Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 56, pp. 1-21.

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

Authors: Mohamed Denche       (Univ. Mentouri, Algeria)
         Abderrahmane Meziani (Univ. Mentouri, Algeria)

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 $L^{p}(0,1)$, but there is no maximal
 decrease at infinity for $p>1$. Furthermore, the studied operator generates
 in $L^{p}(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.
 
  
Submitted May 10, 2006. Published April 17, 2007.
Math Subject Classifications: 47E05, 35K20.
Key Words: Green's function; regular and non regular boundary conditions;
          semi group with singularities; weighted mixed boundary conditions.