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Second-order differential operators with integral boundary conditions and generation of analytic semigroups. (English) Zbl 0984.34014

The class of differential expressions $l\left(u\right)={u}^{\text{'}\text{'}}+{q}_{1}\left(x\right)u+{q}_{0}\left(x\right)x$ in $\left(a,b\right)$ with the integral boundary conditions

${B}_{i}u={\int }_{a}^{b}{R}_{i}\left(t\right)u\left(t\right)dt+{\int }_{a}^{b}{S}_{i}\left(t\right){u}^{\text{'}}\left(t\right)dt=0,\phantom{\rule{1.em}{0ex}}i=1,2,$

is considered, with ${q}_{0},{R}_{i},{S}_{i}\in C\left(\left[a,b\right];ℂ\right)$ and ${q}_{1}\in {C}^{1}\left(\left[a,b\right];ℂ\right)$. Suppose that the boundary conditions are regular, i.e., one of the following conditions is satisfied: ${S}_{1}\left(a\right){S}_{2}\left(b\right)-{S}_{1}\left(b\right){S}_{2}\left(a\right)\ne 0$; ${S}_{1}=0$ and ${R}_{1}\left(a\right){S}_{2}\left(b\right)+{R}_{1}\left(b\right){S}_{2}\left(a\right)\ne 0$; ${S}_{2}=0$ and ${R}_{2}\left(a\right){S}_{1}\left(b\right)+{R}_{2}\left(b\right){S}_{1}\left(a\right)\ne 0$; ${S}_{1}=0$, ${S}_{2}=0$ and ${R}_{1}\left(a\right){R}_{2}\left(b\right)-{R}_{1}\left(b\right){R}_{2}\left(a\right)\ne 0$. As usual, the linear operator ${L}_{1}$ on ${L}^{1}\left(a,b\right)$ is associated with $l$, where the domain of ${L}_{1}$ is $D\left({L}_{1}\right)=\left\{u\in {W}^{2,1}\left(a,b\right):{B}_{i}\left(u\right)=0,\phantom{\rule{4pt}{0ex}}i=1,2\right\}$.

It is shown that ${L}_{1}$ is the generator of an analytic semigroup ${\left\{{e}^{t{L}_{1}}\right\}}_{t\ge 0}$ of bounded linear operators on ${L}_{1}\left(a,b\right)$. The detailed proof uses the usual techniques of the location of the spectrum and estimates on the resolvent as an integral operator with the Green function as kernel.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 47D06 One-parameter semigroups and linear evolution equations 47D03 (Semi)groups of linear operators 34B27 Green functions