## 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:

 [1] G.D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter , Trans. Amer. Math. Soc. 9 (1908), 219-231. JSTOR: · JFM 39.0386.01 · doi:10.2307/1988652 [2] ——–, Boundary value problems and expansion problems of ordinary differential equations , Trans. Amer. Math. Soc. 9 (1908), 373-395. JSTOR: · JFM 39.0386.02 · doi:10.2307/1988661 [3] J.M. Gallardo, Generation of analytic semigroups by second-order differential operators with nonseparated boundary conditions , Rocky Mountain J. Math. 30 (3) (2000), 869-899. · Zbl 0994.47042 · doi:10.1216/rmjm/1021477250 [4] T. Kato, Perturbation theory for linear operators , Springer-Verlag, New York, 1966. · Zbl 0148.12601 [5] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems , Birkhäuser, New York, 1995. · Zbl 0816.35001 [6] M.A. Naimark, Linear differential operators , Vols. I, II, Ungar Publishing, 1967. · Zbl 0219.34001
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