Boundary-value problems for second-order differential operators with nonlocal boundary conditions

Abstract

In this paper, we study a second-order differential operator combining weighting integral boundary condition with another two-point boundary condition. Under certain conditions on the weighting functions, called regular and non regular cases, we prove that the resolvent decreases with respect to the spectral parameter in Lp(0, 1), but there is no maximal decrease at infinity for p > 1. Furthermore, the studied operator generates in Lp(0, 1), an analytic semi group for p = 1 in the regular case, and an analytic semi group with singularities for p > 1, in both cases, and for p = 1, in the non regular case only. The obtained results are then used to show the correct solvability of a mixed problem for parabolic partial differential equation with non regular boundary conditions.