## 3 Main Results
If \(w_{k_{j}}\rightharpoonup x^{\star}\in\mathcal{H}\) and \(\lim_{j\to\infty}\|w_{k_{j}}-v_{k_{j}}\|=0\), then \(x^{\star}\in(D+E)^{-1}(0)\)._  
Proof. : The proof is similar to the proof of Lemma 9. **Lemma 14**. : _Let \(\{p_{k}\}\) be the sequence generated by Algorithm 2. Then, \(\{p_{k}\}\) is bounded._  
Proof. : Now, due to the boundedness of \(\{\vartheta_{k}\}\) and Lemma 11, there exists \(K\in\mathbb{N}\), such that \(\vartheta_{k}\bigg{(}1-\frac{(s_{k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}} \bigg{)}>0\), for all \(k\geq K\). This, together with (61), yields  
\[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]2  
The remaining part of the proof is similar to that of Lemma 10. Therefore, we omit it here and this completes the proof of the Lemma. **Theorem 2**. : _Suppose that Assumption 1 holds and \(\{p_{k}\}\) is the sequence defined by Algorithm 2. Then, \(\{p_{k}\}\) converges strongly to the unique solution of problem (8)._  
Proof. : The proof of the theorem will be divided into three steps. **Claim (i):**  
\[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]3 \[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]4  
Indeed, using (29), (30) and (61), we have  
\[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]5 \[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]6 \[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]7  
From (63), it implies that  
\[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]8 \[\|z_{k}-p^{*}\|^{2} \leq (1-\theta_{k})\|w_{k}-p^{*}\|^{2}-\theta_{k}\Bigg{(}1-\frac{(s_{ k}+s)^{2}\lambda_{k}^{2}}{\lambda_{k+1}^{2}}\Bigg{)}\|w_{k}-v_{k}\|^{2} \tag{54}\]9  
**Claim (ii):**  
\[-\theta_{k}(1-\theta_{k})\|h_{k}-w_{k}\|^{2}.\]0 \[-\theta_{k}(1-\theta_{k})\|h_{k}-w_{k}\|^{2}.\]1  
The proof of **Claim (ii)** is similar to that of **Claim 2** of Theorem 1. Therefore, we omit it here. **Claim (iii):** sequence \(\{\|p_{k}-p^{\star}\|^{2}\}\) converges to zero. For this, recalling Lemma 3 and Remark 3, it suffices to show that \(\limsup_{k\to\infty}\langle\mathcal{F}p^{\star},p^{\star}-p_{k+1}\rangle\leq 0\) for every subsequence \(\{\|p_{k_{j}}-p^{\star}\|\}\) of \(\{\|p_{k}-p^{\star}\|\}\), satisfying  
\[-\theta_{k}(1-\theta_{k})\|h_{k}-w_{k}\|^{2}.\]2  
Now, we assume that \(\|p_{k_{j}}-p^{\star}\|^{2}\) is a subsequence of \(\|p_{k}-p^{\star}\|^{2}\), such that (65) holds.