
3.18
Trang 11as β1n → 0 ByLemma 2.2, we have limn → ∞ T k x n − z n 0 for all k 1, 2, , N, and
hence
for all i 1, 2, , N Then ω w x n ⊂N
i1 FT i N
i1 FT i F.
v ∈ ω w x n ⊂ F Put w : Π F x0∈ F ⊂ C n k, we observe that
φ
x n k , x0
φΠC nk x0, x0
min
y∈C nk φ
y, x0
φw, x0
min
z∈F φ
z, x0
φv, x0
φ
v, x0
lim inf
k → ∞ φ
x n k , x0
lim sup
k → ∞
φ
x n k , x0
φw, x0
φv, x0
Corollary 3.2 Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty closed convex subset of E Let {Ω i}N i1 be a finite family of nonempty closed convex subset of
C such that Ω :N i1Ωi is nonempty and let x0∈ E For C1 C and x1 ΠC1x0, define a sequence
{x n } of C as follows:
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1 β1n Jx n N
i1
β i1 n JΠΩi x n
,
C n1z ∈ C n : φ
z, y n
φz, x n
,
x n1 ΠC n1 x0, n 0, 1, 2, ,
3.22
where {α n }, {β n i } ⊂ 0, 1 satisfy the following conditions:
i 0 α n < 1 for all n ∈ N ∪ {0} and lim sup n → ∞ α n < 1,
ii 0 β i n 1 for all i 1, 2, , N 1, N1 i1 β i n 1 for all n ∈ N ∪ {0} If either
a lim infn → ∞ β1n β i1 n > 0 for all i 1, 2, , N or
b limn → ∞ β1n 0 and lim inf n → ∞ β k1 n β l1 n > 0 for all i / j, k, l 1, 2, , N Then the sequence {x n } converges strongly to ΠΩx0, whereΠΩ is the generalized projection from E onto Ω.
Trang 12Theorem 3.3 Let E be a uniformly convex and uniformly smooth Banach space, and let C be a
nonempty closed convex subset of E Let {T i}N
i1 be a finite family of relatively nonexpansive mappings from C into itself such that F :N i1 FT i is nonempty Let a sequence {x n } defined by
x0∈ C,
y n J−1
α n Jx n1− α n
Jz n
,
z n J−1 β1n Jx n N
i1
β n i1 JT i x n
,
H nz ∈ C : φz, y n
φz, x n
,
W nz ∈ C :x n − z, Jx0− Jx n
0,
x n1 ΠH n ∩W n x0, n 0, 1, 2, ,
3.23
where {α n }, {β n i } ⊂ 0, 1 satisfy the following conditions:
i 0 α n < 1 for all n ∈ N ∪ {0} and lim sup n → ∞ α n < 1,
ii 0 β i n 1 for all i 1, 2, , N 1, N1 i1 β i n 1 for all n ∈ N ∪ {0} If either
a lim infn → ∞ β1n β i1 n > 0 for all i 1, 2, , N or
b limn → ∞ β1n 0 and lim inf n → ∞ β k1 n β l1 n > 0 for all i / j, k, l 1, 2, , N Then the sequence {x n } converges strongly to Π F x0, whereΠF is the generalized projection from E onto F.
Proof From the definition of H n and W n , it is obvious H n and W nare closed and convex for
n ∈ N ∪ {0} Then, as in the proof ofTheorem 3.1, we have
φ
u, z n
φu, x n
3.24
F ⊂ H n for each n ∈ N ∪ {0} We note by 21, Proposion 2.4 that each FTi is closed and
and
lim
n → ∞ x n1 − y n limn → ∞ x n1 − x n 0. 3.25
Since J is uniformly norm-to-norm continuous on bounded sets, we have
lim
n → ∞ Jx n1 − Jy n lim
n → ∞ Jx n1 − Jx n 0. 3.26
...a finite family of relatively nonexpansive mappings in Banach spaces by using the hybrid method in mathematical programming
Theorem 3.1 Let E be a uniformly convex and uniformly... This inequality is affine in v and hence the set K is closed and convex
3 Main result
In this section, we prove strong convergence theorems for finding a common fixed point of. .. nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let T be a mapping from C into itself, and let FT be the set of all fixed points
mapping if the