In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.
Let be a real Hilbert space with inner product and induced norm . In this paper, we consider the variational inequality problem (VIP) of finding a point such that
where is a nonempty closed convex subset of , and is a nonlinear operator. We denote by the solution set of the VIP (1).
Variational inequality theory, which was first introduced independently by Fichera  and Stampacchia , is a vital tool in mathematical analysis, and has a vast application across several fields of study, such as optimisation theory, engineering, physics, operator theory, economics, and many others (see [3,4, 5,6] and references therein). Over the years, several iterative methods have been formulated and adopted in solving VIP (1) (see [7,8,9, 10,11] and references therein). There are two common approaches to solving the VIP, namely, the regularised methods and the projection methods. These approaches usually require that the nonlinear operator in VIP (1) has certain monotonicity. In this study, we adopt the projection method and consider the case in which the associated nonlinear operator is pseudomonotone (see definition below) – a larger class than monotone mappings.
Now, we review some nonlinear operators in nonlinear analysis.
A mapping is said to be
-strongly monotone on if there exists a constant such that(2)
-inverse strongly monotone on if there exists a constant such that
Monotone on , if(3)
-strongly pseudomonotone on , if there exists a constant such that(4)
Pseudomonotone on , if(5)
Lipschitz-continuous on , if there exists a constant such that(6)
If , then is said to be a contraction mapping.
Sequentially weakly continuous on , if for each sequence ,
From the above definitions, we observe that and . However, the converses are not generally true. Moreover, if is - strongly monotone and - Lipschitz continuous, then is - inverse strongly monotone (see [12,13]).
The simplest known projection method for solving VIP is the gradient method (GM), which involves a single projection onto the feasible set per iteration. However, the algorithm only converges weakly under some strict conditions that the operator is either strongly monotone or inverse strongly monotone, but fails to converge if is monotone. The classical gradient projection algorithm proposed by Sibony  is given as follows:
where is strongly monotone and -Lipschitz continuous, with step size .
Korpelevich  and Antipin  proposed the extragradient method (EGM) for solving VIP (1), thereby relaxing the conditions placed in (7). The initial algorithm proposed by Korpelevich was employed in solving saddle point problems, but was later extended to VIPs in both Euclidean space and infinite dimensional Hilbert spaces. The EGM method is given as follows:
where , is monotone and -Lipschitz continuous, and denotes the metric projection from onto . If the set is nonempty, then the algorithm only converges weakly to an element in .
Over the years, EGM has been of interest to several researchers. Also, many results and variants have been developed from this method, using the assumptions of Lipschitz continuity, monotonicity, and pseudomonotonicity, see [17,18, 19,20] and references therein.
Due to the extensive amount of time required in executing the EGM method, as a result of calculating two projections onto the closed convex set in each iteration, Censor et al.  proposed the subgradient extragradient method (SEGM) in which they replaced the second projection onto by a projection onto a half-space, thus, making computation easier and convergence rate faster. The SEGM is presented as follows:
where . The authors only obtained a weak convergence result for the proposed method. However, they later introduced a hybrid SEGM in  and obtained a strong convergence result. Likewise, Tseng , in the bid to improve on the EGM, proposed Tseng’s extragradient method (TEGM), which only requires one projection per iteration, as follows:
where is monotone, -Lipschitz continuous, and . The TEGM (10) converges to a weak solution of the VIP with the assumption that is nonempty. The TEGM is also known as the forward-backward method. Recently, some authors have carried out some interesting works on the TEGM (see [22,23] and references therein).
In this work, we consider the inertial algorithm, which is a two-step iteration process and a technique for accelerating the speed of convergence of iterative schemes. The inertial extrapolation technique was derived by Polyak  from a dynamic system called the heavy ball with friction. Due to its efficiency, the inertial technique has become a centre of attraction and interest to many researchers in this field. Over the years, researchers have studied the inertial algorithm and applied it to solve different optimisation problems, see [25,26, 27,28] and references therein.
Very recently, Tan and Qin  proposed the following Tseng’s extragradient algorithm for solving pseudomonotone VIP:
where is a contraction and is a pseudomonotone, Lipschitz continuous, and sequentially weakly continuous mapping. The authors proved a strong convergence result for the proposed method under mild conditions on the control parameters.
Another area of interest in this study is the fixed point theory. Let be a nonlinear map. The fixed point problem (FPP) is to find a point (called the fixed point of ) such that
In this work, we denote the set of fixed points of by . Our interest in this study is to find a common element of the fixed point set, , and the solution set of the variational inequality, . That is, the problem of finding a point such that
Many algorithms have been proposed over the years and in recent times for solving the common solution problem (13) (see [30,31,32, 33,34,35, 36,37,38, 39,40] and references therein). Common solution problem of this type has drawn the attention of researchers because of its potential application to mathematical models whose constraints can be expressed as FPP and VIP. This arises in areas like signal processing, image recovery, and network resource allocation. An instance of this is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services is mathematically related (see [37,41,42] and references therein).
Recently, Cai et al.  proposed the following inertial Tseng’s extragradient algorithm for approximating the common solution of pseudomonotone VIP and FPP for nonexpansive mappings in real Hilbert spaces:
where is a contraction, is a nonexpansive mapping, is pseudomonotone, -Lipschitz and sequentially weakly continuous, and . They proved a strong convergence result for the proposed algorithm under some suitable conditions.
One of the major drawbacks of Algorithm (14) is the fact that the step size of the algorithm depends on the Lipschitz constant of the cost operator. In many cases, this Lipschitz constant is unknown or even difficult to estimate. This makes it difficult to implement algorithms of this nature.
Very recently, Thong and Hieu  proposed an iterative scheme for finding a common element of the solution set of monotone variational inequality and set of fixed points of demicontractive mappings as follows:
where is monotone and -Lipschitz continuous, is a demicontractive mapping such that is demiclosed at zero, and is a contraction. The authors proved a strong convergence result under suitable conditions for the proposed method.
Motivated by the above results and the ongoing research activities in this direction, in this paper our aim is to introduce an effective iterative technique, which employs the efficient combination of the inertial technique, TEGM together with the viscosity method for finding a common solution of FPP of demicontractive mappings and pseudomonotone VIP with Lipschitz continuous and sequentially weakly continuous operator in Hilbert spaces. In line with this goal, we construct an algorithm with the following features:
Our algorithm approximates the solution of a more general class of VIP and FPP.
The proposed method only requires one projection per iteration onto the feasible set, which guarantees the minimal cost of computation.
Moreover, our method is computationally efficient. It employs an efficient self-adaptive step size technique which makes the algorithm independent of the Lipschitz constant of the cost operator.
We employ the combination of the inertial technique together with the viscosity method, which are two of the efficient techniques for accelerating the rate of convergence of iterative schemes.
Furthermore, by several numerical experiments, we demonstrate the efficiency of our proposed method over many other existing methods in related literature.
The remainder of this paper is organised as follows. In Section 2, useful definitions and lemmas employed in the study are presented. In Section 3, we present the proposed algorithm and highlight some of its notable features. Section 4 presents the convergence analysis of the proposed method. In Section 5, we carry out some numerical experiments to illustrate the computational advantage of our method over some of the existing methods in the literature. Finally, in Section 6 we give a concluding remark.
Let be a real Hilbert space and be a nonempty closed convex subset of . We denote the weak and strong convergence of sequence to by , as and , as .
The metric projection [46,47], is defined, for each , as the unique element such that
It is a known fact that is nonexpansive, i.e. . Also, the mapping is firmly nonexpansive, i.e.
for all . Some results on the metric projection map are given below.
 Let C be a nonempty closed convex subset of a real Hilbert space H. For any and , Then,
[48,49] Let C be a nonempty, closed, and convex subset of a real Hilbert space H, . Then:
A mapping is said to be
Nonexpansive on , if there exists a constant such that
Quasi-nonexpansive on , if and
-strictly pseudocontractive on with , if
-demicontractive with if
It is known that every strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive. The class of demicontractive mappings includes all the other classes of mappings defined above (see ).
Next, we give some examples of the class of demicontractive mappings, as shown in [23,50].
Let be the real line and . Define on by:
Then is demicontractive.
Consider a mapping defined such that,
We have the following lemmas which will be employed in our convergence analysis.
 For each , and , we have the following results:
 Let be a sequence of nonnegative real numbers, be a sequence in with , and be a sequence of real numbers. Assume that
if for every subsequence of satisfying , then .
 Assume that is a nonlinear operator with . Then, is said to be demiclosed at zero if for any in H, the following implication holds: and .
 Assume that D is a strongly positive bounded linear operator on a Hilbert space H with coefficient and . Then .
 Let be -demicontractive with and set , . Then,
is a closed convex subset of H.
3 Proposed algorithm
In this section, we propose an inertial viscosity-type Tseng’s extragradient algorithm with self adaptive step size and highlight some of its important features. We establish the convergence of the algorithm under the following conditions:
The feasible set is closed, convex, and nonempty.
The solution set denoted by is nonempty.
The mapping is pseudomonotone, -Lipschitz continuous on , and sequentially weakly continuous on .
The mapping is a -demicontractive map such that is demiclosed at zero.
is a strongly positive bounded linear operator with coefficient .
is a contraction with coefficient such that .
such that and .
The positive sequence satisfies for some .
Inertial TEGM with self-adaptive stepsize
Given . Select initial data , and set .
Given the ( )th and nth iterates, choose such that with defined by(16)
If , then set and go to Step 5. Else go to Step 4.
Set and return to Step 1.
Below are some of the interesting features of our proposed algorithm.
Observe that Algorithm 3.1 involves only one projection onto the feasible set per iteration, which makes the algorithm computationally efficient.
The step size in (17) is self-adaptive and supports easy and simple computations, which makes it possible to implement our algorithm without prior knowledge of the Lipschitz constant of the cost operator.
We also point out that in Step 1 of the algorithm, the inertial technique employed can easily be implemented in numerical computation, since the value of is known prior to choosing .
It can easily be seen from (16) and condition (B1) that
4 Convergence analysis
First, we establish some lemmas which will be employed in the convergence analysis of our proposed algorithm.
The sequence generated by (17) is a nonincreasing sequence and .
It follows from (17) that . Hence, is nonincreasing. Also, since is Lipschitz continuous, we have
which implies that
Consequently, we obtain
Combining this together with (17), we obtain
Since is nonincreasing and bounded below, we can conclude that
Let and be two sequences generated by Algorithm 3.1, and suppose that conditions (A1)–(A3) hold. If there exists a subsequence of convergent weakly to and , then
Using the property of the projection map and , we obtain
which implies that
From this we obtain
Since converges weakly to , we have that is bounded. Then, from the Lipschitz continuity of and , we obtain that and are also bounded. Since , from (18) it follows that
Moreover, we have that
Since , then by the Lipschitz continuity of we have . This together with (19) and (20) gives
Now, choose a decreasing sequence of positive numbers such that as . For any , we represent the smallest positive integer with such that:
It is clear that the sequence is increasing since is decreasing. Furthermore, for any , from , we can assume (otherwise, is a solution) and set:
Consequently, we have , for each . From (21), one can easily deduce that
By the pseudomonotonicity of , we have
which implies that
Next, we show that . Indeed, since and , we obtain . Since , we obtain . By the sequentially weakly continuity of on , we have . We can assume that (otherwise, is a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have
By the fact that and as , we obtain
and this implies that . Now, by the facts that is Lipschitz continuous, sequences are bounded and , we conclude from (22) that
Consequently, we have
Thus, by Lemma 2.11, as required.□
Let sequences and be two sequences generated by Algorithm 3.1 such that conditions (A1)–(A3) hold. Then, for all we have
By applying the definition of , we have
Clearly, if , then inequality (25) holds. Otherwise, from (17) we have
It then follows that
Thus, the inequality (25) is valid both when and . Now, from the definition of and applying Lemma 2.6 we have
Since , then by the projection property, we obtain
So, from (25), (26), and (27), we have
Now, from , we have that
Then, by the pseudomonotonicity of , we obtain
Combining (28) and (29), we have that
Moreover, from the definition of and (25), we obtain
which completes the proof.□
Assume conditions and hold. Then, the sequence generated by Algorithm 3.1 converges strongly to an element , where is a solution of the variational inequality
We divide the proof of Theorem 4.4 as follows:
Claim 1. The sequence generated by Algorithm 3.1 is bounded.
First, we show that is a contraction of . For all , we have
It shows that is a contraction. Thus, by the Banach contraction principle there exists an element such that . Next, setting and applying (23) we have
By the condition on , from this we obtain
From Lemma 4.1, we have that
This implies that there exists such that for all . Hence, from (31) we obtain
Also, by definition of and triangle inequality,
From Remark 3.3, we have as . Thus, there exists a constant that satisfies:
So, from (32), (33), and (34) we obtain
Now, by applying Lemma 2.6 and (35), we have
Hence, the sequence is bounded, and so , , are also bounded.
Claim 2. The following inequality holds for all and
Using the Cauchy-Schwartz inequality and Lemma 2.6, we obtain
Now, by applying Lemma 2.6, (30), and (36) we have