Integrated semigroups and their applications to the abstract Cauchy problem.

*(English)*Zbl 0675.47030Summary: This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators \(A\) on a Banach space \(E\) with nonempty resolvent set for which the abstract Cauchy problem \(u'(t)=Au(t)\), \(u(0)=x\) has unique, exponentially bounded solutions for every initial value \(x\in D(A^ n)\).

Investigating these operators we are led to the class of “integrated semigroups”. Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups.

The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups.

We indicate how integrated semigroups can be used studying second order Cauchy problems \[ u''(t)-A_ 1u'(t)-A_ 2u(t)=0, \] operator valued equations \(U'(t)=A_ 1U(t)+U(t)A_ 2\) and nonautonomous equations \(u'(t)=A(t)u(t)\).

Investigating these operators we are led to the class of “integrated semigroups”. Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups.

The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups.

We indicate how integrated semigroups can be used studying second order Cauchy problems \[ u''(t)-A_ 1u'(t)-A_ 2u(t)=0, \] operator valued equations \(U'(t)=A_ 1U(t)+U(t)A_ 2\) and nonautonomous equations \(u'(t)=A(t)u(t)\).

##### MSC:

47D03 | Groups and semigroups of linear operators |

34G10 | Linear differential equations in abstract spaces |

47D99 | Groups and semigroups of linear operators, their generalizations and applications |