Positivity 8: 11–30, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Semigroups Generated by Ordinary Differential
Operators in L
and MICHELE CAMPITI
Department of Economic Sciences, University of Bari, Via C. Rosalba 53, I-70124 Bari, Italy.
Department of Mathematics, Polytechnic of Bari, Via E. Orabona 4, I-70125 Bari, Italy.
(Received 11 December 2000; accepted 5 January 2003)
Abstract. In this paper we give necessary and sufﬁcient conditions for the existence of a C
semigroup in L
I (I real interval) generated by a second-order differential operator when suitable
boundary conditions at the endpoints are imposed.
AMS Mathematical Classiﬁcation: 47D05, 34B05
Key words: generators of semigroups, second-order differential operators, boundary conditions
1 Introduction and preliminaries
Let I =r
+ be a real interval and consider the
second-order differential operator
x x ∈ I (1.1)
where I→ R are continuous functions and x> 0 for every x ∈I.
Necessary and sufﬁcient conditions in order for B to be the generator of a
-semigroup in C
I have been given by Clément and Timmermans  when
Ventcel’s boundary conditions are imposed at the endpoints, and by Timmermans
in  on the maximal domain. More recently, the existence of a C
been characterized in  also in the case of Neumann’s type boundary conditions at
the endpoints. However, all the results obtained in ,  and  are very closely
related to the pioneer work by Feller .
The importance of these results was pointed out in the same paper by Feller
 considering the diffusion approximation of some models in population genetics
describing the evolution of the alleles in a population of diploid organisms. From
an analytic point of view, it is natural to consider the so-called forward and back-
ward Kolmogorov equations associated with the diffusion approximation of these
models. Under suitable assumption on the diffusion model, the following forward