*(English)*Zbl 0984.34014

The class of differential expressions $l\left(u\right)={u}^{\text{'}\text{'}}+{q}_{1}\left(x\right)u+{q}_{0}\left(x\right)x$ in $(a,b)$ with the integral boundary conditions

is considered, with ${q}_{0},{R}_{i},{S}_{i}\in C([a,b];\u2102)$ and ${q}_{1}\in {C}^{1}([a,b];\u2102)$. Suppose that the boundary conditions are regular, i.e., one of the following conditions is satisfied: ${S}_{1}\left(a\right){S}_{2}\left(b\right)-{S}_{1}\left(b\right){S}_{2}\left(a\right)\ne 0$; ${S}_{1}=0$ and ${R}_{1}\left(a\right){S}_{2}\left(b\right)+{R}_{1}\left(b\right){S}_{2}\left(a\right)\ne 0$; ${S}_{2}=0$ and ${R}_{2}\left(a\right){S}_{1}\left(b\right)+{R}_{2}\left(b\right){S}_{1}\left(a\right)\ne 0$; ${S}_{1}=0$, ${S}_{2}=0$ and ${R}_{1}\left(a\right){R}_{2}\left(b\right)-{R}_{1}\left(b\right){R}_{2}\left(a\right)\ne 0$. As usual, the linear operator ${L}_{1}$ on ${L}^{1}(a,b)$ is associated with $l$, where the domain of ${L}_{1}$ is $D\left({L}_{1}\right)=\{u\in {W}^{2,1}(a,b):{B}_{i}\left(u\right)=0,\phantom{\rule{4pt}{0ex}}i=1,2\}$.

It is shown that ${L}_{1}$ is the generator of an analytic semigroup ${\left\{{e}^{t{L}_{1}}\right\}}_{t\ge 0}$ 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 |