Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 902437, 12 pages
doi:10.1155/2012/902437
Research Article
Approximation of Common Fixed Points of
a Sequence of Nearly Nonexpansive Mappings and
Solutions of Variational Inequality Problems
D. R. Sahu,1 Shin Min Kang,2 and Vidya Sagar1
1
2
Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
Correspondence should be addressed to Shin Min Kang, smkang@gnu.ac.kr
Received 10 April 2012; Accepted 14 May 2012
Academic Editor: Yonghong Yao
Copyright q 2012 D. R. Sahu 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.
We introduce an explicit iterative scheme for computing a common fixed point of a sequence of
nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which
is also a solution of a variational inequality problem. We prove a strong convergence theorem for
a sequence generated by the considered iterative scheme under suitable conditions. Our strong
convergence theorem extends and improves several corresponding results in the context of nearly
nonexpansive mappings.
1. Introduction
Let C be a nonempty subset of a real Hilbert space H with inner product ·, · and norm · ,
respectively. A mapping T : C → H is called the following:
1 monotone if
T x − T y, x − y ≥ 0 ∀x, y ∈ C,
1.1
2 η-strongly monotone if there exists a positive real number η such that
2
T x − T y, x − y ≥ ηx − y
∀x, y ∈ C,
1.2
3 k-Lipschitzian if there exists a constant k > 0 such that
T x − T y ≤ kx − y
∀x, y ∈ C,
1.3
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4 nonexpansive if
T x − T y ≤ x − y ∀x, y ∈ C,
1.4
5 nearly nonexpansive 1, 2 with respect to a fixed sequence {an } in 0, ∞ with an → 0
if
n
T x − T n y ≤ x − y an
∀x, y ∈ C and n ∈ N.
1.5
In 3, Moudafi proposed viscosity approximation methods of selecting a particular
fixed point of a given nonexpansive mapping in Hilbert spaces see 4 for further developments in both Hilbert and Banach spaces. Let f be a contraction on H. Starting with an
arbitrary initial x1 ∈ H, define a sequence {xn } recursively by
xn1 αn fxn 1 − αn T xn
∀n ∈ N,
1.6
where {αn } is a sequence in 0, 1. It is proved in 4 that under appropriate conditions imposed on {αn }, the sequence {xn } generated by 1.6 strongly converges to the unique solution x∗ ∈ C of the variational inequality
I − f x∗ , x − x∗ ≥ 0 ∀x ∈ C,
1.7
where C FT , the set of fixed points of T .
In 2006, Marino and Xu 5 introduced the viscosity iterative method for nonexpansive
mappings. Starting with an arbitrary initial x1 ∈ H, define a sequence {xn } recursively by
xn1 αn γfxn I − αn AT xn
∀n ∈ N.
1.8
They proved that the sequence {xn } generated by 1.8 converges strongly to the unique
solution of the variational inequality
A − γf x∗ , x − x∗ ≥ 0 ∀x ∈ C,
1.9
which is the optimality condition for the minimization problem
1
min Ax, x − hx,
x∈C 2
1.10
where h is a potential function for γf i.e., h x γfx for all x ∈ H, and A is a strongly
positive bounded linear operator on H; that is, there is a constant γ > 0 such that
Ax, x ≥ γx2
∀x ∈ H.
1.11
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3
The applications of the iterative method 1.8 have been studied by some researchers see
6, 7.
Also, Wang 8, 9 and Wang and Hu 10 introduced the iterative method for nonexpansive mappings.
Recently, Tian 11 proposed an implicit and an explicit schemes on combining the
iterative methods of Marino and Xu 5 and Yamada 12. He also proved the strong convergence of these two schemes to a fixed point of a nonexpansive mapping T defined on a real
Hilbert space under suitable conditions.
More recently, Ceng et al. 13 introduced an implicit and an explicit schemes using the
properties of projection for finding the fixed points of a nonexpansive mapping defined on the
closed convex subset of a real Hilbert space. They also proved the strong convergence of the
sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping
which is also a solution of a variational inequality defined on the set of fixed points.
Aoyama et al. 14 proved strong convergence of an iterative scheme for a sequence of
nonexpansive mappings as follows.
Theorem 1.1. Let X be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and C be a nonempty closed convex subset of X. Let {Tn } be a sequence of nonexpansive
mappings from C into itself such that ∞
/ ∅. Let T be a mapping from C into itself defined
n n1 FT
∞
by T x limn → ∞ Tn x for all x ∈ C and FT n1 FTn . Let {xn } be a sequence in C generated by
the following iterative process:
x1 x ∈ C,
xn1 αn x 1 − αn Tn xn
∀n ∈ N,
1.12
where {αn } is a sequence in 0, 1 satisfying the following conditions:
a limn → ∞ αn 0 and ∞
n1 αn ∞;
∞
b either n1 |αn1 − αn | < ∞ or αn ∈ 0, 1 and limn → ∞ αn1 /αn 1;
c ∞
n1 sup{Tn z − Tn1 z : z ∈ B} < ∞ for any bounded subset B of C.
Then, the sequence {xn } converges strongly to Qx, where Q is the sunny nonexpansive retraction from X onto FT .
Let C be a nonempty subset of a real Hilbert space H. Let T : {Tn } be a sequence
of mappings from C into itself. We denote by FT the set of common fixed points of the
sequence T, that is, FT ∞
n1 FTn . Fix a sequence {an } in 0, ∞ with an → 0, and let
{Tn } be a sequence of mappings from C into H. Then, the sequence {Tn } is called a sequence
of nearly nonexpansive mappings 15 with respect to a sequence {an } if
Tn x − Tn y ≤ x − y an
∀x, y ∈ C, n ∈ N.
1.13
It is obvious that the sequence of nearly nonexpansive mappings is a wider class of
sequence of nonexpansive mappings.
In this paper, inspired by Aoyama et al. 14, Ceng et al. 13, and Sahu et al. 15, we
introduce an explicit iterative scheme and prove a strong convergence theorem for computing
an element of FT, the set of common fixed points of a sequence T {Tn } of nearly
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Journal of Applied Mathematics
nonexpansive mappings which is also a solution of a variational inequality over FT. Our
result generalizes and improves the results of Ceng et al. 13, Tian 11, and many other
related works.
2. Preliminaries
Throughout this paper, we denote by I the identity operator of H. Also, we denote by → and
the strong convergence and weak convergence, respectively. The symbol N stands for the
set of all natural numbers.
Let C be a nonempty closed convex subset of a real Hilbert space H. Then, for any
x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that
x − PC x ≤ x − y ∀y ∈ C.
2.1
The mapping PC is called the metric projection from H onto C see 1.
Let C be a nonempty subset of a real Hilbert space H and T1 , T2 : C → H be two
mappings. We denote BC, the collection of all bounded subsets of C. The deviation between
T1 and T2 on B ∈ BC, denoted by DB T1 , T2 , is defined by
DB T1 , T2 sup{T1 x − T2 x : x ∈ B}.
2.2
The following lemmas will be needed to prove our main result.
Lemma 2.1 see 16. The metric projection mapping PC is characterized by the following properties:
a PC x ∈ C for all x ∈ H;
b x − PC x, PC x − y ≥ 0 for all x ∈ H and y ∈ C;
c x − y2 ≥ x − PC x2 y − PC x2 for all x ∈ H and y ∈ C;
d PC x − PC y, x − y ≥ PC x − PC y2 for all x, y ∈ H.
Lemma 2.2 see 13. Let V : C → H be an L-Lipschitzian mapping and F : C → H be a kLipschitzian and η-strongly monotone operator. Then, for 0 ≤ γL < μη,
2
2.3
x − y, μF − γV x − μF − γV y ≥ μη − γL x − y ∀x, y ∈ C.
That is, μF − γV is strongly monotone with coefficient μη − γL.
Lemma 2.3 see 12. Let C be a nonempty subset of a real Hilbert space H. Suppose that λ ∈ 0, 1
and μ > 0. Let F : C → H be a k-Lipschitzian and η-strongly monotone operator on C. Define the
mapping G : C → H by
Gx x − λμFx
∀x ∈ C.
Then G is a contraction that provided μ < 2η/k2 . More precisely, for μ ∈ 0, 2η/k2 ,
Gx − Gy ≤ 1 − λτx − y ∀x, y ∈ C,
where τ 1 −
1 − μ2η − μk2 ∈ 0, 1.
2.4
2.5
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5
Lemma 2.4 see 1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let
T : C → C be a nonexpansive mapping. Then I − T is demiclosed at zero; that is, if {xn } is a sequence
in C weakly converging to some x ∈ C and the sequence {I − T xn } converges strongly to 0, then
x ∈ FT .
Lemma 2.5 see 17. Assume that {tn } is a sequence of nonnegative real numbers such that
tn1 ≤ 1 − αn tn αn βn
∀n ∈ N,
2.6
where {αn } and {βn } are sequences of nonnegative real numbers which satisfy the following conditions:
∞
a {αn }∞
n1 αn ∞;
n1 ⊂ 0, 1 and
b lim supn → ∞ βn ≤ 0, or
b’ ∞
n1 αn βn is convergent.
Then limn → ∞ tn 0.
Lemma 2.6 see 18. Let C be a nonempty closed convex subset of a real Hilbert space H and
1. Let T1 , T2 , T3 , . . . , TN : C → C be nonexpansive
λi > 0. i 1, 2, 3, . . . , N such that N
i1 λi N
∅, and let T N
mappings such that i1 FTi /
i1 λi Ti . Then T is nonexpansive from C into itself
FT
.
and FT N
i
i1
3. Main Result
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H
be a k-Lipschitzian and η-strongly monotone operator and V : C → H be an L-Lipschitzian
mapping. Let T {Tn } be a sequence of nearly nonexpansive mappings from C into itself with
respect to a sequence {an } such that FT / ∅ and T be a mapping from C into itself defined by
2
T x lim
n → ∞ Tn x for all x ∈ C. Suppose that FT FT, 0 < μ < 2η/k and 0 ≤ γL < τ, where
τ 1 − 1 − μ2η − μk2 . For an arbitrary x1 ∈ C, consider the sequence {xn } in C generated by
the following iterative process:
xn1
x1 ∈ C,
PC αn γV xn I − αn μF Tn xn
∀n ∈ N,
3.1
where {αn } is a sequence in 0, 1 satisfying the conditions:
a limn → ∞ αn 0 and ∞
n1 αn ∞;
∞
b either n1 |αn1 − αn | < ∞ or limn → ∞ αn1 /αn 1;
c either ∞
n1 DB Tn , Tn1 < ∞ or limn → ∞ DB Tn , Tn1 /αn1 0 for each B ∈ BC;
d limn → ∞ an /αn 0.
Then, the sequence {xn } converges strongly to x ∈ FT, where x is the unique solution of variational
inequality
μF − γV x,
x − y ≤ 0 ∀y ∈ FT.
3.2
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Proof. Let T be a mapping from C into itself defined by T x limn → ∞ Tn x for all x ∈ C. It
is clear that T is a nonexpansive mapping. So, we have FT /
∅. Now, we proceed with the
following steps.
Step 1. {xn } is bounded. Let z ∈ FT. From 3.1, we have
xn1 − z PC αn γV xn I − αn μF Tn xn − PC z
≤ αn γV xn I − αn μF Tn xn − z
≤ αn γV xn − μFz I − αn μF Tn xn − I − αn μF Tn z
≤ αn γLxn − z αn γV − μF z 1 − αn τxn − z an ≤ 1 − αn τ − γL xn − z αn γV − μF z 1 − αn τan
≤ 1 − αn τ − γL xn − z αn γV − μF z an .
3.3
Note that limn → ∞ an /αn 0, so there exists a constant K > 0 such that
αn γV − μF z an
≤K
αn
∀n ∈ N.
3.4
Thus, we have
xn1 − z ≤ 1 − αn τ − γL xn − z αn K
K
≤ max xn − z,
∀n ∈ N.
τ − γL
3.5
Hence, {xn } is bounded. So {Tn xn } and {V xn } are bounded.
Step 2. xn1 − xn → 0 as n → ∞. From 3.1, we have
xn1 − xn PC αn γV xn I − αn μF Tn xn −PC αn−1 γV xn−1 I − αn−1 μF Tn−1 xn−1 ≤ αn γV xn I − αn μF Tn xn − αn−1 γV xn−1 I − αn−1 μF Tn−1 xn−1 ≤αn γV xn − V xn−1 γαn − αn−1 V xn−1
I − αn μF Tn xn − I − αn μF Tn xn−1
Tn xn−1 − Tn−1 xn−1 αn−1 μFTn−1 xn−1 − αn μFTn xn−1 ≤αn γLxn − xn−1 γαn − αn−1 V xn−1 1 − αn τTn xn − Tn xn−1 Tn xn−1 − Tn−1 xn−1 μαn−1 FTn−1 xn−1 − αn FTn xn−1 Journal of Applied Mathematics
7
≤ 1 − αn τ − γL xn − xn−1 DB Tn , Tn−1 1 − αn τan
γαn − αn−1 V xn−1 μαn−1 FTn−1 xn−1 − FTn xn−1 − αn − αn−1 FTn xn−1 ≤ 1 − αn τ − γL xn − xn−1 DB Tn , Tn−1 1 μαn−1 k
M|αn − αn−1 | an ,
3.6
for some constant M > 0. Thus, using Lemma 2.5, we get xn1 − xn → 0 as n → ∞.
Step 3. We have xn − T xn → 0 as n → ∞. Note that
xn − Tn xn ≤ xn − xn1 xn1 − Tn xn xn − xn1 PC αn γV xn I − αn μF Tn xn − PC Tn xn ≤ xn − xn1 αn γV xn I − αn μF Tn xn − Tn xn xn − xn1 αn γV xn − μFTn xn −→ 0 as n −→ ∞.
3.7
Since
xn − T xn ≤ xn − Tn xn Tn xn − T xn ≤ xn − Tn xn DB Tn , T ,
3.8
it follows that limn → ∞ xn − T xn 0.
γV − μFx
≤ 0. Let us choose a subsequence
Step 4. We have lim supn → ∞ xn − x,
{xnk } of {xn } such that
γV − μF x lim xnk − x,
γV − μF x .
lim sup xn − x,
n→∞
k→∞
3.9
Without loss of generality, we may assume that xnk z ∈ C. By using Lemma 2.4, we get
that z ∈ FT . Note that FT FT, it follows that z ∈ FT. Hence from 3.2, we get the
following:
lim sup xn − x,
γV − μF x z − x,
γV − μF x ≤ 0.
n→∞
3.10
Step 5. We have xn → x as n → ∞. Set yn αn γV xn I − αn μFTn xn and γn αn τ − γL. Noticing that xn1 PC yn . From 3.1, we have
2 yn − x,
xn1 − x PC yn − yn , PC yn − x
xn1 − x
≤ yn − x,
xn1 − x
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Journal of Applied Mathematics
xn1 − x
αn γV xn − μF x,
I − αn μF Tn xn − I − αn μF Tn x,
xn1 − x
αn γV xn − V x,
xn1 − x
αn γV x − μF x,
xn1 − x
I − αn μF Tn xn − I − αn μF Tn x,
xn1 − x
≤αn γLxn − xx
αn γV − μF x,
xn1 − x
n1 − x
1 − αn τxn − x
an xn1 − x
1 − αn τ − γL xn − xx
n1 − x
αn γV − μF x,
xn1 − x 1 − αn τan xn1 − x
1 ≤ 1 − αn τ − γL
2 xn1 − x
2
xn − x
2
αn γV − μF x,
xn1 − x an xn1 − x.
3.11
Hence, we have
1 − αn τ − γL
2αn ≤
2
γV − μF x,
xn1 − x
xn − x
xn1 − x
1 γn
1 αn τ − γL
2
2an
xn1 − x
1 γn
2αn ≤ 1 − αn τ − γL xn − x
2
γV − μF x,
xn1 − x
1 γn
3.12
2an
xn1 − x
1 γn
2an
1 − γn xn − x
2 γn δn xn1 − x,
1 γn
where
δn 1 γn
2
τ − γL
γV − μF x,
xn1 − x .
3.13
Noticing that limn → ∞ an /αn 0, it follows from Lemma 2.5 that limn → ∞ xn x.
This
completes the proof.
Now, we derive the main result of Ceng et al. 13, Theorem 3.2 as the following
corollary.
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be
a k-Lipschitzian and η-strongly monotone operator and V : C → H be an L-Lipschitzian mapping.
Let T : C → C be a nonexpansive mapping such that FT /
∅. Suppose that 0 < μ < 2η/k2 and
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9
0 ≤ γL < τ, where τ 1 − 1 − μ2η − μk2 . For an arbitrary x1 ∈ C, consider the sequence {xn }
generated by the following iterative process:
xn1
x1 ∈ C,
PC αn γV xn I − αn μF T xn ∀n ∈ N,
3.14
where {αn } is a sequence in 0, 1 satisfying the conditions (a) and (b) of Theorem 3.1.
Then, the sequence {xn } converges strongly to x ∈ FT , where x is the unique solution of the
following variational inequality:
μF − γV x,
x − y ≤ 0 ∀y ∈ FT .
3.15
Again, we derive the result of Tian 11, Theorem 3.2 as the following corollary.
Corollary 3.3. Let H be a real Hilbert space. Let f be an α-contraction on H and F : H → H be
a k-Lipschitzian and η-strongly monotone operator. Let T : H → H be a nonexpansive mapping
such that FT / ∅. Suppose that 0 < μ < 2η/k2 and 0 < γα < τ, where τ μη − μk2 /2. For an
arbitrary x1 ∈ H, consider the sequence {xn } generated by the following iterative process:
xn1
x1 ∈ H,
αn γfxn I − αn μF T xn
∀n ∈ N,
3.16
where {αn } is a sequence in 0, 1 satisfying the conditions a and b of Theorem 3.1.
Then, the sequence {xn } converges strongly to x ∈ FT , where x is the unique solution of the
following variational inequality:
μF − γf x,
x − y ≤ 0
∀y ∈ FT .
3.17
The following result obtains immediately from Theorem 3.1.
Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be
a k-Lipschitzian and η-strongly monotone operator and V : C → H be an L-Lipschitzian mapping.
∅ and T be
Let {Tn } be a sequence of nonexpansive mappings from C into itself such that ∞
n1 FTn /
T
x
for
all
x
∈
C.
Suppose
that
0
<
μ < 2η/k2
a mapping from C into itself defined
by
T
x
lim
n
→
∞
n
and 0 ≤ γL < τ, where τ 1 − 1 − μ2η − μk2 . For an arbitrary x1 ∈ C, consider the sequence
{xn } in C generated by the following iterative process:
xn1
x1 ∈ C,
PC αn γV xn I − αn μF Tn xn
∀n ∈ N,
where {αn } is a sequence in 0, 1 satisfying the conditions (a)–(c) of Theorem 3.1.
3.18
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Journal of Applied Mathematics
Then, the sequence {xn } converges strongly to x ∈
of the following variational inequality:
∞
n1
FTn , where x is the unique solution
∞
FTn .
μF − γV x,
x − y ≤ 0 ∀y ∈
n1
3.19
4. Application
Recall that the so-called problem of image recovery is essentially to find a common element
of finitely many nonexpansive retracts C1 , C2 , . . . , Cr of C with ri1 Ci / ∅. It is easy to see that
every nonexpansive retraction Pi of C onto Ci is a nonexpansive mapping of C into itself.
There is no doubt that the problem of image recovery is equivalent to finding a common
fixed point of finitely many nonexpansive mappings P1 , P2 , . . . , Pr of C into itself. Applying
our main result, we obtain the following result which improves a number of results connected
to the problem of image recovery.
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C →
H be a k-Lipschitzian and η-strongly monotone operator and V : C → H be an L-Lipschitzian
mapping. Let λi > 0i 1, 2, 3, . . . , N such that N
i1 λi 1 and T1 , T2 , T3 , . . . , TN : C → C be
N
∅. Suppose that 0 < μ < 2η/k2 and 0 ≤ γL < τ,
nonexpansive mappings such that i1 FTi /
where τ 1 − 1 − μ2η − μk2 . For an arbitrary x1 ∈ C, consider the sequence {xn } in C generated
by the following iterative process:
x1 ∈ C,
xn1 PC
N
αn γV xn I − αn μF
λi Ti xn
∀n ∈ N,
4.1
i1
where {αn } is a sequence in 0, 1 satisfying the conditions a and b of Theorem 3.1.
is the unique solution
Then, the sequence {xn } converges strongly to x ∈ N
i1 FTi , where x
of the following variational inequality:
N
FTi .
μF − γV x,
x − y ≤ 0 ∀y ∈
4.2
i1
Then T is nonexpansive mapping from C into itself. Thus, using
Proof. Define T N
i1 λi Ti .
Lemma 2.6, we get FT N
i1 FTi . Therefore, the proof follows from Corollary 3.2.
5. Numerical Example
For showing the effectiveness and convergence of the sequence generated by the considered
iterative scheme, we discuss the following example.
Example 5.1. Let H R and C 0, 1. Let T be a self-mapping defined by T x 1 − x for all
x ∈ C. Let F, V : C → H be two mappings defined by Fx 4x and V x 2x for all x ∈ C,
Journal of Applied Mathematics
11
0.6
0.5
xn
0.4
0.3
0.2
0.1
0
0
10
20
30
Number of iterations (n)
40
50
x1 = 0
x1 = 0.2
x1 = 0.4
Figure 1
where F is a k-Lipschitzian and η-strongly monotone, and V is an L-Lipschitzian mapping.
We take 0 < μ < 2η/k2 and 0 ≤ γL < τ, and we have μ 1/4,τ 1 and γ 1/4. Define {αn }
in 0, 1 by αn 1/n 1. Without loss of generality, we may assume that an 1/n3/2 for all
n ∈ N. For each n ∈ N, define Tn : C → C by
1 − x,
Tn x an ,
if x ∈ 0, 1,
if x 1.
5.1
In 15, it is proved that T {Tn } is a sequence of nearly nonexpansive mappings
from C into itself such that FT {1/2} and T x limn → ∞ Tn x for all x ∈ C, where T is
nonexpansive mapping.
It can be observed that all the assumptions of Theorem 3.1 are satisfied and the
sequence {xn } generated by 3.1 converges to a unique solution 1/2 of variational inequality
3.2 over FT. The graphical presentation of the convergence of the sequence {xn }
generated by the iterative scheme 3.1 is given in Figure 1.
Acknowledgments
The authors would like to thank the referees for useful comments and suggestions.
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