## 2 Preliminaries  
Let \(\mathcal{K}\) be a nonempty, closed, and convex subset of a real Hilbert space \(\mathcal{H}\). We represent the weak and strong convergence of \(\{p_{k}\}\) to \(p\) by \(p_{k}\rightharpoonup p\) and \(p_{k}\to p\), respectively. For every point \(p\in\mathcal{H}\), the unique nearest point, which is denoted by \(\mathcal{P}_{\mathcal{K}}p\), exists in \(\mathcal{K}\), such that \(\|p-\mathcal{P}_{\mathcal{K}}p\|\leq\|p-q\|\), \(\forall q\in\mathcal{K}\). The mapping \(\mathcal{P}_{\mathcal{K}}\) is called the metric projection of \(\mathcal{H}\) onto \(\mathcal{K}\) and it is known to be nonexpansive. **Lemma 1** ([28]). : _Let \(\mathcal{H}\) be a real Hilbert space and \(\mathcal{K}\) a nonempty closed convex subset of \(\mathcal{H}\). Suppose \(p\in\mathcal{H}\) and \(q\in\mathcal{K}\). Then \(q=P_{\mathcal{K}}p\iff\langle p-q,q-w\rangle\geq 0\), \(\forall w\in\mathcal{K}\)._  
**Lemma 2** ([28]). : _Let \(\mathcal{H}\) be a real Hilbert space. Then for every \(p,q\in\mathcal{H}\) and \(\sigma\in\mathbb{R}\), we have_  
_(i)_ \(\|p+q\|^{2}\leq\|p\|^{2}+2\langle q,p+q\rangle\)_;_  
_(ii)_ \(\|p+q\|^{2}=\|p\|^{2}+2\langle p,q\rangle+\|q\|^{2}\)_;_  
_(iii)_ \(\|\sigma u+(1-\sigma)v\|^{2}=\sigma\|p\|^{2}+(1-\sigma)\|q\|^{2}-\sigma(1- \sigma)\|p-q\|^{2}\)_._  
**Lemma 3** ([29]). : _Let \(\{a_{k}\}\) be a sequence of non-negative real numbers, such that_  
\[a_{k+1}\leq(1-\
u_{k})a_{k}+\
u_{k}b_{k},\ \forall\,k\geq 1,\]  
_where \(\{\
u_{k}\}\subset(0,1)\) with \(\sum_{k=0}^{\infty}\
u_{k}=\infty\). If \(\limsup_{k\to\infty}b_{k}\leq 0\) for every subsequence \(\{a_{k_{j}}\}\) of \(\{a_{k}\}\), the following inequality holds:_  
\[\liminf_{j\to\infty}(a_{k_{j+1}}-a_{k_{j}})\geq 0,\]  
_then \(\lim_{k\to\infty}a_{k}=0\)._  
**Definition 1**. : _Let \(\mathcal{H}\) be a real Hilbert space and \(\mathcal{F}:\mathcal{H}\to\mathcal{H}\) be a mapping. Then, \(\mathcal{F}\) is called (1) \(L\)-Lipschitz continuous, if \(L>0\) exists, such that_  
\[\|\mathcal{F}p-\mathcal{F}q\|\leq L\|p-q\|,\ \forall p,q\in\mathcal{H}.\]  
_If \(L\in[0,1)\), then \(\mathcal{F}\) is a contraction. (2) \(\eta\)-strongly monotone, if there exists a constant \(\eta>0\), such that_  
\[\langle p-q,\mathcal{F}p-\mathcal{F}q\rangle\geq\eta\|p-q\|^{2},\ \forall p,q\in\mathcal{H}.\]  
_(3) \(\eta\)-inverse strongly monotone (\(\eta\)-co-coercive), if there exists a constant \(\eta>0\), such that_  
\[\langle p-q,\mathcal{F}p-\mathcal{F}q\rangle\geq\eta\|\mathcal{F}p-\mathcal{F }q\|^{2},\ \forall p,q\in\mathcal{H}.\]  
_(4) Monotone, if_  
\[\langle\mathcal{F}p-\mathcal{F}q,p-q\rangle\geq 0,\,\forall p,q\in\mathcal{H}.\]  
**Definition 2**. : _Let \(E:\mathcal{H}\to 2^{\mathcal{H}}\) be a multi-valued operator. Then (a) The graph of \(E\) is defined by_  
\[\text{Graph}(E)=\{(p,q)\in\mathcal{H}\times\mathcal{H}:p\in\mathcal{H},\,q\in E (p)\}.\]  
_(b) Operator \(E\) is said to be monotone if_  
\[\langle p-q,y-z\rangle\geq 0,\,\forall y,z\in\mathcal{H},\,p\in E(y),\,q\in E (z).\]  
_(c) Operator \(E\) is said to be maximal monotone if \(E\) is monotone and its graph is not a proper subset of the graph of any of the monotone operators.__._  
_For all_ \(p\in\mathcal{H}\)_, the resolvent of_ \(E\) _is a single-valued mapping_ \(J_{\lambda}^{E}:\mathcal{H}\to\mathcal{H}\) _defined by_  
\[J_{\lambda}^{E}(p)=(I+\lambda E)^{-1}(p),\] _where_ \(\lambda>0\) _and_ \(I\) _is an identity operator on_ \(\mathcal{H}\)_._  
**Lemma 4** ([30]). : _Let \(E:\mathcal{H}\to 2^{\mathcal{H}}\) be a maximal monotone mapping and \(D:\mathcal{H}\to\mathcal{H}\) be a monotone and \(L\)-Lipschitz continuous operator. Then, the mapping \(D+E\) is a maximal monotone mapping._  
**Lemma 5** ([31]). : _Suppose that \(\varrho>0,\alpha\in(0,1)\), and \(\mathcal{F}:\mathcal{H}\to\mathcal{H}\) is \(\eta\)-strongly monotone and \(L_{1}\) continuous, such that \(0<\eta\leq L_{1}\).