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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 $l(u)=u''+q_1(x)u+q_0(x)x$ in $(a,b)$ with the integral boundary conditions $$B_iu=\int_a^bR_i(t)u(t) dt+\int_a^bS_i(t)u'(t) dt=0,\quad i=1,2,$$ is considered, with $q_0,R_i,S_i\in C([a,b];\bbfC)$ and $q_1\in C^1([a,b];\bbfC)$. Suppose that the boundary conditions are regular, i.e., one of the following conditions is satisfied: $S_1(a)S_2(b)-S_1(b)S_2(a)\ne 0$; $S_1=0$ and $R_1(a)S_2(b)+R_1(b)S_2(a)\ne 0$; $S_2=0$ and $R_2(a)S_1(b)+R_2(b)S_1(a)\ne 0$; $S_1=0$, $S_2=0$ and $R_1(a)R_2(b)-R_1(b)R_2(a)\ne 0$. As usual, the linear operator $L_1$ on $L^1(a,b)$ is associated with $l$, where the domain of $L_1$ is $D(L_1)=\{u\in W^{2,1}(a,b):B_i(u)=0,\ i=1,2\}$. It is shown that $L_1$ is the generator of an analytic semigroup $\{e^{tL_1}\}_{t\ge 0}$ of bounded linear operators on $L_1(a,b)$. 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
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##### References:
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