## 4 Applications
### Application to the Bilevel Variational Inequality Problem
If \(\{p_{k}\}\), a sequence generated by Algorithm 4:_```
Initialization: Choose \(\phi>0\), \(\lambda_{1}>0\), \(\varrho\in\left(0,\frac{2\eta}{L_{1}^{2}}\right)\) and \(\{\theta_{k}\}\subset[a,b]\subset(0,1]\). Let \(p_{0}\), \(p_{1}\in\mathcal{H}\) and set \(k=1\). Iterative steps: Calculate the next iteration point \(p_{k+1}\) as follows: Step 1: Choose \(\phi_{k}\), such that \(\phi_{k}\in[0,\tilde{\phi}_{k}]\), where \[\tilde{\phi}_{k}= \begin{cases}\min\Bigl{\{}\frac{k-1}{k+\phi-1},\frac{e_{k}}{ \|p_{k}-p_{k-1}\|}\Bigr{\}},&\text{if }\ p_{k}\
eq p_{k-1},\\ \frac{k-1}{k+\phi-1},&\text{otherwise}.\end{cases} \] Step 2: Set \[w_{k}=p_{k}+\phi_{k}(p_{k}-p_{k-1}),\] and compute \[v_{k}=P_{\mathcal{K}}(I-\lambda_{k}D)w_{k},\] Step 3: Compute \[z_{k}=(1-\theta_{k})w_{k}+\theta_{k}(v_{k}+\lambda_{k}(Ew_{k}-Ev_{k})).\] Step 4: Compute \[p_{k+1}=(I-\alpha_{k}\mathcal{G})z_{k},\ \forall k\geq 1.\]  Update \[\lambda_{k+1}= \begin{cases}\min\Bigl{\{}\frac{(s_{k}+s)\|w_{k}-p_{k}\|}{\|Dw_{k }-Dv_{k}\|},\lambda_{k}+q_{k}\Bigr{\}},&\text{if }\ Dw_{k}\
eq Dv_{k},\\ \lambda_{k}+q_{k},&\text{otherwise}.\end{cases} \]  Put \(k:=k+1\) and return to Step 1. ```

**Algorithm 4** A modified accelerated Tseng splitting method. _Then, sequence \(\{p_{k}\}\) converges strongly to a unique solution of the (BVIP) (71)._

### Application to the Convex Minimization Problem

Let \(h:\mathcal{H}\to\mathbb{R}\) be a convex differentiable function and \(g:\mathcal{H}\to\mathbb{R}\) be a proper lower-semi-continuous and convex function. Then, the convex minimization problem (CMP) is formulated as follows:

\[\text{find }p^{\star}\in\mathcal{H}\text{ such that }h(p^{\star})+g(p^{\star})=\lim_{p\in \mathcal{H}}\{h(u)+g(u)\}. \tag{74}\]

It is well-known that problem (74) is a special case of the MIP; that is, it is equivalent to the problem: \(0\in\tri h(p^{\star})+\partial g(p^{\star})\). It is a known fact that if \(\tri h\) is \(\bigl{(}\frac{1}{L}\bigr{)}\)-Lipschitz continuous, then it is \(L\)-inverse strongly monotone and \(\partial g\) is a maximal monotone operator. The solution set of the CMP (74) is denoted by \((\tri h+\partial g)^{-1}(0)\). In Theorems 1 and 2, we set \(D=\tri h\) and \(E=\partial g\), \(\mathcal{F}(p)=p-f(p)\), where \(f:\mathcal{H}\to\mathcal{H}\) is a \(\gamma\)-contraction mapping. It is not hard to see that \(\mathcal{F}:\mathcal{H}\to\mathcal{H}\) is \((1+\gamma)\) Lipschitz continuous and \((1-\gamma)\)-strongly monotone. Consequently, if we take \(\varrho=1\), then, we obtain the following corollaries from Theorems 1 and 2, respectively. **Corollary 3**. : _Let \(\mathcal{H}\) be a real Hilbert space, \(\triangledown h:\mathcal{H}\rightarrow\mathcal{H}\) be a \(L\)-Lipschitz continuous operator, and \(\partial g:\mathcal{H}\to 2^{\mathcal{H}}\) be a maximal monotone operator. Assume that \((\triangledown h+\partial g)^{-1}(0)\
eq\emptyset\) and that assumptions \((A_{4})\)-\((A_{5})\) hold. If \(\{p_{k}\}\), a sequence generated by Algorithm 5:_

```
Initialization: Choose \(\phi>0\), \(\lambda_{1}>0\), \(0<c_{1}<c_{1}^{\prime}<2\) and \(\gamma\in[0,5-\sqrt{2})\). Let \(p_{0},p_{1}\in\mathcal{H}\) and set \(k=1\). Iterative steps: Calculate the next iteration point \(p_{k+1}\) as follows: Step 1: Choose \(\phi_{k}\), such that \(\phi_{k}\in[0,\bar{\phi}_{k}]\), where \[\bar{\phi}_{k}= \begin{cases}\min\Bigl{\{}\phi,\frac{e_{k}}{\|p_{k}-p_{k-1}\|} \Bigr{\}},&\text{if}\;\;p_{k}\
eq p_{k-1},\\ \phi,&\text{otherwise}.\end{cases} \] Step 2: Compute \[w_{k} =p_{k}+\phi_{k}(p_{k}-p_{k-1}),\] \[v_{k} =(I+\lambda_{k}\partial g)^{-1}(w_{k}-\lambda_{k}\triangledown hw _{k}).\] Step 3: Compute \[z_{k}=w_{m}-m_{k}r_{k},\] where \[r_{k}=w_{k}-v_{k}-\lambda_{k}(\triangledown hw_{k}-\triangledown hv _{k})\] and \[m_{k}= \begin{cases}(c_{1}+t_{k})\frac{(p_{k}-p_{k}r_{k})}{\|r_{k}\|^{2}},& \text{if}\;r_{k}\
eq 0,\\ 0,&\text{otherwise}.\end{cases} \] Step 4: Compute \[p_{k+1}=(I-\alpha_{k})z_{k}+\alpha_{k}f(z_{k}),\;\forall k\geq 1.\]  Update \[\lambda_{k+1}=