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 \(q_0,R_i,S_i\in C([a,b];\mathbb{C})\) and \(q_1\in C^1([a,b];\mathbb{C})\). Suppose that the boundary conditions are regular, i.e., one of the following conditions is satisfied: \(S_1(a)S_2(b)-S_1(b)S_2(a)\neq 0\); \(S_1=0\) and \(R_1(a)S_2(b)+R_1(b)S_2(a)\neq 0\); \(S_2=0\) and \(R_2(a)S_1(b)+R_2(b)S_1(a)\neq 0\); \(S_1=0\), \(S_2=0\) and \(R_1(a)R_2(b)-R_1(b)R_2(a)\neq 0\). 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 \(\{e^{tL_1}\}_{t\geq 0}\) 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.


34B15 Nonlinear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47D06 One-parameter semigroups and linear evolution equations
47D03 Groups and semigroups of linear operators
34B27 Green’s functions for ordinary differential equations
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