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Boundary-value problems for second-order differential operators with nonlocal boundary conditions. (English) Zbl 1131.47044

Summary: 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 nonregular 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 semigroup for \(p=1\) in the 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 parabolic partial differential equation with nonregular boundary conditions.

MSC:

47E05 General theory of ordinary differential operators
35K20 Initial-boundary value problems for second-order parabolic equations
47N20 Applications of operator theory to differential and integral equations
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