Combining Strong Convergence, Values Fast Convergence and Vanishing of Gradients for a Proximal Point Algorithm Using Tikhonov Regularization in a Hilbert Space

In a real Hilbert space $\mathcal{H}$. Given any function $f$ convex differentiable whose solution set ${\mathrm{arg\hspace{0.17em}min}}_{\mathcal{H}}f$ is nonempty, by considering the Proximal Algorithm $x_{k+1}={\text{prox}}_{{\text{b}}_{k}f}(d{x}_k)$, where $0<d<1$ and $({\text{b}}_k)$ is nondecreasing function, and by assuming some assumptions on $({\text{b}}_k)$, we will show that the value of the objective function in the sequence generated by our algorithm converges in order $\mathcal{O}\left(\frac{1}{{\beta }_k}\right)$ to the global minimum of the objective function, and that the generated sequence converges strongly to the minimum norm element of ${\mathrm{arg\hspace{0.17em}min}}_{\mathcal{H}}f$, we also obtain a convergence rate of gradient toward zero. Afterward, we extend these results to non-smooth convex functions with extended real values.