Abstract
In this paper, we continue to investigate the convergence analysis of Tsengtype forwardbackwardforward algorithms for solving quasimonotone variational inequalities in Hilbert spaces. We use a selfadaptive technique to update the step sizes without prior knowledge of the Lipschitz constant of quasimonotone operators. Furthermore, we weaken the sequential weak continuity of quasimonotone operators to a weaker condition. Under some mild assumptions, we prove that Tsengtype forwardbackwardforward algorithm converges weakly to a solution of quasimonotone variational inequalities.
1. Introduction
Let be a real Hilbert space endowed with inner product and corresponding norm . Let be a nonempty closed and convex subset of . Let be an operator. Our purpose of this paper is to investigate the following Stampacchiatype variational inequality (shortly, VI).
Find such that
Denote the solution set of (1) by .
Variational inequality problem (1) was introduced by Stampacchia [1] in 1964. Now it is wellknown that variational inequality problem (1) provides a natural, convenient, and unified framework for the study of a large number of problems in economics, operation research, and engineering (see [2–5]). Variational inequality (1) contains, as special cases, such wellknown problems in mathematical programming as systems of nonlinear equations, optimization problems ([3, 6]), complementarity problems ([7–9], and fixedpoint problems ([10–20]). Many iterative algorithms for solving variational inequalities and related problems have been proposed and investigated (see, for example, [1, 6, 9, 16, 21–40]). Among them, one of the influential algorithms for solving ) is the projectiongradient algorithm ([28, 39, 40]) which defines a sequence by where is the orthogonal projection operator onto and is the step size.
The projectiongradient algorithm guarantees the convergence of the sequence defined by (2) if is strongly (pseudo)monotone (see [8, 41]) or is inverse strongly monotone (see [3, 42]). However, if is plain monotone, then the sequence generated by (2) does not necessarily converge. Consequently, Korpelevich [43] proposed an extragradient algorithm which generates a sequence by
This algorithm guarantees the convergence of the sequence defined by (3) if is pseudomonotone. Since then, Korpelevich’s algorithm has attracted so much attention by many scholars, who modified it in several different forms (see, e.g., [34, 44–47]). Especially, Vuong [31] proved that Korpelevich’s extragradient method has weak convergence provided that is sequentially weakly continuous and pseudomonotone.
A challenging task when devise efficient algorithms for solving variational inequalities is to avoid to compute the projection operators at each iteration because the computation of the projection operator may be very expensive. In this respect, Tseng [30] modified extragradient algorithm with the following form:
Boţ et al. [48] approach the solution of from a continuous perspective by means of trajectories generated by the following dynamical system of forwardbackwardforward type: where and .
Note that (5) has its roots and the existence and uniqueness of the trajectory generated by (5) has been obtained (see [49]). The explicit time discretization of the dynamical system (5) yields the following Tsengtype forwardbackwardforward algorithm:
Bot et al. ([48]) proved that the sequence generated by (6) converges weakly to an element in provided is pseudomonotone and sequentially weakly continuous. On the other hand, for solving (1) and related problems, some selftechniques have been used to relax the step size without prior knowledge of the Lipschitz constant of the operator (see [50–53]).
Let be the solution set of the dual variational inequality of (1), that is,
Note that is closed convex. If is convex and is continuous, then .
To prove the convergence of the sequence , a common assumption has been used, that is, which is a direct consequence of the pseudomonotonicity of . But this conclusion (that is, ) is false, if is quasimonotone.
In this paper, we introduce a selfadaptive Tsengtype forwardbackwardforward algorithm to solve quasimonotone variational inequalities (1). The algorithm is designed such that the step sizes are dynamically chosen and its convergence is guaranteed without prior knowledge of the Lipschitz constant of . Moreover, we replace the sequential weak continuity imposed on by a weaker condition. We show that the proposed algorithm converges weakly to a solution of quasimonotone variational inequalities under some additional conditions.
2. Preliminaries
Let be a nonempty convex and closed subset of a real Hilbert space . Use “” and “” to denote weak convergence and strong convergence, respectively. Let be an operator. Recall that is said to be (i)strongly monotone if there exists a positive constant such that (ii)inverse strongly monotone if there exists a positive constant such that (iii)monotone if (iv)pseudomonotone if (v)quasimonotone if
It is easy to see that .
But the reverse assertions are not true in general.
Example 1 (see [50]). Let and . Let be defined by for all , where . Then, is pseudomonotone on . But is not monotone on .
Example 2 (see [33]). The function defined by is quasimonotone on , but not pseudomonotone on .
An operator is said to be Lipschitz continuous if there exists a positive constant such that
If , then is said to be nonexpansive.
An operator is said to be sequentially weakly continuous if for given sequence : implies that .
For , there exists a unique point in , denoted by satisfying
Moreover, has the following property:
3. Main Results
In this section, we present our main results.
Let be a real Hilbert space and a nonempty closed convex set. Assume that the following conditions are satisfied:
(C1) The operator is quasimonotone on .
(C2) The operator is Lipschitz continuous on
(C3) and is a finite set.
Assume that the operator possesses the following property: for any given sequence ,
Remark 1. If the operator is sequentially weakly continuous, then satisfies the above property (17).
Next, we propose a selfadaptive Tsengtype forwardbackwardforward algorithm for solving the quasimonotone variational inequality (1).

Remark 2. If , that is, , then . In what follows, we assume that . In this case, we can obtain an infinite sequence generated by Algorithm 1.
Remark 3. According to the definition (3.4) of , is monotonically decreasing and therefore converges. Set . It is obvious that .
Next, we prove the convergence of the sequence generated by Algorithm 1.
Theorem 4. Suppose that the conditions (C1)(C3) and (17) are satisfied. Then, the sequence generated by Algorithm 1 converges weakly to a point in .
Proof. Let . Set . Then, we have
Since , from (16) and (3.2), we achieve . It follows that
Using and , we obtain
By (18), (19), and (20), we receive
From (3.4), we have . This together with (21) implies that
Note that . So, there exists an integer such that when . Hence, from (22), we deduce when .
In terms of (3.3), we get
Thus, the sequence is monotonically decreasing and exists. So, the sequence is bounded.
By virtue of (22) and (23), we have
It follows that
Since , , and exist, it follows from (25) that
Since is Lipschitz, from (26), we obtain
Thanks to (3.3), we derive
Based on (26)–(28), we deduce
According to (16) and (3.2), we have
It follows that
Since is bounded, by (26), is also bounded. At the same time, using the Lipschitz continuity of , is bounded. Combining (26), (27), and (31), we attain
Since is bounded, there exists a subsequence of such that as . By virtue of (32), we have
Next, we consider two possible cases.
Case 1. . Since and satisfies (17), we deduce that .
Case 2. In this case, such that for all . From (33), we obtain
Choose a positive strictly decreasing sequence such that as . Thanks to (34), there exists a strictly increasing subsequence with the property that and
It follows that
Set for all . Thus, we have for each . From (36), we deduce
Since is quasimonotone on , by (37), we get
Observe that . Since is Lipschitz continuous, as . Thus, taking the limit as in (38), we obtain that
So, .
Next, we prove has finite weak cluster points in . First, we show that has at most one weak cluster point in . Let and be two distinct weak cluster points of . There exist two sequences and of satisfying as and as . Note that for all ,
Since and exist, by (40), we conclude that exists, denoted by . Thus,
Since and , from (41), we have
which implies that and hence, . Therefore, has at most one weak cluster point in . By the condition (C3), is a finite set. Therefore, has finite weak cluster points in .
Let be the finite weak cluster points of in . Set and
Taking any weak cluster point , there exists a subsequence of such that as . Then, we have
Observe that ,
According to (44) and (45), there exists a large enough positive integer such that when ,
Set
In the light of (46) and (47), we have when .
Now, we show that for a large enough . Assume that there exists a subsequence of such that . By the boundedness of , there exists a subsequence of convergent weakly to . Without loss of generality, we still denote the subsequence as . According to assumptions, , so for any . Therefore, there exists a subsequence of such that ,
Thus,
which implies that . This is impossible. So, for a large enough positive integer , when .
Next, we show that has a unique weak cluster point in . First, from (29), there exists a positive integer such that for all . Assume that has at least two weak cluster points in . Then, there exists such that and , where and , that is,
Therefore,
Combining (51) and (52), we achieve
At the same time, we have
Based on (53) and (54), we deduce
This leads to a contradiction. Thus, has a unique weak cluster point in . Therefore, converges weakly to a point in . This completes the proof.☐

Corollary 5. Suppose that the conditions (C1)(C3) and (17) are satisfied. Then, the sequence generated by Algorithm 2 converges weakly to a point in .
Remark 6. If is pseudomonotone, then Theorem 4 and Corollary 5 hold.
Remark 7. If the operator is sequentially weakly continuous and also satisfies conditions (C1)(C3), then Theorem 4 and Corollary 5 still hold.
Remark 8. Our main purpose is to solve (1); hence, a natural condition is . In order to prove our main theorem, we assume that . Note that . This means that even if , does not necessarily hold. A question is under what conditions holds. In fact, we have the following results: (i)If is pseudomonotone on and , then (ii)If is quasimonotone on , and , then (see [35])
Data Availability
No data were used to support this study.
Conflicts of Interest
All authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.