Electron. J. Diff. Eqns., Vol. 2007(2007), No. 56, pp. 1-21.

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

Mohamed Denche, Abderrahmane Meziani

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 greater than 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 greater than 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.

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Mohamed Denche
Laboratoire Equations Differentielles
Departement de Mathematiques
Faculte des Sciences, Universite Mentouri
25000 Constantine, Algeria
email: denech@wissal.dz
Abderrahmane Meziani
Laboratoire Equations Differentielles
Departement de Mathematiques
Faculte des Sciences, Universite Mentouri
25000 Constantine, Algeria
email: mezianiar@yahoo.fr

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