1. Introduction
The fixed point problem is a powerful instrument for understanding the fields of chemistry, physics, engineering, and economics in various mathematical models (see [
1,
2,
3,
4]). The fixed point problem is expressed as follows:
where
H is a real Hilbert space, and
is a mapping. The set of fixed points of the mapping
T is denoted by
. To find fixed points of a nonexpansive mapping
T can be performed by using the widest methods, which was proposed by Mann [
5]:
where
C is a nonempty closed convex subset of
H, and
. In [
6], the author showed that if
T has a fixed point and
, then the sequence
generated by Equation (
2) converges weakly to a fixed point of
T.
On the contrary, after a paper by Blum and Oettli [
7] was published, the equilibrium problem began to garner attention and has been utilized for studying a range of mathematical problems, such as variational inequality problems, optimization problems, minimax problems, saddle point problems, and Nash equilibrium problems (see [
7,
8,
9,
10]). The following is a formulation of the equilibrium problem:
where
is a bifunction. The solution set of the equilibrium problem (
3) is represented by
. An acknowledged approach to solve the equilibrium problem (
3) is the proximal point method during which
f is a monotone bifunction (see [
11]). Unfortunately, in this circumstance, if
f is a pseudomonotone bifunction, which is a weaker property than a monotone bifunction, the proximal point method cannot be assured. To maneuver past this obstacle, Tran et al. [
12] presented the following extragradient method to solve the equilibrium problem when
f is a pseudomonotone and Lipschitz-type continuous bifunction:
where
and
are Lipschitz constants of
f, and
. They showed that the sequence
generated by Equation (
4) converges weakly to a solution of the equilibrium problem. It is important to note that when the feasible set
C has a complicated structure, the computing performance of the algorithm is impacted by the extragradient method, a two-step iteration technique. Specifically, each iteration requires solving the optimization problems two times on the feasible set
C in order to find
and
. To gain over this drawback, Hieu [
13] expanded the following subgradient extragradient method to solve the equilibrium problem when
f is a pseudomonotone and Lipschitz-type continuous bifunction:
where
and
are Lipschitz constants of
f,
,
with
such that
. The author showed that the sequence
generated by Equation (
5) converges strongly to
. It is highlighted that in the second step, the optimization problem on the feasible set
C is transformed to the half-space
by the subgradient extragradient method in order to achieve
for each iteration. As a result, by only having the optimization problem solved on the feasible set
C once in order to acquire
, the computing performance of the extragradient method is improved by the subgradient extragradient method. Additionally, the step sizes of the previously stated algorithms are dependent on the values
and
. This indicates that the prior knowledge of the values
and
is necessary for these algorithms. In practical applications, it may be difficult to acquire this kind of data.
At this point, the inertial method, which originates from a discrete version of a second-order dissipative dynamic system [
14,
15], has been extensively studied to accelerate algorithmic convergence properties, for instance, see [
16,
17,
18,
19,
20] and the references therein. The key characteristic of this method is that the previous two iterates are applied to establish the next iterate point. In 2023, by combining the techniques of subgradient extragradient and inertial methods in conjunction with the Mann-type method, Panyanak et al. [
21] presented the following algorithm to solve the equilibrium and fixed point problems, when
T is a
-demicontractive mapping and
f is a pseudomonotone and Lipschitz-type continuous bifunction:
where the parameter
is chosen in
with
and the step size
is given as
when
,
,
,
such that
,
,
,
with
,
, and
. They showed that the sequence
generated by Equation (
6) converges strongly to
. It is apparent that Equation (
6) effectively addresses the unknown information of the Lipschitz constants of
f through the use of the adaptive step size. Furthermore, the adaptive step size criteria are provided, which depend on previously known information for the uncomplicated calculation of updating the step size of each iteration.
In 2016, Dinh et al. [
22] generated the following split equilibrium and fixed point problems:
where
and
are real Hilbert spaces;
and
are nonempty closed convex subsets of
and
, respectively;
and
are mappings;
and
are bifunctions; and
is a bounded linear operator. Dinh et al. [
22] presented the following algorithm by utilizing the extragradient and proximal point methods to solve the split equilibrium and fixed point problems (
7), where
S and
T are nonexpansive mappings,
is a pseudomonotone and Lipschitz-type continuous bifunction, and
is a monotone bifunction:
where
and
are Lipschitz constants of
,
,
,
with
,
such that
, and
, and
is the adjoint operator of
A. They showed that the sequence
generated by Equation (
8) converges weakly to a solution of the split equilibrium and fixed point problems (
7). This kind of problem has attracted a lot of attention due to its broad applications in intensity-modulated radiation therapy, image restoration, network resource allocation, signal processing, phase retrievals, and other real-world applications (see, e.g., [
23,
24,
25]).
Inspired by the above results, Petrot et al. [
26] presented the following algorithm by applying the concept of the extragradient method to solve the split equilibrium and fixed point problems (
7), where
S and
T are nonexpansive mappings and
and
are pseudomonotone and Lipschitz-type continuous bifunctions:
where
and
are Lipschitz constants of
;
and
are Lipschitz constants of
,
with
,
with
,
with
,
with
,
with
such that
; and
h is a
-contraction mapping. They showed that the sequence
generated by Equation (
9) converges strongly to a solution of the split equilibrium and fixed point problems (
7). It is noteworthy that the step size
of this algorithm is determined by the operator norm of the bounded linear operator
A. In order to overcome this drawback, Ezeora et al. [
27] presented the following algorithm by employing the ideas of extragradient and inertial methods to solve the split equilibrium and fixed point problems (
7), where
S and
T are nonexpansive mappings and
and
are pseudomonotone and Lipschitz-type continuous bifunctions:
where the parameter
is chosen in
with
and for
small enough, the step size
is given as
when
and
are Lipschitz constants of
;
and
are Lipschitz constants of
,
,
,
,
with
,
with
,
with
,
such that
, and
; and
h is a
-contraction mapping. They showed that the sequence
generated by Equation (
10) converges strongly to a solution of the split equilibrium and fixed point problems (
7). We observe that, in order to guarantee the convergence of Equation (
10), the maximum values of the scalars
and
depend on the Lipschitz constants of
and
, respectively.
Motivated by the advantageous method presented in Equation (
10) by Ezeora et al. [
27], this paper continues to focus on methods that solve the split equilibrium and fixed point problems (
7). We consider an updated iterative algorithm that does not require prior knowledge of either the Lipschitz constants of the bifunctions or the operator norm of the bounded linear operator. This approach aims to find the minimum-norm solution for the split equilibrium and fixed point problems (
7). Numerical experiments and comparisons with some recently developed algorithms are conducted to evaluate the performance of the introduced algorithm.
This paper is arranged as follows:
Section 2 reviews some preliminary definitions and properties for use later on.
Section 3 presents the Mann-type inertial accelerated subgradient extragradient algorithm and the corresponding strong convergence result. In
Section 4, we discuss the performance of the proposed algorithm in comparison to some recently developed algorithms through numerical experiments. In
Section 5, we close this paper with some conclusions.
2. Preliminaries
We provide some important definitions and qualifications in this section that are applied throughout the work. Let H be a real Hilbert space endowed with inner product , and its corresponding norm . For a sequence in H, we denote the strong convergence and the weak convergence of a sequence to a point by and , respectively. The sets of the real numbers and the natural numbers are represented by and , respectively.
We begin with some definitions and results concerning the nonlinear mappings.
Definition 1. Let be a mapping. The mapping T is called nonexpansive if Remark 1. We notice that if T is a nonexpansive mapping, is closed and convex; see [28]. Definition 2. Let be a mapping. The mapping T is called demiclosed at if for each sequence , , and , we have .
Lemma 1 ([
28]).
Let be a nonexpansive mapping having a fixed point. Then, is demiclosed at 0. In what follows, we collect some fundamental concepts which are needed for the sequels. For each
, it is known that
and
for each
such that
; see [
17].
For a point
, the metric projection of a point
x onto
C is represented by
in the sense of
where
C is a nonempty closed convex subset of
H.
Lemma 2 ([
28,
29]).
Let C be a nonempty closed convex subset of H. Then, the following results hold:- (i)
For each , is well-defined and singleton;
- (ii)
if and only if , ;
- (iii)
is a nonexpansive mapping.
Let
be a function. The subdifferential of
f at
is given by
The function f is called subdifferentiable at y if .
Lemma 3 ([
29]).
For each , the subdifferentiable of a continuous convex function f is a weakly closed and bounded convex set. Lemma 4 ([
9]).
Let C be a convex subset of H. Suppose that is subdifferentiable on C. Then, is a solution to the following convex problem: if and only if , where is the normal cone of C at . The following technical lemmas are necessary to obtain the convergence results.
Lemma 5 ([
30]).
Let and be sequences of non-negative real numbers and be a sequence of real numbers satisfying the following relation: where with . If and , then . Lemma 6 ([
31]).
Let be a sequence of real numbers in which there exists a subsequence of with , for any . Then, there exists a non-decreasing sequence of positive integers with , and the following relations hold: for any (sufficiently large) numbers . Indeed, is the largest number k in the set satisfying the following relation: We end this section by providing some definitions and properties relating to the equilibrium problems.
Definition 3. Let C be a nonempty closed convex subset of H and be a bifunction. The bifunction f is called
- (i)
- (ii)
- (iii)
Lipschitz-type continuous on H if there exists constants and satisfying
Remark 2. Note that a monotone bifunction is a pseudomonotone bifunction. However, in general, the converse is not true; see, e.g., [32]. Let C be a nonempty closed convex subset of a real Hilbert space H and be a bifunction. In this paper, we take into account the following assumptions:
- (A1)
For each fixed , is sequentially weakly upper semicontinuous on C, that is, if is a sequence in C with , then ;
- (A2)
For each fixed , is convex, subdifferentiable, and lower semicontinuous on H;
- (A3)
f is psuedomonotone on C;
- (A4)
f is Lipschitz-type continuous on H.
Remark 3. - (i)
It is well-known that the solution set is closed and convex, where the bifunction f satisfies assumptions ; see [12,33,34]. - (ii)
We observe that , for each , where the bifunction f satisfies assumptions and ; see [17]. - (iii)
For each fixed , the subdifferential of a bifunction at is provided by
3. Main Results
Let
and
be real Hilbert spaces and
and
be nonempty closed convex subsets of
and
, respectively. We start by recalling the split equilibrium and fixed point problems:
where
and
are bifunctions,
and
are mappings,
is a bounded linear operator, and
is the adjoint operator of
A. We observe that the setting of problem (
13) differs from that of problem (
7). Specifically, in problem (
13), the domain of each operator is the entire space, whereas in problem (
7), the domain of the considered operator is a closed convex subset of the corresponding space. Consequently, the fixed point sets of the operators
T and
S in problem (
13) are independent of the sets
and
, respectively. This distinction can lead to different applications and interpretations for problems (
7) and (
13). The solution set of the split equilibrium and fixed point problems (
13) is, henceforth, represented by
in the form of
It is noteworthy that, by using Remark 1, 3 (i), and the linearity property of the operator
A, the solution set
is closed and convex; see [
35].
The following algorithm (Algorithm 1) is presented to solve the split equilibrium and fixed point problems (
13).
Algorithm 1. Mann-type inertial accelerated subgradient extragradient algorithm |
Initialization. Select , , , , , , with , with , with , with , with , with , , such that , , , and , for some positive constants a and b. Put and arbitrarily. |
Step 1. Select satisfying , where
and calculate
|
Step 2. Compute
|
Step 3. Select and create a half-space
where
|
Step 4. Compute
|
Step 5. Calculate
|
Step 6. Compute
|
Step 7. Select and create a half-space
where
|
Step 8. Compute
|
Step 9. Construct by using the following expression:
|
Step 10. Calculate
and
|
Step 11. Set and return to Step 1. |
Remark 4. - (i)
The auxiliary sequences and in Algorithm 1 are introduced to account for bias in the bifunctions and in steps 2 and 6, respectively. These auxiliary biases contribute to the acceleration of Algorithm 1. We highlight that the superior numerical behavior of Algorithm 1 can be affected by choosing sequences and . Take note that the accelerated subgradient extragradient method contained in Algorithm 1 reduces to the situation as presented in [13,21] if and , for each . - (ii)
The self-adaptive step sizes , , and are implemented without requiring prior knowledge of the Lipschitz constants of the bifunctions and or the operator norm of the bounded linear operator A, respectively. This demonstrates that Algorithm 1 automatically updates the iteration step sizes , , and by utilizing some previously existing data. We emphasize that the step sizes , , and in Algorithm 1 reduce to the non-increasing step sizes in the case of , , and for each , as presented in [21,36]. These may have a significant impact on the numerical results; see Section 4 for further discussion. - (iii)
It is important to note that step 9 in Algorithm 1 is improved as the metric projection onto the half-space instead of the metric projection onto the nonempty closed convex subset , as highlighted in [22,26,27]. This is an advantage of Algorithm 1 given that the metric projection onto the half-space has an explicit formula. Meanwhile, the metric projection onto may be difficult if has a complicated structure. - (iv)
In step 3, the nonemptiness of the set is always guaranteed. Indeed, by the definition of and Lemma 4, one sees that Thus, there exist and such that This ensures that there exists a solution within , confirming its nonemptiness. Similarly, in step 7, the nonemptiness of the set is guaranteed by utilizing the definition of and Lemma 4.
The following lemmas are very useful for analyzing the convergence of Algorithm 1.
Lemma 7. Let and be bifunctions that satisfy –, and be nonexpansive mappings, be a bounded linear operator, and be the adjoint operator of A. Let and . Assume that the sequences , , and are generated by (14), (15), and (16), respectively. Then, the following results hold: Proof. The proof of this Lemma follows the technique in (Lemma 3.1 of [
37]). Firstly, we note that
This, together with the definition of
and the assumptions on the parameter
, yields
Thus, the sequence
has a lower bound as
by induction.
Additionally, in view of the definition of , we obtain for each . Then, by induction, the sequence has an upper bound as , where . Consequently, we conclude that is a bounded sequence and .
In what follows, we set
Combining it with the definition of sequence
, we have
Thus, the series
is convergent. Next, we assert the convergence of the series
. Suppose that
. We observe that
This implies that
Then, by taking
in (
17), we have
, as
. This is a contradiction with the boundedness property of
. Owing to the convergence of the series
and
, by taking
in (
17), we deduce that
and
.
On the other hand, from the Lipschitz-type continuity of
on
and of
on
, there exists some positive constants
and
, respectively, such that
and
These, together with the definitions of
and
, and the conditions on the sequences
and
, yield
and
Hence, by induction, we find that the sequences
and
have a lower bound as
and
, respectively. Similar to the above technique, we can show that
and
This completes the proof. □
Lemma 8. Let and be bifunctions that satisfy –, and be nonexpansive mappings, be a bounded linear operator, and be the adjoint operator of A. Assume that the solution set Ω
is nonempty. Let and . If , , , , , and are created by the process of Algorithm 1, then the following results hold: and . Proof. First, we show that
for every
. Let
be fixed and
. Since
, it follows that there exists
such that
So, from
, we obtain
This implies that
. Hence, we conclude that
for every
. Similarly, since
, we can show that
for every
. As a result, we ensure that Algorithm 1 is well-defined.
Afterwards, by utilizing the above facts, we display the results of the Lemma. Let
. That is,
,
, and
,
. Due to
and the subdifferentiability of
, we obtain
It follows from
that
Likewise, by using the definition of
and
, we obtain
Combined with relation (
20), one sees that
Furthermore, from the definition of
and Lemma 4, we obtain
Then, there exists
and
such that
This, together with the subdifferentiability of
, yields
Additionally, from
, we obtain
It follows from equality (
22) that
Due to relation (
23), we obtain
In particular, from
, one sees that
Combined with the pseudomonotonic of
, we have
So, relations (
21) and (
26) imply that
On the other hand, we observe from the definition of
that
Using this together with relation (
27), we obtain
Owing to the above relation, we have the following facts:
This implies that
Hence, by applying the choice of the parameter
, we deduce that
Similarly, we can show that
This completes the proof. □
We are presently equipped to consider the strong convergence of Algorithm 1.
Theorem 1. Let and be bifunctions that satisfy –, and be nonexpansive mappings, be a bounded linear operator, and be the adjoint operator of A. Assume that the solution set Ω is nonempty. Then, the sequence that is generated by Algorithm 1 converges strongly to the minimum-norm element of Ω.
Proof. Let
. That is,
,
, and
,
. We start by regarding the definition of
and using the nonexpansivity of
as follows:
Consider,
Using this one together with relation (
30), we have
It follows from the nonexpansivity of
S and the definition of
that
Combined with the condition on the parameter
and Lemma 7, we obtain
So, there exists
such that
On the other hand, from the assumption of the parameter
, the fact that
, and Lemma 7, we have
Thus, there exists
such that
Moreover, considering the choice of the parameter
, the fact that
, and Lemma 7, we obtain
Then, there exists
such that
Choose
. Let us now consider the case for every
satisfying
. So, by applying relations (
33) and (
34) to Lemma 8, we have the following relations:
and
Using this together with relation (
31), we have
This, together with the assumption on the parameter
and the fact (
32), yields
On the other hand, we note from the definition of
, the nonexpansivity of
T, and relation (
35) that
It follows from the definition of
that
Owing to the condition on the sequence
, we obtain
This, together with the facts
and
, yields
Then, there exists a constant
such that
Combined with relations (
38) and (
39), we have
It follows that
is a bounded sequence. As such, the sequence
is bounded.
On the other hand, in view of the definition of
, one sees that
Furthermore, from the definition of
and (
12), we obtain
Thus, by utilizing relations (
35), (
38), (
42), and (
43), the nonexpansivity of
T, and Lemma 8, we have
Now, we are ready to assert that the sequence converges strongly to by investigating the proof in two cases.
Case 1. Suppose that
, for every
. This means the sequence
is non-increasing. Since the sequence
is bounded, we can confirm that the limit of
exists. Combining with relation (
45) the facts that
and
, and the properties of the sequences
and
, we have
and
These imply that
In addition, from
, we obtain
This, together with (
49), yields
It follows from (
47) that
Consider
So, by using (
48) and the fact that
, we have
Combining it with (
51), we obtain
On the other hand, in view of relation (
37), one sees that
Thus, by utilizing (
51) and (
53), and the existence of
, we have
Furthermore, from Lemma 8 and the nonexpansivity of
S, we obtain
So, by using (
55) and the above relation, we obtain
and
These imply that
Since
, it follows from (
55) and (
58) that
Now, let
and
be a subsequence of
with
, as
. By using (
50)–(
52) and (
54), we have
,
,
, and
, as
. It follows that
, as
. Combining (
56) and (
58), we have
and
as
. Since
and
are closed convex subsets,
and
are weakly closed. Hence,
and
.
Next, due to relations (
21), (
25) and (
28), we obtain
for each
. So, by utilizing the facts (
46), (
47) and (
49), and the boundedness of
, we find that the right-hand side of the inequality (
60) tends to zero. It follows from the sequentially weakly upper semicontinuity of
and
that
Similarly, we can show that
for each
. Using this one together with (
56)–(
58), and the boundedness of
, we find that the right-hand side of the inequality (
61) tends to zero. Thus, by using the sequentially weakly upper semicontinuity of
and
, we have
On the other hand, since
, as
, and considering (
48), by the demiclosedness at zero of
, we have
. Similarly, since
, as
, and considering (
59), it follows from the demiclosedness at zero of
that
. Hence, we can conclude that
.
Put
. Relation (
35) and the nonexpansivity of
T imply that
From the definition of
, one sees that
It follows from (
11) that
Using this together with relations (
38) and (
62), we obtain
Consider
where
. It follows from relation (
64) that
Indeed, from
and the properties of
, we have
Therefore, by using (
40), (
48), (
66), (
67), and Lemma 5, we obtain
Case 2. Suppose that there exists a subsequence
of
satisfying
According to Lemma 6, there exists a non-decreasing sequence
with
, and
Using this one together with relation (
45), we have
Following the line proof of Case 1, we can check that
and
This, together with relation (
69), yields
It follows from relation (
69), again, that
Then, by using (
40), (
72) and (
74), we have
Therefore, the sequence
converges strongly to
. This completes the proof. □