Denche, Mohamed; Meziani, Abderrahmane Boundary-value problems for second-order differential operators with nonlocal boundary conditions. (English) Zbl 1131.47044 Electron. J. Differ. Equ. 2007, Paper No. 56, 21 p. (2007). 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. Cited in 1 Document 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 Keywords:Green’s function; regular boundary conditions; non-regular boundary conditions; semigroup with singularities; weighted mixed boundary conditions PDFBibTeX XMLCite \textit{M. Denche} and \textit{A. Meziani}, Electron. J. Differ. Equ. 2007, Paper No. 56, 21 p. (2007; Zbl 1131.47044) Full Text: EuDML EMIS