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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 \(q_0,R_i,S_i\in C([a,b];\mathbb{C})\) 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.

MSC:

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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References:

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