Regularization Methods for a Class of
Variational Inequalities in Banach Spaces
Nguyen Buong
1
and Nguyen Thi Hong Phuong
2
1
Vietnamese Academy of Science and Technology,
Institute of Information Technology,
18, Hoang Quoc Viet, Hanoi, Vietnam.
Email:
2
9/5, Tran Quoc Hoan, Cau Giay, Ha Noi, Viet Nam.
Abstract In this paper, we introduce two regularization methods, based on the Browder-
Tikhonov and iterative regularizations, for finding a solution of variational inequalities
over the set of common fixed points of an infinite family of nonexpansive mappings on
real reflexive and strictly convex Banach spaces with a uniformly Gˆateaux differentiable
norm.
Key words Regularization · nonexpansive mapping · fixed point · variational inequal-
ity.
AMS 2000 Mathematics Subject Classification: 47J05, 47H09, 49J30.
1
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 2
1. INTRODUCTION AND PRELIMINARIES
Let E be a Banach space with the dual space E
∗
. For the sake of simplicity, the
norms of E and E
∗
are denoted by the symbol .. We write x, x
∗

and, in general, is multi-valued. It is well known that J
q
(tx) = tJ
q
(x), for all t > 0
and x ∈ E, and J
q
(−x) = −J
q
(x). When q = 2, J
2
is called the normalized duality
mapping and is usually denoted by J. In the case that E ≡ H, a Hilbert space, we
have J = I, the identity mapping.
Let C be a nonempty, closed and convex subset of E and T, F : C → E be two
nonlinear mappings. Recall that a mapping T , satisfying the condition
T x − T y ≤ x − y,
for all x, y ∈ C, is said to be nonexpansive. Put Fix(T ) = {x ∈ C : x = T x}, the set of
fixed points of T . In addition, if T : C → C, then T is called a nonexpansive mapping on
C. The mapping F is said to be η-strongly accretive and γ-strictly pseudocontractive,
iff it satisfies, respectively, the following conditions:
F (x) − F (y), j(x − y) ≥ ηx − y
2
,
and
F (x) − F (y), j(x − y) ≤ x − y
2
− γ(I − F)x − (I − F)y
2
,

(I − λF(p
∗
)), (2)
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 3
where Q
C
is a sunny nonexpansive retraction from any point of the space onto C. The
sunny nonexpansive retraction is not easy to compute, due to the complexity of the
feasible set. To overcome this drawback in a Hilbert space, where the retraction is
a metric projection, in [12], Yamada assumed that the feasible set C were the set of
common fixed points of a finite family of nonexpansive mappings {T
i
}
N
i=1
, proposed the
following iterative algorithm
u
k+1
= T
[k+1]
u
k
− λ
k+1
µF (T
[k+1]
u
k
), (3)

|λ
k
− λ
k+N
| < ∞,
the sequence {u
k
}
k∈N
of (3) converges strongly to p
∗
in (1). Next, in [13], Xu and Kim,
by replacing condition (L3) by
(L4) lim
k→∞
(λ
k
− λ
k+N
)/λ
k+N
= 0,
proved also a strong convergence result.
In the case that C = ∩
∞
i=1
F ix(T
i
), where {T
i

)I,
U
k,k−1
= α
k−1
T
k−1
U
k,k
+ (1 − α
k−1
)I,
. . . . . . . . . . . . . . . . . . . . . . . . . .
U
k,2
= α
2
T
2
U
k,3
+ (1 − α
2
)I,
W
k
= U
k,1
= α
1

)F
k
(x
k
) + γ
k
W
k
F
k
(x
k
),
where F
k
= I − λ
k
F , λ
k
∈ (0, 1), satisfying (L1) and (L2), and γ
k
∈ [γ, 1/2], converges
strongly to the unique element p
∗
in (1).
Very recently, in [15], Wang obtained the same result, under the conditions that
λ
k
F (x
k

following equation
A
k
(x
k
) + ε
k
F (x
k
) = 0, A
k
= I − V
k
, (4)
where
V
k
= V
1
k
, V
i
k
= T
i
T
i+1
· · · T
k
, T

[A
k
(z
k
) + ε
k
F (z
k
)] k ≥ 1, (7)
where the iteration parameter β
k
satisfies some conditions, for any z
1
∈ E.
We recall the following facts which will be used to prove our result.
Let E be a real normed linear space. Let S
1
(0) := {x ∈ E : x = 1}. The space E
is said to have a Gˆateaux differentiable norm (or to be smooth) if the limit
lim
t→0
x + ty − x
t
exists for each x, y ∈ S
1
(0). The space E is said to have a uniformly Gˆateaux differen-
tiable norm if the limit is attained uniformly for x ∈ S
1
(0). Assume that dim(E) ≥ 2.
The modulus of smoothness of E is the function ρ

, 1 < p < ∞,
are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth while L
p
or l
p
or W
p
m
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 5
is p-uniformly smooth if 1 < p ≤ 2, 2-uniformly smooth if p ≥ 2. The space E is said
to be strictly convex, iff for x, y ∈ S
1
(0) with x = y, we have
(1 − λ)x + λy < 1 ∀λ ∈ (0, 1).
It is well-known (see, [20]) that if E is smooth, then the normalized duality mapping is
single valued; and if the norm of E is uniformly Gˆateaux differentiable, then the nor-
malized duality mapping is norm to weak star uniformly continuous on every bounded
subset of E. In the sequel, we shall denote the single valued normalized duality map-
ping and single valued generalized duality mapping by j and j
q
, respectively.
Let µ be a continuous linear functional on l
∞
and let (a
1
, a
2
, ) ∈ l
∞
. We write

k
≤ µ
k
(a
k
) ≤ lim sup
k→∞
a
k
for all (a
1
, a
2
, ) ∈ l
∞
. If a = (a
1
, a
2
, ) ∈ l
∞
, b = (b
1
, b
2
, ) ∈ l
∞
and a
k
→ c

= min
u∈C
µ
k
x
k
− u
2
iff µ
k
u − z, j(x
k
− z) ≤ 0 for all u ∈ C.
Lemma 2 [22]. Let {a
k
}, {b
k
} and {c
k
} be the sequences of positive numbers satisfying
the conditions
(i) a
k+1
≤ (1 − b
k
)a
k
+ c
k
, b

+ qy, j
q
(x) + c
q
y
q
.
2. MAIN RESULTS
By using the techniques in [25-27] for W -mapping, we have the following lemmas.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 6
Lemma 3 Let C be a closed and convex subset of a real strictly convex Banach space E
and let {T
i
}
k
i=1
, k ≥ 1, be k nonexpansive mappings on C such that the set of common
fixed points F := ∩
k
i=1
F ix(T
i
) = ∅. Let a, b and α
i
, i = 1, 2, ·· ·, k, be real numbers such
that 0 < a ≤ α
i
≤ b < 1, and let V
k
be a mapping, defined by (5). Then, F ix(V

1
T
2
· · · T
k
z − p
= T
1
T
2
· · · T
k
z − T
1
p
≤ T
2
· · · T
2
z − p
= T
2
· · · T
N
z − T
2
p
. . . . . . . . . . . . . .
≤ T
k−1

z − p).
Since E is strictly convex and α
k
∈ (a, b) with a, b ∈ (0, 1), we obtain that
T
k
z − p = z − p,
and hence T
k
z = z. So, z ∈ F ix(T
k
) for each z ∈ F ix(V
k
). Moreover, this implies that
[(1 − α
k−1
)I + α
k−1
T
k−1
]T
k
z − p = [(1 − α
k−1
)I + α
k−1
T
k−1
]z − p.
Now, from (9) it follows that

i
}
∞
i=1
be an infinite family of nonexpansive mappings on C such that the set of common fixed
points F := ∩
∞
i=1
F ix(T
i
) = ∅. Let V
k
be a mapping, defined by (5)-(6). Then, for each
x ∈ C and i ∈ N, the set of all positive integers, lim
k→∞
V
i
k
x exists.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 7
Proof. Let p ∈ F and x ∈ C such that p = x. Then, for k ∈ N with fixed k ≥ i, we
have
V
i
k+1
x − V
i
k
x = T
i

≤ 2α
k+1
x − p.
Since

∞
i=1
α
i
< ∞, we have lim
n,m→∞

m
j=n
α
j
= 0. So, for any ε > 0, there exists
k
0
∈ N with k
0
≥ i such that, for any n, m with m > n > k
0
, we have
m−1

j=n
α
j+1
<

j=n
α
j+1
< ε.
Therefore, {V
i
k
x}, for each fixed i, is a Cauchy sequence in the Banach space E and
hence lim
k→∞
V
i
k
x exists.
Now, we can define the mappings
V
i
∞
x := lim
k→∞
V
i
k
x, and V x := lim
k→∞
V
k
x = lim
k→∞
V

V
k
x − V
k
y = V
1
k
x − V
1
k
y
= (1 − α
1
)(V
2
k
x − V
2
k
y) + α
1
(T
1
V
2
k
x − T
1
V
2

k
y
≤ V
k
k
x − V
k
k
y ≤ x − y,
which together with V x − V y = x − y implies that
V
i
∞
x − V
i
∞
y = V
i+1
∞
x − V
i+1
∞
y = x − y.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 8
So, we have
(1 − α
i
)(V
i+1
∞

∞
x − T
i
V
i+1
∞
y = T
i
V
i+1
∞
x − y,
= V
i+1
∞
x − V
i+1
∞
y = V
i+1
∞
x − y,
and hence x = T
i
V
i+1
∞
x and x = V
i+1
∞

} converges strongly to p
∗
, solving (1).
Proof. It is easy to see that the mapping A
k
+ ε
k
F , for each ε
k
> 0, is ε
k
-strongly
accretive on E. So, it is m-accretive. Therefore, (4) has a unique solution x
k
, for each
ε
k
> 0. Next we show that the set {x
k
} is bounded. Indeed, for any p ∈ C, we have
A
k
(p) = 0, and hence, from (4) it follows that
A
k
(x
k
) − A
k
(p), j(x

k
} is bounded. So, are the sets {V
k
x
k
} and {F (x
k
)}. Without any loss of generality,
we assume that x
k
, V
k
x
k
, F (x
k
) ≤ M
1
, for some positive constant M
1
and all
k ≥ 1.
On the other hand, since A
k
(x
k
) = ε
k
F
k

0
. Taking D = {x
k
: k ≥ 1} and i = 1, we have
V
k
x
k
− V x
k
 ≤ sup
x∈D
V
k
x − V x ≤ ε.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 9
This implies that V
k
x
k
− V x
k
 → 0, as k → ∞. Further, since
A(x
k
) ≤ A
k
(x
k
) + V

ϕ(u)} = ∅.
It is easy to see that C
∗
is a bounded, closed, and convex subset of E. From (11), we
have that
ϕ(V ˜p) = µ
k
x
k
− V ˜p
2
= µ
k
V x
k
− V ˜p
2
≤ µ
k
x
k
− ˜p
2
= ϕ(˜p)
which implies that V C
∗
⊂ C
∗
, that is C
∗

k
x
k
− ˜p
2
= 0. Hence, there exists a subsequence
{x
k
i
} of {x
k
} which strongly converges to ˜p as i → ∞. Again, from (8) and the norm
to weak star continuous property of the normalized duality mapping j on bounded
subsets of E, we obtain that
F (p), j(˜p − p) ≤ 0 ∀p ∈ C. (14)
Since p and ˜p belong to F, a closed and convex subset, by replacing p in (14) by
sp + (1 − s)˜p for s ∈ (0, 1), using the well-known property j(s(˜p − p)) = sj(˜p − p) for
s > 0, dividing by s and taking s → 0, we obtain
F (˜p), j(˜p − p) ≤ 0 ∀p ∈ C.
The uniqueness of p
∗
in (1) guarantees that ˜p = p
∗
. So, all the sequence {x
k
} converges
strongly to p
∗
as k → ∞. This completes the proof.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 10

β
k
= lim
k→∞
α
k+1
β
k
ε
2
k
= 0,
∞

k=0
ε
k
β
k
= ∞, lim sup
k→∞
c
q
β
q−1
k
(2 + ε
k
)
q

k
)]
q
= z
k
− x
k
− β
n
[(I − V
k
)z
k
− (I − V
k
)x
k
+ ε
k
(F (z
k
) − F (x
k
)]
q
≤ z
k
− x
k


k
)z
n
− (I − V
k
)x
k
+ ε
k
(F (z
k
) − F (x
k
))
q
,
where
(I − V
k
)z
k
− (I − V
k
)x
k
, j
q
(z
k
− x

(z
k
− x
k
) ≥ ηz
k
− x
k

q
.
Therefore,
z
k+1
− x
k

q
≤ z
k
− x
k

q
[1 − qβ
n
ε
k
+ c
q

)
q
]
1/q
.
As c
q
β
q
k
(2 + ε
k
)
q
< β
k
ε
k
and (1 − t)
γ
≤ 1 − γt, for 0 < γ < 1, then
z
k+1
− x
k
 ≤ z
k
− x
k
[1 − (q − 1)β

k+1
x
k+1
, j(x
k
− x
k+1
)
− ε
k
F (x
k
) − ε
k+1
F (x
k+1
), j(x
k
− x
k+1
)
≤ x
k
− x
k+1

2
+ 2b
k+1
x

x
k
− x
k+1
 ≤ 2b
k+1
(M
1
+ p)/ε
k
+ M
1
(ε
k
− ε
k+1
)/ε
k
.
Hence,
z
k+1
− x
k+1
 ≤ z
k+1
− x
k
 + x
k+1

= z
k
− x
k
, b
k
= (q − 1)β
k
ε
k
/q, c
k
= 2α
k+1
(M
1
+ p)/ε
k
+ M
1
(ε
k
− ε
k+1
)/ε
k
.
we obtain that lim
k→∞
z

0
<
1
c
1/(q−1)
q
(2 + ε
0
)
q/(q−1)
satisfy all the necessary conditions in Theorem 4 for the case q = 2. For the case
1 < q < 2, ε
k
= (1 + k)
−p
with p < (q − 1)/(2q) and β
k
= γ
0
ε
1/(q−1)
k
also satisfy all the
necessary conditions in Theorem 2.2.
This research is founded by Vietnamese National Foundation of Science and Tech-
nology Development under grant number 101.01-2011.17.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 12
References
[1] Antman S., The influence of elasticity in analysis: modern developments// Bull.
Amer. Math. Soc. 1983, V. 9, N. 3, P. 267-291

for variational inequalities// Appl. Math. Comput. 216 2010, V. 216, P. 822-829
[15] Wang Sh. , Convergence and weaker control conditions for hybrid iterative algo-
rithms, FPTA 2011, doi: 10.1186/1687-1812-2011-3.
Regularization Fixed Point Iteration for Nonlinear Ill-Posed 13
[16] Alber Ya.I., On the stability of iterative approximations to fixed points of nonex-
pansive mappings// J. Math. Anal. Appl. 2007, V.328, P. 958-971.
[17] Alber Ya.I., Ryazantseva Ir., Nonlinear iil-posed problems of monotone type//
Springer-Verlage, 2006.
[18] Bakushinskii A., Gancharskii A., Iil-posed problems: Theory and applications//
Kluwer Academic Publishers, 1994.
[19] Browder F.E., Existence and approximation of solutions of nonlinear variational
inequations// Proc. Nat. Acad. Sci. USA, 1966, V. 56, P. 1080-1086.
[20] Cioranescu I., Geometry of Banach spaces, Duality mappings and nonlinear prob-
lems// Kluwer Acad. Publ., Dordrecht, 1990.
[21] Takahashi W., Ueda Y., On Reich’s strong convergence theorem for resolvents of
accretive operators// J. Math. Anal. Appl. 1984, V. 104, P. 546–553.
[22] Vasin V.V., Ageev A.L. Incorrect problems with priori information, Ekaterenburg,
Nauka, 1993 (in Russian).
[23] Xu H.K., Inequalities in Banach spaces with applications// Nonlinear Anal. 1991,
V. 16, N. 12, P. 1127-1138.
[24] Suzuki T., Strong convergence of approximated sequences for nonexpansive map-
pings in Banach spaces// Proc. Amer. Math. Soc. 2007, V. 135, P. 99-106.
[25] Takahashi W., Weak and strong convergence theorems for families of nonexpansive
mappings and their applications// Ann. Univ. Mariae Curie-Sklodowska Sect. A.
1997, V. 51, P. 277-292
[26] Takhashi W., Shimoji K., Convergence theorems for nonexpansive mappings and
feasibility problems// Math. Comput. Model. 2000, V. 32, P. 1463-1471.
[27] Shimoji K., Takhashi W. , Strong convergence to common fixed points of infinite
nonexpansive mappings and applications// Taiwanese J. Math. 2001, V. 5, N. 2,
P. 387-404.