## 5 Numerical Experiments  
In this section, we present some numerical experiments to illustrate the numerical behavior of Algorithms 1 (in short, OAURN1) and 2 (in short, OAURN2). Moreover, we compare them with Algorithm 3.3 by [PERSON] et al. [36] (in short, ADIA Alg. 3.3) and Algorithm 3.1 by [PERSON] et al. [15] (in short, AOM Alg. 3.1). We choose the parameters of all the methods as follows:  
* In the proposed Algorithms 1 and 2, we set \(\phi=0.73\), \(\lambda_{1}=2.5\), \(s=0.59\), \(c_{1}=0.67\), \(\hat{\sigma}_{k}=0.89\), \(\alpha_{k}=\frac{1}{2k+1}\), \(\epsilon_{k}=\frac{1}{(2k+1)^{3}}\), \(s_{k}=\frac{1}{k+1}\), \(\varrho=\frac{1.7\eta}{L_{1}^{2}}\) and \(q_{k}=\frac{1}{(k+1)^{1.1}}\). * In ADIA Alg. 3.3, we set \(\tau_{k}=\frac{1}{2k+1}\), \(\sigma_{k}=\mu_{k}=0.5(1-\tau_{k})\), \(\epsilon_{k}=\gamma_{k}=s_{k}=\frac{1}{(2k+1)^{2}}\), \(u=\frac{1}{2}\), \(\lambda_{k}=\frac{1}{6}\) and \(a=0.9\). * In AOM Alg. 3.1, we set \(\alpha_{k}=\frac{1}{2k+1}\), \(\delta_{k}=\xi_{k}=0.5(1-\alpha_{k})\), \(\epsilon_{k}=\frac{1}{(2k+1)^{3}}\), \(\theta=0.73\), \(\lambda_{1}=2.5\), \(\phi=0.59\), \(\alpha=0.145\), \(\beta=0.465\), \(Sp=\frac{2}{3}p\), \(Tp=\frac{2}{3}p\), \(f(p)=\frac{1}{2}p\) and \(\phi_{k}=\frac{1}{(k+1)^{1.1}}\). **Example 1**. : _Let \(\mathcal{H}=L_{2}([0,1])\) and let the operators \(D,E,F:\mathcal{H}\rightarrow\mathcal{H}\) be defined by_  
\[D(p)=3p(t),\ \ E(p)=6p(t),\ \ \text{and}\ \ \mathcal{F}(p)=0.5p(t),\ \forall t \in[0,1].\]  
_It is not hard to verify that \(D\) is \(\frac{1}{2}\)-inverse strongly monotone, \(E\) is a maximal monotone operator, and \(\mathcal{F}\) is \(0.5\)-strongly monotone and \(0.5\)-Lipschitz continuous. For this experiment, we take the stopping \(E_{k}=\|p_{k+1}-p_{k}\|<10^{-5}\) and consider the following cases:_  
_Case I: \(p_{0}(t)=t\) and \(p_{1}(t)=1+t^{2}\);_  
_Case II: \(p_{0}(t)=2s\) and \(p_{1}(t)=\sin(t)\);_  
_Case III: \(p_{0}(t)=t^{3}+t\) and \(p_{1}(t)=t^{3}+3t\),_  
_Case IV: \(p_{0}(t)=t+2\) and \(p_{1}(t)=\cos(t)\)._  
Figure 4: Graphs of SNR for the methods OAURN1, OAURN2, ADIA Alg. 3.3, and AOM Alg 3.1 for the Hand X-ray image. _The obtained numerical results are presented in the following Table 2 and Figure 5. It can be seen that our method outperforms the compared methods._  
**Example 2**. : _Let \(H=(\ell_{2}(\mathbb{R}),\|\cdot\|_{\ell_{2}})\), where \(\ell_{2}(\mathbb{R})=\{p=(p_{1},p_{2},p_{3},\cdots),p_{j}\in\mathbb{R}:\sum_{j= 1}^{\infty}|u_{j}|<\infty\}\) and \(\|p\|_{\ell_{2}}=(\sum_{j=1}^{\infty}|p_{j}|^{2})^{\frac{1}{2}}\), \(\forall p\in\ell_{2}(\mathbb{R})\). We now define the operators \(D,E,\mathcal{F}:\ell_{2}(\mathbb{R})\rightarrow\ell_{2}(\mathbb{R})\) by_  
\[Dp=0.5p,\ \ Ep=8p,\ \ \text{and}\ \ \mathcal{F}(p)=0.8p,\ \ \forall p\in\mathcal{H}.\]  
_It is easy to check that \(D\) is 2-inverse strongly monotone, \(E\) is a maximal monotone operator, and \(\mathcal{F}\) is 0.8-strongly monotone and 0.8-Lipschitz continuous. For this experiment, we take the stopping \(E_{k}=\|p_{k+1}-p_{k}\|<10^{-8}\) and consider the following cases:_  
_Case A: \(p_{0}=\left(\frac{1}{4},\frac{1}{8},\frac{1}{9},\cdots\right)\) and \(p_{1}=\left(1,\frac{1}{2},\frac{1}{3},\cdots\right)\)._  
_Case B: \(p_{0}=\left(\frac{1}{5},\frac{1}{7},\frac{1}{10},\cdots\right)\) and \(p_{1}=\left(1,\frac{1}{3},\frac{1}{5},\cdots\right)\)._  
_Case C: \(p_{0}=\left(1,\frac{1}{8},\frac{1}{10},\cdots\right)\) and \(p_{1}=\left(\frac{1}{6},\frac{1}{5},\frac{1}{5},\cdots\right)\)._