1. Introduction
In a real Hilbert space
H, one employs
and
to stand for its inner product and norm. Let
be the projection operator from the space
H onto a nonempty convex and closed set
C, where
. Let us denote by
the set of all fixed points of an operator
. The notations →,
, and ⇀ will be used to stand for the strong convergence, the set of real numbers, and the weak convergence, respectively. A self-operator
is named
-strictly pseudocontractive if
such that
In particular, whenever , S is called nonexpansive. This means that the class of nonexpansive mappings is a proper subclass of the one of strict pseudocontractions. Recall that an operator is called
- (i)
Lipschitz with module
L if
such that
- (ii)
monotone if ;
- (iii)
- (iv)
strongly monotone with module
if
s.t.
- (v)
sequentially weakly continuous if .
It is not hard to see that the pseudomonotone operators may not be monotone. In addition, recall that the operator is -strictly pseudocontractive with constant iff the following inequality holds: . It is obvious that if S is a -strict pseudocontraction, then S satisfies Lipschitz condition . For each point , we know that there exists a unique nearest point in C, denoted by , such that . The operator is called the metric projection of H onto C.
Consider an operator
. The classical monotone variational inequality problem (VIP) consists of finding
s.t.
. The solution set of such a VIP is denoted by VI(
). Korpelevich [
1] first designed an extragradient method with two projections
with
, which has been one of the most popular methods for dealing with the VIP up until now. If
, it was shown in [
1] that
weakly converges to a vector in
. The gradient (reduced) type iterative schemes are under the spotlight of investigators of applied mathematicians and engineers in the communities of nonlinear and optimization. Based on this approach, a number of authors have conducted various investigations on efficient iterative algorithms; for examples, see [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11].
Let both the operators
A and
B be inverse-strongly monotone from
C to
H and the self-mapping
be
-strictly pseudocontractive. In 2010, via the extragradient approach, Yao et al. [
12] designed an efficient, fast algorithm for obtaining a feasibility point in a common solution set:
where
is a
-contractive map with
, and
are four sequences in
s.t.
,
,
,
,
, and
. They claimed the strong convergence of the sequence in
H.
In the extragradient approach, one has to compute two projection operators onto
C. It is clear that the projection operator onto the convex set
C is closely related to a minimum distance problem. In the case where
C is a general convex set, the computation of two projections might be prohibitively time-consuming. Via Korpelevich’s extragradient approach, Censor et al. [
13] suggested a subgradient algorithm, in which the second projection operator onto the subset
C is changed onto a half-space. Recently, numerous methods of reduced-gradient-type are focused and extensively investigated in both infinite and infinite dimensional spaces; see, for example [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23]. Based on inertial effects, Thong and Hieu [
24] proposed an inertial subgradient method, and also proved the weak convergence of their algorithms. In addition, the authors [
25] investigated subgradient-based fast algorithms with inertial effects.
Inspired by the above research works in [
12,
24,
25,
26,
27], we are concerned with hybrid-adaptive step-sizes Tseng’s extragradient algorithms, that are more advantageous and more subtle than the above iterative algorithms because they involve solving the VIP with Lipschitzian, pseudomonotone operators, and the common fixed-point problem of a finite family of strict pseudocontractions in Hilbert spaces. By imposing some appropriate weak assumptions on parameters, one obtains a norm solution of the problems, which solves a certain hierarchical variational inequality. The outline of this article is organized below. In
Section 2, a toolbox containing definitions and preliminary results is provided. In
Section 3, we propose and investigate the iterative algorithms and their convergence criteria. In
Section 4, theorems of norm solutions are employed as illustrating examples to support the convergence criteria.
3. Results
From now on, one can always assume that our feasibility set is consistent.
Put
and return to Step 1 (Algorithm 1), where
and
are such that
;
;
;
;
;
;
;
; the pseudomonotone self-operator
A is Lipschitz continuous with module
L and sequentially weakly continuous on
H;
T is a
-strictly pseudocontractive self-operator on
H; and
is a
-contraction operator, where
, from
H to
C.
Algorithm 1: Initial Step: Fix two initials and in H and set . |
Iteration Steps: calculate iterative sequence as follows: Step 1. Given the iterates and , choose s.t. , where Step 2. Let and calculate . Step 3. Calculate , where . Update |
Remark 1. We show. It follows from (1) that. Since, one sees that.
Lemma 5. Letbe generated by (2). Then,is a nonincreasing sequence withand.
Proof. By borrowing (2), one concludes that
. One also has
Note that . So, . □
Lemma 6. Let, andbe three iterative vector sequences defined by Algorithm 1. We havewhere. Proof. Fixing
arbitrarily, one asserts
and
. This yields
Thanks to
, we have
By using the fact that
, one obtains that
. Hence,
Moreover, from (2), it follows that
Combining (4) and (5), we obtain
Using the convexity of the norm function, we get
This completes the proof. □
Lemma 7. Let, andbe three iterative sequences, which are bounded, defined by Algorithm 1. Suppose that there exists a subsequenceof the weakly convergent sequencesuch that. If, then.
Proof. Algorithm 1 shows
. Utilizing Remark 1, we have
. This, together with the assumption
, yields that
Since
is bounded and
, one asserts that
is bounded. Note that (4) yields
Hence,
is bounded, where
. By Algorithm 1, we also get
So, it follows from the boundedness of
and
that
which indicates
tends to 0 as
n tends to the infinity. Using Algorithm 1 again, we get
which immediately leads to
Since
, and
tend to 0 as
n tends to the infinity and
, we obtain
which, together with Lemma 1, ensures that
From the restriction on the operator
A, we have
Using the boundedness of and Lipschitzian property of A, we get the boundedness of . Note that and the boundedness of . Inequality (8) deduces . Borrowing the facts that and A is Lipschitz continuous with moudle L, one concludes that , which combines with (8) and sends us to the situation .
One now focuses on
. Thanks to the weak convergence
, as
, one reaches
. Without loss of generality, we may assume
for all
k. Since by the assumption
we have
for all
, we deduce from (7) that
as
. An application of Lemma 1 is to yield
for all
j. This amounts to
Let
be a decreasing real sequence in
converging to 0 and let
for all
where
is the smallest integer satisfying the above inequality. Note that sequence
is increasing and
. It follows that
where
. This sends us to
, which guarantees
On the other hand, one has
and
as
. This infers
, which lies in
as
k goes to the infinity. So,
as
k goes to the infinity. This shows that
z is not a solution. In the sense of norms, one obtains
. This further concludes that
which reaches that
as
.
Finally, one focuses on the desired point z. (11), the boundedness of sequences and , and the fact that as , yield that for all x in C. Lemma 3 asserts that the desired point z is a solution to the VIP, e.g., . Therefore, we have from (9) that . The proof is complete. □
Theorem 1. Letbe a vector sequence constructed by Algorithm 1 and letbe bounded. Suppose thatis in Ω
, which uniquely solves , . Then, Proof. Noticing condition (iv) on
, one may assume that
, which is a subset of
. Using the Banach Fixed Point Theory, one deduces that a unique point
in
H s.t.
. Hence, there is a solution
to the HVI problem
for any point
x in
. If
, then
and
So, as In order to prove the sufficiency of the theorem, one supposes and . Then, we divide the proof of the sufficiency into several steps. □
Step 1. One proves the boundedness of
. In fact, taking an arbitrary
, one has
and (6), that is,
Since
, there exists an integer
with
From Remark 1, we have
. This ensures that
s.t.
Combining (14), (15), and (16), we have
Note that
is bounded,
,
, and
Hence, we know that
and
are both bounded. From
and
, we conclude that
where
for some
. By using (17), one concludes
which, together with Lemma 2 and
, yields
This indicates that all the vector sequence and are bounded sequences.
Step 2. We claim
for some
. Indeed, using Lemma 2, Lemma 6, and the convexity of
, we have from
that
,
where
for some
. In addition, from (17) we get
where
for some
. Substituting (19) for (18), we obtain that for all
,
where
. This immediately implies that for all
,
Step 3. One proves
where
M is some appropriate constant.
where
. From the convexity of
, one arrives at
which yields that
From (17) and (22) we know that
. Hence, we have,
, that
which immediately yields
Step 4. We claim strong convergence of vector sequence
to the unique solution of HVI (12),
. One lets
, and use (23) to obtain
From (21),
, and
, we obtain
This immediately implies that
. From the Lipschitzian property of
A, we have
. Consequently,
Since
with
, from (25) and the boundedness of
, we get
and hence,
Obviously, combining (25) and (26) guarantees that
which indicates that
Let vector sequence
be a subsequence of original sequence
. From its boundedness, one asserts that
Without loss of generality, one lets
. (28) implies
On the other hand, one has
. This indicates
. Lemma 7 guarantees that
is in
. Therefore, (12) and (29) amount to
Note that
. It follows that
. It is clear that
By utilizing Lemma 4, one concludes easily. The proof is complete.
4. Applications
In this section, our main results are applied to solve the VIP and CFPP in an illustrating example. The initial point is randomly chosen in . Take , and .
We first provide an example of Lipschitzian, pseudomonotone operator A satisfying the boundedness of and strictly pseudocontractive operator with . Let and with the inner product and induced norm . Then, f is a -contractive map with and because for all .
Let
and
be defined as
and
for all
. Now, we first show that
A is Lipschitzian, pseudomonotone operator with
, such that
is bounded. Indeed, for all
, we have
This implies that
A is Lipschitzian operator with
. Next, we verify that
A is pseudomonotone. For any given
, it is clear that the relation holds:
Furthermore, it is easy to see that
is strictly pseudocontractive with constant
. Indeed, we observe that for all
,
It is clear that
for all
. In addition, it is clear that
and
because the derivative
. Therefore,
. In this case, Algorithm 1 can be rewritten below:
where, for each
,
and
are chosen as in Algorithm 1. Then, by Theorem 1, we know that
iff
and
.