## 1 Introduction  
Let \(\mathcal{K}\) be a nonempty, closed, and convex subset of a real Hilbert space \(\mathcal{H}\) with inner product \(\langle\cdot,\cdot\rangle\) and induced norm \(\|\cdot\|\). Let \(\mathcal{F}:\mathcal{K}\rightarrow\mathcal{H}\) be an operator. The classical variational inequality problem (VIP) is derived as follows: Find \(p^{\star}\in\mathcal{K}\), such that  
\[\langle\mathcal{F}p^{\star},q-p^{\star}\rangle\geq 0,\forall q\in\mathcal{K}. \tag{1}\]  
We denote by \(V(\mathcal{K},\mathcal{F})\) the solution set of the VIP (1). Problem (1) has a wide range of applications; several methods for solving this problem have been developed by many researchers (see [1, 2, 3] and the references in them). On the other hand, the monotone inclusion problem (MIP) is formulated as follows: Find \(p^{\star}\in\mathcal{H}\), such that  
\[0\in(D+E)p^{\star}, \tag{2}\]  
where \(\mathcal{H}\) is a real Hilbert space, \(E:\mathcal{H}\rightarrow\mathcal{H}\) is a single-valued monotone operator, and \(D:\mathcal{H}\to 2^{\mathcal{H}}\) is a maximal monotone operator. We denote by \((D+E)^{-1}(0)\) the solution set of the MIP (2); it is referred to as the set of zero points of \(D+E\). Several optimization problems can be reformulated into the MIP (2). Some of these problems include convex minimization problems, equilibrium problems, image/signal processing problems, DC programming problems, split feasibility problems, and variational inequality problems; see [4; 5; 6]. The numerous applications of this problem have attracted the attention of a large number of researchers in the last few years, and many methods for solving the problem have also been developed; see [7; 8; 9]. One of the first methods for solving this problem is the forward-backward algorithm (FBA), which is defined by sequence \(\{p_{k}\}\) as follows:  
\[p_{k+1}=(I+\lambda_{k}E)^{-1}(I-\lambda_{k}D)p_{k}, \tag{3}\]  
where \(\lambda_{k}>0\) is the step size, and \((I-\lambda_{k}D)\) and \((I+\lambda_{k}E)^{-1}\) are denoted as forward and backward operators (also referred to as resolvent operators), respectively. The FBA for solving the MIP was independently studied by [PERSON] and [PERSON] [6], and [PERSON] [10]. In recent years, the convergence analysis and modifications of this method have been deeply exploited by many authors; see [4; 11; 12] and the references in them. We should note that the weak convergence result of method (3) requires the operator (\(D\)) to be strongly monotone; that is a strong assumption. In order to weaken this restriction, several methods have been developed by a large number of researchers; see [6; 11; 13] and the reference therein. One of the first methods considered to weaken this assumption was introduced by [PERSON] [13]. This method is called the Tseng splitting algorithm; it is also known as the forward-backward-forward method. Precisely, this method is defined as follows:  
\[ \begin{cases}p_{1}\in\mathcal{H},\\ q_{k}=(I+\lambda_{k}E)^{-1}(I-\lambda_{k}D)p_{k},\\ p_{k+1}=q_{k}+\lambda_{k}(Dp_{k}-Dq_{k}),\end{cases}  \tag{4}\]  
where \(\{\lambda_{k}\}\) is the step size, which can be updated automatically by the Armijo-type line-search technique. The author proves the weak convergence result of method (4) when operator \(D\) is Lipschitz continuous and monotone, and operator \(E\) is a maximal monotone operator. In [14], [PERSON] and [PERSON] merge the FBA (3) and the projection and contraction method to obtain an iterative method that also surmounts the limitations of the FBA. Precisely, this method is defined as follows:  
\[ \begin{cases}p_{1}\in\mathcal{H},\\ q_{k}=(I+\lambda_{k}E)^{-1}(I-\lambda_{k}D)p_{k},\\ p_{k+1}=p_{k}-\gamma\delta_{k}m_{k},\end{cases}  \tag{5}\]  
where \(m_{k}=p_{k}-q_{k}-\lambda_{k}(Dp_{k}-Dq_{k})\), \(\delta_{k}=\frac{\langle p_{k}-q_{k},m_{k}\rangle}{\|m_{k}\|^{2}}\), \(\gamma\in(0,2)\); \(\{\lambda_{k}\}\) is a control sequence, operator \(D\) is monotone-Lipschitz continuous, and \(E\) denotes the maximal monotone operator.