In this paper, we introduce a general iterative algorithm for finding a common element of the common fixed point set of an infinite family of
λ_{i}
strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in
q
uniformly smooth Banach spaces. Then, we analyze the strong convergence of the iterative sequence generated by the proposed iterative algorithm under mild conditions.
AMS Mathematics Subject Classification : 47H09, 47H10, 47H17.
1. Introduction
Throughout this paper, we denote by
E
and
E^{∗}
a real Banach space and the dual space of
E
respectively. Let
C
be a subset of
E
and
T
be a mapping on
C
. We use
F
(
T
) to denote the set of fixed points of
T
. Let
q
> 1 be a real number. The (generalized) duality mapping
J_{q}
:
E
→ 2
^{E∗}
is defined by
for all
x
∈
E
, where ⟨·, ·⟩ denotes the generalized duality pairing between
E
and
E
^{∗}
. In particular,
J
=
J
_{2}
is called the normalized duality mapping and
J_{q}
(
x
) = ∥
x
∥
^{q}
^{−2}
J
_{2}
(
x
) for
x
≠ 0. If
E
is a Hilbert space, then
J
=
I
where
I
is the identity mapping. It is well known that if
E
is smooth, then
J_{q}
is singlevalued, which is denoted by
j_{q}
. Among nonlinear mappings, nonexpansive mappings and strict pseudocontractions are two kinds of the most important nonlinear mappings. The study of them has a very long history (see
[1

16
,
19

31]
and the references therein). Recall that a mapping
T
:
C
→
E
is nonexpansive if
A mapping
T
:
C
→
E
is
λ
strict pseudocontractive in the terminology of Browder and Petryshyn (see
[2
,
3
,
4]
), if there exists a constant
λ
> 0 such that
for every
x, y
∈
C
and for some
j_{q}
(
x
−
y
) ∈
J_{q}
(
x
−
y
). It is clear that (1.1) is equivalent to the following inequality
Remark 1.1.
The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g.,
[2
,
3
,
4
,
20
,
22]
). However, their iterative methods are far less developed though Browder and Petryshyn
[24]
initiated their work in 1967. As a matter of fact, strictly pseudocontractive mappings have more powerful applications in solving inverse problems (see, e.g.,
[32]
). Therefore it is interesting to develop the theory of iterative methods for strictly pseudocontractive mappings.
In the early sixties, Stampacchia
[33]
first introduced variational inequality theory, which has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences (see
[7
,
8
,
9
,
10
,
11
,
34]
and the references therein). In 1968, Brezis
[34]
initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximalpoint mappings due to Moreau
[35]
. Variational inclusions include variational, quasivariational, variationallike inequalities as special cases. It can be viewed as innovative and novel extension of the variational principles and thus, has wide applications in the fields of optimization and control, economics and transportation equilibrium and engineering sciences. Recently, some new and interesting problems, which are called to be system of variational inequalitys/inclusions received many attentions. System of variational inequalitys/inclusions can be viewed as natural and innovative generalizations of the variational inequalities/inclusions and it can provide new insight regarding problems being studied and can stimulate new and innovative ideas for solving problem.
Ceng et al.
[26]
proposed the following new system of variational inequality problem in a Hilbert space
H
: find
x
^{∗}
,
y
^{∗}
∈
C
such that
where
λ, μ
> 0 are two constants,
A,B
:
E
→
E
are two nonlinear mappings. This is called the new system of variational inequalities. If we add up the requirement that
x
^{∗}
=
y
^{∗}
and
A
=
B
, then problem (1.3) reduces to the classical variational inequality problem: find
x
^{∗}
∈
C
such that
In order to find the solutions of the system of variational inequality problem (1.3), Ceng et al.
[26]
studied the following approximation method. Let the mappings
A,B
:
C
→
H
be inversestrongly monotone,
S
:
C
→
C
be nonexpansive. Suppose that
x
_{1}
=
u
∈
C
and {
x_{n}
} is generated by
They proved that the iterative sequence defined by the relaxed extragradient method (1.5) converges strongly to a fixed point of
S
, which is a solution of the system of variational inequality (1.3).
On the other hand, in order to find the common element of the solutions set of a variational inclusion and the set of fixed points of a nonexpansive mapping
T
, Zhang et al.
[6]
introduced the following new iterative scheme in a Hilbert space
H
. Starting with an arbitrary point
x
_{1}
=
x
∈
H
, define sequence {
x_{n}
} by
where
A
:
H
→
H
is an
α
cocoercive mapping,
M
:
H
→ 2
^{H}
is a maximal monotone mapping,
S
:
H
→
H
is a nonexpansive mapping and {
α_{n}
} is a sequence in [0,1]. Under mild conditions, they obtained a strong convergence theorem.
Motivated by Zhang et al.
[6]
and Zeng et al.
[26]
, Qin et al.
[8]
considered the following new system of variational inclusion problem in a uniformly convex and 2uniformly smooth Banach space: find (
x
^{∗}
,
y
^{∗}
) ∈
E
×
E
such that
The following problems are special cases of problem (1.7).
(1) If
A
=
B
and
M
_{1}
=
M
_{2}
=
M
, then problem (1.7) reduces to the problem: find (
x
^{∗}
,
y
^{∗}
) ∈
E
×
E
such that
(2) If
x
^{∗}
=
y
^{∗}
, problem (1.7) reduces to the problem: find
x
^{∗}
∈
E
such that
Qin et al.
[8]
also introduced the following scheme for finding a common element of the solution set of the general system (1.7) and the fixed point set of a
λ
strict pseudocontraction. Starting with an arbitrary point
x
_{1}
=
u
∈
E
, define sequences {
x_{n}
} by
And they proved a strong convergence theorem under mild conditions.
One question arises naturally: Can we extend Theorem 2.1 of Zhang et al.
[6]
, Theorem 3.1 of Qin et al.
[8]
, Theorem 3.1 of Zeng et al.
[26]
from Hilbert spaces or 2uniformly smooth Banach spaces to more broad
q
uniformly smooth Banach spaces? We put forth another question: Can we get some more general results even without the condition of uniform convexity of Banach spaces ? However, the condition of uniform convexity of Banach spaces is necessary in Theorem 3.1 of Qin et al.
[8]
, Yao et al.
[36]
and so on.
The purpose of this article is to give the affirmative answers to these questions mentioned above. Motivated by Zhang et al.
[6]
, Qin et al.
[8]
, Yao et al.
[9]
, Hao
[10]
, J. C. Yao
[11]
, and Takahashi et al.
[12]
, we consider a relaxed extragradienttype method for finding common elements of the solution set of a general system of variational inclusions for inversestrongly accretive mappings and the common fixed point set of an infinite family of
λ_{i}
strict pseudocontractions. Furthermore, we obtain strong convergence theorems under mild conditions to improve and extend the corresponding results.
2. Preliminaries
The norm of a Banach space
E
is said to be Gâteaux differentiable if the limit
exists for all
x, y
on the unit sphere
S
(
E
) = {
x
∈
E
: ∥
x
∥ = 1}. If, for each
y
∈
S
(
E
), the above limit is uniformly attained for
x
∈
S
(
E
), then the norm of
E
is said to be uniformly Gâteaux differentiable. The norm of
E
is said to be Fréchet differentiable if, for each
x
∈
S
(
E
), the above limit is attained uniformly for
y
∈
S
(
E
).
Let
ρ_{E}
: [0, 1) → [0, 1) be the modulus of smoothness of
E
defined by
A Banach space
E
is said to be uniformly smooth if
Let
q
be a fixed real number with 1 <
q
≤ 2. Then a Banach space
E
is said to be
q
uniformly smooth, if there exists a fixed constant
c
> 0 such that
ρ_{E}
(
t
) ≤
ct^{q}
. It is well known that
E
is uniformly smooth if and only if the norm of
E
is uniformly Fréchet differentiable. If E is
q
uniformly smooth, then
q
≤ 2 and
E
is uniformly smooth, and hence the norm of
E
is uniformly Fréchet differentiable. In particular, the norm of
E
is Fréchet differentiable.
Recall that, a mapping
T
:
C
→
E
is said to be
L
Lipschitz if for all
x, y
∈
C
, there exists a constant
L
> 0 such that
In particular, if 0 <
L
< 1, then
T
is called contractive and if
L
= 1, then
T
reduces to a nonexpansive mapping.
For some
η
> 0,
T
:
C
→
E
is said to be
η
strongly accretive, if for all
x, y
∈
C
, there exists
η
> 0,
j_{q}
(
x
−
y
) ∈
J_{q}
(
x
−
y
) such that
For some
μ
> 0,
T
:
C
→
E
is said to be
μ
inverse strongly accretive, if for all
x, y
∈
C
there exists
j_{q}
(
x
−
y
) ∈
J_{q}
(
x
−
y
) such that
A setvalued mapping
T
:
D
(
T
) ⊆
E
→ 2
^{E}
is said to be
m
accretive if for any
x, y
∈
D
(
T
), there exists
j
(
x
−
y
) ∈
J
(
x
−
y
), such that for all
u
∈
T
(
x
) and
v
∈
T
(
y
),
A setvalued mapping
T
:
D
(
T
) ⊆
E
→ 2
^{E}
is said to be
m
accretive if
T
is accretive and (
I
+
ρT
)(
D
(
T
)) =
E
for every (equivalently, for some scalar
ρ
> 0), where
I
is the identity mapping.
Let
M
:
D
(
M
) → 2
^{E}
be
m
accretive. Denote by
J_{M,ρ}
the resolvent of
M
for
ρ
> 0:
It is known that
J_{M,ρ}
is a singlevalued and nonexpansive mapping from
E
to
which will be assumed convex (this is so provided
E
is uniformly smooth and uniformly convex).
Let {
T_{n}
} be a family of mappings from a subset
C
of a Banach space
E
into itself with
We say that {
T_{n}
} satisfies the
AKTT
condition (see
[17]
), if for each bounded subset
B
of
C
,
The following proposition supports {
T_{n}
} satisfying AKTTcondition.
Proposition 2.1.
Let C be a nonempty convex subset of a real quniformly smooth Banach space E. Assume that
is a countable family of λ_{i} strict pseudocontractions with
{
λ_{i}
} ⊂ (0, 1)
and
inf{
λ_{i}
:
i
≥ 1} > 0
such that
For each n
∈ ℕ,
define T_{n}
:
C
→
C by
Let
be a family of nonnegative numbers with k
≤
n such that
Then the following results hold:

(1)Each Tnis a λ strict pseudocontraction.

(2)

(3) {Tn}satisfies AKTTcondition.

(4)If T:C→C is defined by
Proof
. (1) and (2) can be deduced directly from Lemma 2.11 in
[20]
. And the argument of (3) and (4) is similar to the section 4 (Applications) in
[17]
and so it is omitted. □
In order to prove our main results, we need the following lemmas.
Lemma 2.2
(
[16]
).
Let C be a closed convex subset of a strictly convex Banach space E. Let T
_{1}
and T
_{2}
be two nonexpansive mappings from C into itself with F
(
T
_{1}
) ∩
F
(
T
_{2}
) ≠ ∅.
Define a mapping S by
where λ is a constant in (0, 1). Then S is nonexpansive and F
(
S
) =
F
(
T
_{1}
) ∩
F
(
T
_{2}
).
Lemma 2.3
(
[19]
).
Let
{
α_{n}
}
be a sequence of nonnegative numbers satisfying the property:
where
{
γ_{n}
}, {
b_{n}
}, {
c_{n}
}
satisfy the restrictions:

(i)

(ii)

(iii) lim supn→∞cn≤ 0.
Then
lim
_{n}
_{→∞}
α_{n}
= 0.
Lemma 2.4
(
[18]
).
Let q
> 1.
Then the following inequality holds:
for arbitrary positive real numbers a, b.
Lemma 2.5
(
[19]
).
Let E be a real quniformly smooth Banach space, then there exists a constant C_{q}
> 0
such that
In particular, if E is a real 2uniformly smooth Banach space, then there exists a best smooth constant K
> 0
such that
Lemma 2.6
(
[17
,
23]
).
Suppose that
{
T_{n}
}
satisfy the AKTTcondition such that

(i) {Tnx}converges strongly to some point in C for each x∈C.

(ii)Furthermore, if the mapping T:C→C is defined by Tx= limn→∞Tnx for all x∈C.
Then
lim
_{n}
_{→∞}
sup
_{ω}
_{∈}
_{B}
∥
Tω
−
T_{n}ω
∥ = 0
for each bounded subset B of C
.
Lemma 2.7
(
[22]
).
Let C be a nonempty convex subset of a real quniformly smooth Banach space E and T
:
C
→
C be a λstrict pseudocontraction. For α
∈ (0, 1),
we define T_{α}x
= (1 −
α
)
x
+
αTx. Then, as α
∈ (0,
μ
],
is nonexpansive such that F
(
T_{α}
) =
F
(
T
).
Lemma 2.8
(
[23]
).
Let C be a nonempty, closed and convex subset of a real quniformly smooth Banach space E which admits weakly sequentially continuous generalized duality mapping j_{q} from E into E
^{∗}
( i.e., if for all
{
x_{n}
} ⊂
E with x_{n}
⇀
x, implies that
Let T
:
C
→
C be a nonexpansive mapping. Then, for all
{
x_{n}
} ⊂
C, if x_{n}
⇀
x and x_{n}
−
Tx_{n}
→ 0,
then x
=
Tx
.
Lemma 2.9
(
[23]
).
Let C be a nonempty, closed and convex subset of a real quniformly smooth Banach space E. Let V
:
C
→
E be a kLipschitzian and ηstrongly accretive operator with constants
and
Then for each
the mapping S
:
C
→
E defined by S
:= (
I
−
tμV
)
is a contraction with a constant
1 −
tτ
.
Lemma 2.10
(
[23]
).
Let C be a nonempty, closed and convex subset of a real quniformly smooth Banach space E. Let Q_{C} be a sunny nonexpansive retraction from E onto C, V
:
C
→
E be a kLipschitzian and ηstrongly accretive operator with constants k, η
> 0,
f
:
C
→
E be a LLipschitzian mapping with constant L
≥ 0
and T
:
C
→
C be a nonexpansive mapping such that F
(
T
) ≠ ∅.
Let
Then the sequence
{
x_{t}
}
defined by
has following properties:

(i) {xt}is bounded for each

(ii) limt→0∥xt−Txt∥ = 0.

(iii) {xt}defines a continuous curve frominto C.
Lemma 2.11
(
[1]
).
Let C be a closed convex subset of a smooth Banach space E. Let
be a nonempty subset of C. Let
be a retraction and let j, j_{q} be the normalized duality mapping and generalized duality mapping on E, respectively. Then the following are equivalent:

(a)Q is sunny and nonexpansive.

(b) ∥Qx−Qy∥2≤ ⟨x−y, j(Qx−Qy)⟩, ∀x,y∈C.

(c) ⟨x−Qx, j(y−Qx)⟩ ≤ 0, ∀x∈C, y∈.

(d) ⟨x−Qx, jq(y−Qx)⟩ ≤ 0, ∀x∈C, y∈.
Lemma 2.12.
Let C be a nonempty, closed and convex subset of a real quniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping j_{q} from E into E
^{∗}
.
Let Q_{C} be a sunny nonexpansive retraction from E onto C, V
:
C
→
E be a kLipschitzian and ηstrongly accretive operator with constants k, η
> 0,
f
:
C
→
E be a LLipschitzian mapping with constant L
≥ 0
and T
:
C
→
C be a nonexpansive mapping such that F
(
T
) ≠ ∅.
Suppose that
Let
{
x_{t}
}
be defined by (2.1) for each
Then
{
x_{t}
}
converges strongly to x
^{∗}
∈
F
(
T
),
which is the unique solution of the following variational inequality:
Proof
. We firstly show the uniqueness of a solution of the variational inequality (2.2). Suppose that both
are solutions of (2.2). It follows that
Adding up (2.3) and (2.4), we have
Notice that
Therefore
and the uniqueness is proved. We use
x
^{∗}
to denote the unique solution of (2.2).
Next, we prove that
x_{t}
→
x
^{∗}
as
t
→ 0.
Since
E
is reflexive and {
x_{t}
} is bounded due to Lemma 2.10 (i), there exists a subsequence {
x_{tn}
} of {
x_{t}
} and some point
such that
By Lemma 2.10 (ii), we have lim
_{t}
_{→0}
∥
x_{tn}
−
Tx_{tn}
∥ = 0. Together with Lemma 2.8, we can get that
Setting
y_{t}
=
tγfx_{t}
+(
I
−
tμV
)
Tx_{t}
, where
Then, we can rewrite (2.1) as
x_{t}
=
QCy_{t}
. We claim that
Thanks to Lemma 2.11, we have that
It follows from (2.5) and Lemma 2.9 that
Thus,
which implies that
Using that the duality map
j_{q}
is weakly sequentially continuous from
E
to
E
^{∗}
and noticing (2.6), we get that
Next, we shall prove that
solves the variational inequality (2.2).
Since
x_{t}
=
Q_{C}y_{t}
=
Q_{C}y_{t}
−
y_{t}
+
tγfx_{t}
+ (
I
−
tμV
)
Tx_{t}
, we derive that
Note that for ∀
z
∈
F
(
T
),
It thus follows from Lemma 2.11 and (2.8) that
where
M
= sup
_{n}
_{≥0}
{
μk
∥
x_{t}
−
z
∥
^{q}
^{−1}
} < ∞. Now replacing
t
in (2.9) with
t_{n}
and letting
n
→ ∞, noticing (2.7) and Lemma 2.10 (ii), we obtain
That is,
is a solution of (2.2); Hence
by uniqueness. Therefore
x_{tn}
→
x
^{∗}
as
n
→ ∞. And consequently,
x_{t}
→
x
^{∗}
as
t
→ 0. □
Lemma 2.13
(
[20]
).
Let C be a nonempty closed convex subset of a real quniformly smooth Banach space E. Let A
:
C
→
E be a αinversestrongly accretive operator. Then the following inequality holds:
In particular, if
then I − λA is nonexpansive.
Lemma 2.14.
Let C be a nonempty closed convex subset of a real quniformly smooth Banach space E. Let M_{i}
:
D
(
M_{i}
) → 2
^{E}
be maccretive with
for i=1,2 and
ρ
_{1}
,
ρ
_{2}
be two arbitrary positive constants. Let A,B
:
C
→
E be αinversestrongly accretive and βinversestrongly accretive, respectively. Let G
:
C
→
C be a mapping defined by
If
then G
:
C
→
C is nonexpansive.
Proof
. We have by Lemma 2.13 that for all
x, y
∈
C
,
which implies that
G
:
C
→
C
is nonexpansive. This completes the proof. □
Lemma 2.15.
Let C be a nonempty closed convex subset of a real quniformly smooth Banach space E. Let M_{i}
:
D
(
M_{i}
) → 2
^{E}
be maccretive with
for i=1,2 and
ρ
_{1}
,
ρ
_{2}
be two arbitrary positive constants. Then
, (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C is a solution of general system (1.7) if and only if x
^{∗}
=
Gx
^{∗}
,
where G is defined by Lemma 2.14.
Proof.
Note that
and the above system is equivalent to
This completes the proof. □
3. Main results
Theorem 3.1.
Let C be a nonempty closed convex subset of a strictly convex and real quniformly smooth Banach space E, which admits a weakly sequentially continuous generalized duality mapping j_{q}
:
E
→
E
^{∗}
.
Let Q_{C} be a sunny nonexpansive retraction from E onto C. Assume A,B
:
C
→
E are αinversestrongly accretive and βinversestrongly accretive, respectively. Let M_{i}
:
D
(
M_{i}
) → 2
^{E}
be maccretive with
Suppose that V
:
C
→
E is kLipschitz and ηstrongly accretive with constants k, η
> 0,
f
:
C
→
E is LLipschitz with constant L
≥ 0.
be an infinite family of λ_{n}strict pseudocontractions with
{
λ_{n}
} ⊂ (0, 1)
and
inf{
λ_{n}
:
n
≥ 0} =
λ
> 0,
such that
and
Define a mapping T_{n}x
:= (1 −
σ
)
x
+
σS_{n}x for all x
∈
C and n
≥ 0.
For arbitrarily given x
_{0}
∈
C and δ
∈ (0, 1),
let
{
x_{n}
}
be the sequence generated iteratively by
Assume that
{
α_{n}
}
and
{
γ_{n}
}
are two sequences in (0,1) satisfying the following conditions:
Suppose in addition that
satisfies the AKTTcondition. Let S
:
C
→
C be the mapping defined by Sx
= lim
_{n}
_{→∞}
S_{n}x for all x
∈
C and suppose that
Then
{
x_{n}
}
converges strongly to x
^{∗}
∈
F, which is the unique solution of the following variational inequality
Proof
.
Step 1.
We show that sequences {
x_{n}
} is bounded. By condition (ii) there is a positive number
b
such that lim sup
_{n}
_{→∞}
γ_{n}
<
b
< 1. Applying condition (i) and (ii), we may assume, without loss of generality, that {
γ_{n}
} ⊂ (0,
b
] and
From Lemma 2.9, we deduce that ∥((1 −
γ_{n}
)
I
−
α_{n}μV
)
x
− ((1 −
γ_{n}
)
I
−
α_{n}μV
)
_{y}
∥ ≤ ((1 −
γ_{n}
) −
α_{n}τ
) ∥
x
−
y
∥ for ∀
x, y
∈
C
. For
x
^{∗}
∈
F
, it follows from Lemma 2.15 that
Putting
y
^{∗}
=
J_{M}
_{2}
,
_{ρ}
_{2}
(
x
^{∗}
−
_{ρ}
_{2}
Bx
^{∗}
), then we can get that
x
^{∗}
=
J_{M}
_{1}
,
_{ρ}
_{1}
(
y
^{∗}
−
_{ρ}
_{1}
Ay
^{∗}
).
By Lemma 2.13, we obtain
It follows from (3.3) that
Combining Lemma 2.7 and the condition of
we can deduce that
Substituting (3.5) into (3.4) and simplifying, we have that
It follows that
Hence, {
x_{n}
} is bounded. {
y_{n}
}, {
k_{n}
} and {
z_{n}
} are also bounded.
Step 2.
We shall claim that ∥
x
_{n}
_{+1}
−
x_{n}
∥ → 0, as
n
→ ∞. We observe that
This together with Lemma 2.7 implies that
At the same time, we observe that
Substituting (3.7) into (3.8), we have that
where
satisfying the
AKTT
condition, we deduce that
From (i), (ii), (3.9), (3.10) and Lemma 2.3, we deduce that
Notice that
which implies that
Combining conditions (i), (ii), (3.11) and (3.12), we deduce that
For any bounded subset
B
of
C
, we observe that
Since {
S_{n}
} satisfies the
AKTT
condition, we have that
That is, {
T_{n}
} satisfies the
AKTT
condition. Define a mapping
T
:
C
→
C
by
Tx
= lim
_{n}
_{→∞}
T_{n}x
for all
x
∈
C
. It follows that
Noticing that
we deduce that
S
:
C
→
C
is a
λ
strict pseudocontraction. In view of (3.14), Lemma 2.6 and the condition of 0 <
σ
≤
d
, where
we have that
T
:
C
→
C
is a nonexpansive and
F
(
T
) =
F
(
S
). Hence we have
Let
W
:
C
→
C
be the mapping defined by
In view of Lemma 2.2, we see that
W
is nonexpansive such that
Noting that
we obtain
From Lemma 2.6, we can get that
Combing (3.13), (3.16) and (3.17), we deduce that
Define
x_{t}
=
Q_{C}
[
tγfx_{t}
+ (
I
−
tμV
)
Wx_{t}
]. From Lemma 2.11, we deduce that {
x_{t}
} converges strongly to
x
^{∗}
∈
F
(
W
) =
F
, which is the unique solution of the variational inequality of (3.2).
Step 3.
We show that
where
x
^{∗}
is the solution of the variational inequality of (3.2). To show this, we take a subsequence {
x_{ni}
} of {
x_{n}
} such that
Without loss of generality, we may further assume that
x_{ni}
⇀
z
for some point
z
∈
C
due to reflexivity of the Banach space
E
and boundness of {
x_{n}
}. It follows from (3.18) and Lemma 2.8 that
z
∈
F
(
W
). Since the Banach space
E
has a weakly sequentially continuous generalized duality mapping
j_{q}
:
E
→
E
^{∗}
, we obtain that
Step 4.
Finally we prove that lim
_{n}
_{→∞}
∥
x_{n}
−
x
^{∗}
∥. Setting
h_{n}
=
α_{n}γfx_{n}
+
γ_{n}x_{n}
+[(1−
γ_{n}
)
I
−
α_{n}μV
]
y_{n}
, ∀
n
≥ 0. Then by (3.1) we can write
x
_{n}
_{+1}
=
Q_{C}h_{n}
. It follows from Lemmas 2.3 and Lemmas 2.10 that
which implies
Put
γ_{n}
=
α_{n}
(
τ
−
γL
) and
Applying Lemma 2.2 to (3.19), we obtain that
x_{n}
→
x
^{∗}
∈
F
as
n
→ ∞. This completes the proof. □
Remark 3.1.
Compared with the known results in the literature, our results are very different from those in the following aspects:

(i) The results in this paper improve and extend corresponding results in[7,8,9,10,11,12,13]. Especially, Our results extend their results from 2uniformly smooth Banach spaces or Hilbert spaces to more generalquniformly smooth Banach spaces.

(ii) Our Theorem 3.1 extends one nonexpansive mapping in Theorem 2.1 in[6]or oneλstrict pseudocontraction in Theorem 3.1 in[8]and an infinitely family of nonexpansive mappings in Theorem 3.1 in[10]to an infinite family ofλistrict pseudocontractions. And our Theorem 3.1 gets a common element of the common fixed point set of an infinite family ofλistrict pseudocontractions and the solution set of general system of variational inclusions for two inverse strongly accretive mappings in aquniformly smooth Banach space.

(iii) We byf(xn) replace theuwhich is a fixed element in iterative scheme (1.8), wherefis aLLipschitzian operator. And we also add a Lipschitzian and strong accretive operatorVin our scheme (3.1). In particular, wheneverour scheme (3.1) reduces to (1.8).

(iv) It is worth noting that, the Banach spaceEdoes not have to be uniformly convex in our Theorem 3.1. However, it is very necessary in Theorem 3.1 of Qin et al.[8]and many other literatures.
Remark 3.2.
The variational inequality problem in a
q
uniformly smooth Banach space
E
: finding
x
^{∗}
such that
is also very interesting and important. As we can see that:

(i) IfM:=C, then it follows from Lemma 2.11 that the variational inequality problem (3.20) is equivalent to a fixed point problem: findx∗∈Csuch that it satisfies the following equation:

whereς> 0 is a constant.

(ii) WhenE:=Hwhich is a real Hilbert space,M:=F(T) andμ= 1, problem (3.20) reduces to findingx∗∈Csuch that

which is the optimality condition for the minimization problem:

whereF(T) is the fixed point set of a nonexpansive mappingTandhis a potential function forγf(i.e.,h′(x) =γf(x) for allx∈H). Furthermore, ifγ= 1,V=Iandf(x) =ufor allx∈C, then problem (3.20) reduces to findingx∗∈F(T) such that

which is equivalent to findingx∗∈F(T) such that
Corollary 3.2.
Let C be a nonempty closed convex subset of a strictly convex and 2uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j
:
E
→
E
^{∗}
.
Let Q_{C} be a sunny nonexpansive retraction from E onto C. Assume the mappings A,B
:
C
→
E are αinversestrongly accretive and βinversestrongly accretive, respectively.Let M_{i}
:
D
(
M_{i}
) → 2
^{E}
be maccretive with
Suppose V
:
C
→
E is a kLipschitzian and ηstrongly accretive operator with constants k, η
> 0,
f
:
C
→
E is a LLipschitzian with constant L
≥ 0.
Let
and
0 ≤
γL
<
τ where τ
=
μ
(
η
−
K
^{2}
μk
^{2}
).
Let T
:
C
→
C be a nonexpansive with F
=
F
(
T
) ∩
F
(
G
) ≠ ∅.
For arbitrarily given δ
∈ (0, 1)
and x
_{0}
∈
C
,
let
{
x_{n}
}
be the sequence generated iteratively by
Assume that
{
α_{n}
}
and
{
γ_{n}
}
are two sequences in (0, 1) satisfying the following conditions:
Then
{
x_{n}
}
defined by (3.21) converges strongly to x
^{∗}
∈
F
,
which is the unique solution of the following variational inequality:
4. Conclusion
In this research, a general iterative algorithm is proposed for finding a common element of the common fixed point set of an infinite family of
λ_{i}
strict pesudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in
q
uniformly smooth Banach spaces. Then we analyzed the strong convergence of the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, and extension of the corresponding results in some references.
BIO
Yonggang Pei received MS degree from Henan Normal University. and Ph.D at Shanghai Normal University. Since 2002 he has been at Henan Normal University as a teacher. His research interests include nonlinear optimization and fixed point theorems.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China.
email: ygpei59@gmail.com
Fujun Liu received MS degree from zhengzhou University. Since 2007 he has been at Henan Institute Engineering college as a teacher. His research interests include numerical methods of differential equations and fixed point theorems.
Faculty of Science, Henan Institute of Engineering, Zhengzhou, 451191, China.
email: liufujun1981@126.com
Qinghui Gao received MS degree from Henan Normal University. and Ph.D from Beihang University. She is currently a teacher at Henan Normal University since 2013. Her research interests are computational biology and iterative method.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China.
email: qinghuigaocb@163.com
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