We give some results concerning the real-interpolation method and finite differences. Next, we apply them to estimate the resolvents of finite-difference discretizations of Dirichlet boundary value problems for elliptic equations in space dimensions one and two in analogs of spaces of continuous and Hölder continuous functions. Such results were employed to study finite-difference discretizations of parabolic equations.

We establish a multiplicity result to an eigenvalue problem related to second-order Hamiltonian systems. Under new assumptions, we prove the existence of an open interval of positive eigenvalues in which the problem admits three distinct periodic solutions.

We establish the existence of positive solutions of some $m$-point boundary value problems under weaker assumptions than previously employed. In particular, we do not require all the parameters occurring in the boundary conditions to be positive. Our results allow more general behaviour for the nonlinear term than being either sub- or superlinear.