## 4 Applications
### Application to the Bilevel Variational Inequality Problem  
Let \(\mathcal{H}\) be a real Hilbert space and \(\mathcal{K}\) be a nonempty closed convex subset of \(\mathcal{H}\). Let \(D,\mathcal{F}:\mathcal{H}\to\mathcal{H}\) be two single-valued operators. Then, the bilevel variational inequality problem (BVIP) is formulated as follows:  
\[\text{find }p^{\star}\in VI(\mathcal{K},D)\text{ such that }\langle\mathcal{F}p^{\star},q-p^{\star}\rangle\geq 0, \ \forall q\in VI(\mathcal{K},D), \tag{71}\]  
where \(VI(\mathcal{K},D)\) denotes the solution set of the variational inequality problem (VIP):  
\[\text{find }p^{\star}\in\mathcal{K}\text{ such that }\langle Dp^{\star},w-p^{ \star}\rangle\geq 0,\ \forall w\in\mathcal{K}. \tag{72}\]  
Assume that \(VI(\mathcal{K},D)\
eq\emptyset\), the VIP (72) is equivalent to the following inclusion problem:  
\[\text{find }p^{\star}\in\mathcal{K}\text{ such that }0\in(D+E)p^{\star}, \tag{73}\]  
where \(E:\mathcal{H}\to 2^{\mathcal{H}}\) is the sub-differential of the indicator function and it is a maximal monotone operator [34]. In this case, according to [35], the resolvent of \(E\) is the metric projection \(P_{\mathcal{K}}\); that is, \((I+\lambda_{k}E)^{-1}(p)\). Thus, the following corollaries follow immediately from Theorem 1 and Theorem 2, respectively:  
**Corollary 1**. : _Let \(\mathcal{H}\) be a real Hilbert space and \(\mathcal{K}\) a nonempty closed convex subset of \(\mathcal{H}\); let \(D:\mathcal{H}\to\mathcal{H}\) be a monotone and \(L\)-Lipschitz continuous operator, \(\mathcal{F}:\mathcal{H}\to\mathcal{H}\) be a strongly monotone and \(L_{1}\) monotone operator, and \(P_{\mathcal{K}}:\mathcal{H}\to 2^{\mathcal{H}}\) be a maximal monotone operator. Assumethat \(\{p^{\star}\in VI(\mathcal{K},D):\langle\mathcal{F}p^{\star},q-p^{\star}\rangle \geq 0,\ \forall q\in VI(\mathcal{K},D)\}\
eq\emptyset\) and that assumptions \((A_{4})\)-\((A_{5})\) hold. If \(\{p_{k}\}\), a sequence generated by Algorithm 3:_  
```
Initialization: Choose \(\phi>0\), \(\lambda_{1}>0\), \(0<c_{1}<c_{1}^{\prime}<2\) and \(\varrho\in\left(0,\frac{2\eta}{L_{1}^{2}}\right)\). 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 \[\tilde{\phi}_{k}= \begin{cases}\min\Big{\{}\phi,\frac{c_{k}}{\|p_{k}-p_{k-1} \|}\Big{\}},&\text{if }\ p_{k}\
eq p_{k-1},\\ \phi,&\text{otherwise}.\end{cases} \] Step 2: Compute \[w_{k} =p_{k}+\phi_{k}(p_{k}-p_{k-1}),\] \[v_{k} =P_{\mathcal{K}}(w_{k}-\lambda_{k}Dw_{k}).\] Step 3: Compute \[z_{k}=w_{m}-m_{k}r_{k},\] where \[r_{k}=w_{k}-v_{k}-\lambda_{k}(Dw_{k}-Dv_{k})\] and \[m_{k}= \begin{cases}(c_{1}+t_{k})\frac{\langle w_{k}-v_{k},r_{k}\rangle}{\|r_{k }\|^{2}},&\text{ if }r_{k}\
eq 0,\\ 0,&\text{ otherwise}.\end{cases} \] Step 4: Compute \[p_{k+1}=(I-\alpha_{k}\varrho\mathcal{F})z_{k},\ \forall k\geq 1.\] Update \[\lambda_{k+1}= \begin{cases}\min\Big{\{}\frac{(s_{k}+s)\|w_{k}-v_{k}\|}{\|Dw_{k} -Dv_{k}\|},\lambda_{k}+q_{k}\Big{\}},&\text{if }\ Dw_{k}\
eq Dv_{k},\\ \lambda_{k}+q_{k},&\text{otherwise}.\end{cases} \]  Put \(k:=k+1\) and return to Step 1. ```  
**Algorithm 3** A modified accelerated projection and contraction method. _Then, sequence \(\{p_{k}\}\) converges strongly to a unique solution of the (BVIP) (71)._  
**Corollary 2**. : _Let \(\mathcal{H}\) be a real Hilbert space and \(\mathcal{K}\) be a nonempty closed convex subset of \(\mathcal{H}\); let \(D:\mathcal{H}\to\mathcal{H}\) be a monotone and \(L\)-Lipschitz continuous operator, \(\mathcal{F}:\mathcal{H}\to\mathcal{H}\) be a strongly monotone and \(L_{1}\) monotone operator, and \(P_{\mathcal{K}}:\mathcal{H}\to 2^{\mathcal{H}}\) be a maximal monotone operator. Assume that \(\{p^{\star}\in VI(\mathcal{K},D):\langle\mathcal{F}p^{\star},q-p^{\star} \rangle\geq 0,\ \forall q\in VI(\mathcal{K},D)\}\
eq\emptyset\) and that assumptions \((A_{4})\)-\((A_{5})\) hold.