Boundary value problem for second-order differential operators with integral conditions

Abstract

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

Keywords:

• Green's function
• Nonregular boundary conditions
• Regular boundary conditions
• Semigroup with singularities
• AMS(MOS) Subject Classifications:
• 47E05
• 35K20