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In a real Hilbert space $\mathcal{H}$. Given any function $f$ convexdifferentiable whose solution set $\argmin_{\mathcal{H}}\,f$ is nonempty, byconsidering the Proximal Algorithm $x_{k+1}=\text{prox}_{\b_k f}(d x_k)$, where$0<d<1$ and $(\b_k)$ is nondecreasing function, and by assuming someassumptions on $(\b_k)$, we will show that the value of the objective functionin the sequence generated by our algorithm converges in order $\mathcal{O}\left( \frac{1}{ \beta _k} \right)$ to the global minimum of the objectivefunction, and that the generated sequence converges strongly to the minimumnorm element of $\argmin_{\mathcal{H}}\,f$, we also obtain a convergence rateof gradient toward zero. Afterward, we extend these results to non-smoothconvex functions with extended real values.

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