  This work focuses on convergence analysis of the projected gradient method
for solving constrained convex minimization problem in Hilbert spaces. We show
that the sequence of points generated by the method employing the Armijo
linesearch converges weakly to a solution of the considered convex optimization
problem. Weak convergence is established by assuming convexity and Gateaux
differentiability of the objective function, whose Gateaux derivative is
supposed to be uniformly continuous on bounded sets. Furthermore, we propose
some modifications in the classical projected gradient method in order to
obtain strong convergence. The new variant has the following desirable
properties: the sequence of generated points is entirely contained in a ball
with diameter equal to the distance between the initial point and the solution
set, and the whole sequence converges strongly to the solution of the problem
that lies closest to the initial iterate. Convergence analysis of both methods
is presented without Lipschitz continuity assumption.
