  We investigate the strong and the weak convergence properties of the
following gradient projection algorithm with Tikhonov regularizing term \[
x_{n+1}=P_{Q}(x_{n}-\gamma_{n}\nabla f(x_{n})-\gamma_{n}\alpha_{n}\nabla \phi
(x_{n})), \] where $P_{Q}$ is the projection operator from a Hilbert space
$\mathcal{H}$ onto a given nonempty, closed and convex subset $Q,$
$f:\mathcal{H}% \rightarrow \mathbb{R}$ a regular convex function, $\phi
:\mathcal{H}% \rightarrow \mathbb{R}$ a regular strongly convex function, and
$\gamma_{n}$ and $\alpha_{n}$ are positive real numbers. Following a Lyuapunov
approach inspired essentially from the paper [Comminetti R, Peypouquet J Sorin
S. Strong asymptotic convergence of evolution equations governed by maximal
monotone operators with Tikhonov regularization. J. Differential Equations.
(2001); 245:3753-3763], we establish the strong convergence of $(x_{n})_{n}$ to
a particular minimizer $x^{\ast }$ of $f$ on $Q$ under some simple and natural
conditions on the objective function $f$\ and the sequences $(\gamma_{n})_{n}$
and $(\alpha_{n})_{n}$
