Given a Hilbert space

$\mathscr{H}$, a closed convex subset

$K$ of

$\mathscr{H}$ and a countable family of functions

${F}_{i}:{K}^{2}\to R$ (

$i\in I$), the authors consider the problem of finding

$x\in K$ such that

${F}_{i}(x,y)\ge 0$ for all

$i\in I$ and

$y\in K$, as well as the problem of finding the projection of

$a\in \mathscr{H}$ on

$S$, the solution set of the preceding problem. In order to accomplish these aims, proximal-like block-iterative algorithms, as well as regularization and splitting algorithms, are proposed. For every algorithm, convergence results are established.