Geodesic
Boundary-value problems for second-order differential operators with nonlocal boundary conditions
Electronic journal of differential equations, Tome 2007 (2007)
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.
Classification : 47E05, 35K20
Keywords: Green's function, regular and non regular boundary conditions, semi group with singularities, weighted mixed boundary conditions 
@article{EJDE_2007__2007__a125,
     author = {Denche,  Mohamed and Meziani,  Abderrahmane},
     title = {Boundary-value problems for second-order differential operators with nonlocal boundary conditions},
     journal = {Electronic journal of differential equations},
     year = {2007},
     volume = {2007},
     zbl = {1131.47044},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a125/}
}
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%A Meziani,  Abderrahmane
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Denche,  Mohamed; Meziani,  Abderrahmane. Boundary-value problems for second-order differential operators with nonlocal boundary conditions. Electronic journal of differential equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a125/