2 Preliminaries
From now on, we always assume that H is a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H.
Let be a mapping. stands for the fixed point set of S; that is, .
Recall that
S is said to be nonexpansive iff
S is said to be asymptotically nonexpansive iff there exists a sequence
with
such that
Recall that
S is said to be strictly pseudocontractive iff there exits a positive constant
κ such that
S is said to be asymptotically strictly pseudocontractive iff there exits a positive constant
κ and a sequence
with
such that
Let
be a mapping. Recall that
A is said to be monotone iff
A is said to be inverse-strongly monotone iff there exists a constant
such that
For such a case, A is also said to be α-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are Lipschitz continuous.
A multivalued operator
with the domain
and the range
is said to be monotone if for
,
,
and
, we have
. A monotone operator
T is said to be maximal if its graph
is not properly contained in the graph of any other monotone operator. Let
I denote the identity operator on
H and
be a maximal monotone operator. Then we can define, for each
, a nonexpansive single-valued mapping
by
. It is called the
resolvent of
T. We know that
for all
and
is firmly nonexpansive; see [
17‐
23] and the references therein.
Recently, many authors have investigated the solution problems of nonlinear operator equations or inequalities based on iterative methods; see, for instance, [
24‐
33] and the references therein. In [
19], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator via the following iterative algorithm:
(2.1)
where is a sequence in , is a positive sequence, is a maximal monotone and . They showed that the sequence generated in (2.1) converges weakly to some provided that the control sequence satisfies some restrictions.
Recall that the classical variational inequality is to find an
such that
(2.2)
In this paper, we use
to denote the solution set of (2.2). It is known that
is a solution to (2.1) iff
x is a fixed point of the mapping
, where
is a constant,
I stands for the identity mapping, and
stands for the metric projection from
H onto
C. If
A is
α-inverse-strongly monotone and
, then the mapping
is nonexpansive; see [
28] for more details. It follows that
is closed and convex.
In [
28], Takahashi an Toyoda investigated the problem of finding a common solution of variational inequality problem (2.1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
(2.3)
where is a sequence in , is a positive sequence, is a nonexpansive mapping and is an inverse-strongly monotone mapping. They proved that the sequence generated in (2.3) converges weakly to some provided that the control sequence satisfies some restrictions.
In [
29], Tada and Takahashi investigated the problem of finding a common solution of an equilibrium problem and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
(2.4)
for each , where is a sequence in , is a positive sequence, is a nonexpansive mapping and is a bifunction. They showed that the sequence generated in (2.4) converges weakly to some , where stands for the solution set of the equilibrium problem, provided that the control sequence satisfies some restrictions.
In [
30], Manaka and Takahashi introduced the following iteration:
(2.5)
where is a sequence in , is a positive sequence, is a nonexpansive mapping, is an inversely-strongly monotone mapping, is a maximal monotone operator, is the resolvent of B. They showed that the sequence generated in (2.5) converges weakly to some provided that the control sequence satisfies some restrictions.
In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems involving two monotone operators and fixed point problems involving asymptotically strictly pseudocontractive mappings based on a one-step iterative method. Weak convergence theorems are established in the framework of Hilbert spaces.
In order to obtain our main results in this paper, we need the following lemmas.
Recall that a space is said to satisfy Opial’s property [
34] if, for any sequence
with
, where ⇀ denotes the weak convergence, the inequality
holds for every
with
. Indeed, the above inequality is equivalent to the following:
Let C be a nonempty, closed, and convex subset of H, be a mapping, and be a maximal monotone operator. Then .
Lemma 2.2 Let H be a real Hilbert space.
For any and ,
the following holds:
Let ,
,
and be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, and . Then the limit exists.
Let C be a nonempty closed convex subset of H and S be an asymptotically κ-
strictly pseudocontractive mapping.
Then we have (a)
S is uniformly Lipschitz continuous;
(b)
is demiclosed at zero, that is, if is a sequence in C with and , then .
The following lemma can be obtained from [
37] immediately.
Lemma 2.5 Let H be a real Hilbert space.
The following holds:
where denotes some positive integer, are real numbers with in and .
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of H.
Let be some positive integer and be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence .
Let be an inverse-
strongly monotone mapping with the constant and be a maximal monotone operator on H such that the domain of is included in C for each .
Assume .
Let and are real number sequences in .
Let ,
and be positive real number sequences.
Let be a sequence in C generated in the following iterative process:
(3.1)
where is the resolvent of .
Assume that the sequences ,
,
,
and satisfy the following restrictions:
(a)
and , ;
(c)
, ;
(d)
,
where a, b, c, and d are positive real numbers. Then the sequence generated in (3.1) converges weakly to some point in ℱ.
Proof First, we show
is nonexpansive. In view of the restriction (c), we find that
This proves that
is nonexpansive. Let
. In view of Lemma 2.1, we find that
Putting
, we find that
(3.2)
In view of Lemma 2.2, we find from the restriction (b) that
(3.3)
From (3.2) and (3.3), we have
(3.4)
We draw the conclusion that
exists with the aid of Lemma 2.3. This implies that the sequence
is bounded. In view of Lemma 2.5, we find that
(3.5)
which yields
In view of the restriction (a), we find that
(3.6)
On the other hand, we have
(3.7)
It follows that
This in turn implies that
It follows from the restrictions (b) and (d) that
(3.8)
Notice that
It follows that
(3.9)
This implies that
which finds that
In view of the restriction (a), we find from (3.8) that
(3.10)
Notice that
From (3.6) and (3.10), we obtain that
(3.11)
On the other hand, we have
which yields
This implies from the restriction (c) and (3.11) that
(3.12)
Notice that
This implies from (3.10) and (3.11) that
(3.13)
On the other hand, we have
Since
S is uniformly continuous, we obtain from (3.12) and (3.13) that
(3.14)
Since is bounded, there exists a subsequence of such that . We find that with the aid of Lemma 2.4.
Next, we show
for every
. In view of (3.10), we can choose a subsequence
of
such that
. Notice that
This implies that
That is,
Since
is monotone, we get for any
that
(3.15)
Replacing
n by
and letting
, we obtain from (3.10) that
This means , that is, . Hence we get for every . This completes the proof that .
Suppose there is another subsequence
of
such that
. Then we can show that
in the same way. Assume
. Since
exits for any
. Put
. Since the space satisfies Opial’s condition, we see that
This is a contradiction. This shows that . This proves that the sequence converges weakly to . This completes the proof. □
If , then we have the following.
Corollary 3.2 Let C be a nonempty closed convex subset of H.
Let be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence .
Let be an inverse-
strongly monotone mapping with the constant α,
and B be a maximal monotone operator on H such that the domain of B is included in C.
Assume .
Let ,
,
and be real number sequences in .
Let be a positive real number sequence.
Let be a sequence in C generated in the following iterative process:
where is the resolvent of B.
Assume that the sequences ,
,
,
,
and satisfy the following restrictions:
(a)
and , ;
(d)
,
where a, b, c, and d are positive real numbers. Then the sequence converges weakly to some point in ℱ.
If S is asymptotically nonexpansive, then we find from Theorem 3.1 the following by letting .
Corollary 3.3 Let C be a nonempty closed convex subset of H.
Let be some positive integer and be an asymptotically nonexpansive mapping with the sequence .
Let be an inverse-
strongly monotone mapping with the constant and let be a maximal monotone operator on H such that the domain of is included in C for each .
Assume .
Let ,
and be real number sequences in .
Let ,
and be positive real number sequences.
Let be a sequence in C generated in the following iterative process:
where is the resolvent of .
Assume that the sequences ,
,
,
and satisfy the following restrictions:
(a)
and , ;
(b)
, ;
(c)
,
where a, b and c are positive real numbers. Then the sequence converges weakly to some point in ℱ.
If S is the identity mapping, then we draw from Theorem 3.1 the following.
Corollary 3.4 Let C be a nonempty closed convex subset of H.
Let be some positive integer.
Let be an inverse-
strongly monotone mapping with the constant and let be a maximal monotone operator on H such that the domain of is included in C for each .
Assume .
Let ,
and be real number sequences in .
Let ,
and be positive real number sequences.
Let be a sequence in C generated in the following iterative process:
where is the resolvent of .
Assume that the sequences ,
,
and satisfy the following restrictions:
(a)
and , ;
(b)
, ,
where a, b, and c are positive real numbers. Then the sequence converges weakly to some point in ℱ.
Let
be a proper lower semicontinuous convex function. Define the subdifferential
for all
. Then
∂f is a maximal monotone operator of
H into itself; see [
38] for more details. Let
C be a nonempty closed convex subset of
H and
be the indicator function of
C, that is,
Furthermore, we define the normal cone
of
C at
v as follows:
for any
. Then
is a proper lower semicontinuous convex function on
H and
is a maximal monotone operator. Let
for any
and
. From
and
, we get
where is the metric projection from H into C. Similarly, we can get that .
Corollary 3.5 Let C be a nonempty closed convex subset of H.
Let be some positive integer and be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence .
Let be an inverse-
strongly monotone mapping with the constant for each .
Assume .
Let ,
and be real number sequences in .
Let ,
and be positive real number sequences.
Let be a sequence in C generated in the following iterative process:
Assume that the sequences ,
,
,
and satisfy the following restrictions:
(a)
and , ;
(c)
, ;
(d)
,
where a, b, c, and d are positive real numbers. Then the sequence converges weakly to some point in ℱ.
Proof Putting for every , we see . We can immediately draw from Theorem 3.1 the desired conclusion. □
If S is the identity mapping, then we find from Corollary 3.5 the following.
Corollary 3.6 Let C be a nonempty closed convex subset of H.
Let be some positive integer.
Let be an inverse-
strongly monotone mapping with the constant for each .
Assume .
Let ,
and be real number sequences in .
Let ,
and be positive real number sequences.
Let be a sequence in C generated in the following iterative process:
Assume that the sequences ,
,
,
and satisfy the following restrictions:
(a)
and , ;
(b)
, ;
(c)
,
where a, b, and c are positive real numbers. Then the sequence converges weakly to some point in ℱ.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.