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Zbl 0984.34014
Gallardo, José M.
Second-order differential operators with integral boundary conditions and generation of analytic semigroups.
(English)
[J] Rocky Mt. J. Math. 30, No.4, 1265-1291 (2000). ISSN 0035-7596

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 $D(L_1)=\{u\in W^{2,1}(a,b):B_i(u)=0,\ i=1,2\}$. \par 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.
[Manfred Möller (Johannesburg)]
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
34L15 Estimation of eigenvalues for OD operators
47D06 One-parameter semigroups and linear evolution equations
47D03 (Semi)groups of linear operators
34B27 Green functions

Keywords: second-order differential operator; integral boundary condition; analytic semigroup; Green function

Cited in: Zbl 1149.47034

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