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We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection to the minimization problem of the sum of a convex lower semi-continuous function $f$ and a smooth convex function $g$. For the appropriate setting of the parameters we provide strong convergence of the generated sequence $(x_k)$ to the minimum norm minimizer of our objective function $f+g$. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another settings of the parameters the optimal rate of order $\mathcal{O}(k^{-2})$ for the potential energy $(f+g)(x_k)-\min(f+g)$ can be obtained., Comment: 25 pages. arXiv admin note: text overlap with arXiv:2308.05056

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function $f.$ We show that the generated sequences converge strongly to the minimal norm element from $\argmin f$. We also show that from a practical point of view the Tikhonov regularization does not affect Nesterov's optimal convergence rate of order $\mathcal{O}(n^{-2})$ for the potential energies $f(x_n)-\min f$ and $f(y_n)-\min f$, where $(x_n),\,(y_n)$ are the sequences generated by our algorithm. Further, we obtain fast convergence to zero of the discrete velocity, but also some estimates concerning the value of the gradient of the objective function in the generated sequences., Comment: 17 pages