  Proximal splitting algorithms for monotone inclusions (and convex
optimization problems) in Hilbert spaces share the common feature to guarantee
for the generated sequences in general weak convergence to a solution. In order
to achieve strong convergence, one usually needs to impose more restrictive
properties for the involved operators, like strong monotonicity (respectively,
strong convexity for optimization problems). In this paper, we propose a
modified Krasnosel'ski\u{\i}--Mann algorithm in connection with the
determination of a fixed point of a nonexpansive mapping and show strong
convergence of the iteratively generated sequence to the minimal norm solution
of the problem. Relying on this, we derive a forward-backward and a
Douglas-Rachford algorithm, both endowed with Tikhonov regularization terms,
which generate iterates that strongly converge to the minimal norm solution of
the set of zeros of the sum of two maximally monotone operators. Furthermore,
we formulate strong convergent primal-dual algorithms of forward-backward and
Douglas-Rachford-type for highly structured monotone inclusion problems
involving parallel-sums and compositions with linear operators. The resulting
iterative schemes are particularized to the solving of convex minimization
problems.
