Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. (English) Zbl 0977.34056

Here, nonautonomous linear evolution equations \[ (*)\quad u'= A(t)u,\;t\in\mathbb{R}_+,\qquad (**)\quad u'=A(t)u+f(t),\;t\in\mathbb{R}_+, \] on a Banach space \(X\) are considered. Instead of discussing the operator \(L\) defined by \(Lu:u'-A(t)u\) as in [W. N. Zhang, J. Math. Anal. Appl. 191, No. 1, 180-201 (1995; Zbl 0832.34050) and D. Henry, Lecture Notes in Math., 840, Berlin etc.: Springer-Verlag (1981; Zbl 0456.35001)], etc., the authors investigate the evolution family \({\mathcal U}=\{U(t,s): t\geq s\) in \(\mathbb{R} _+\}\) generated by solutions to \((*)\) so that solutions to \((**)\) can be expressed as \[ u(t)=U(t,s) u(s)+\int_s^tU(t,\xi) f(\xi)d\xi,\;t\geq s\text{ in }\mathbb{R}_+. \] By defining an evolution semigroup \[ {\mathcal T}=\{T(t): t\geq 0\} \text{ such that }\bigl[T(t) v\bigr](s)= \begin{cases} U(s,s-t) v(s-t), & s\geq t,\\ U(s,0)v(0), & 0\leq s\leq t,\end{cases} \] the authors characterize exponential stability (in theorems 2.2 and 3.2), exponential expansiveness (in theorem 2.5), and exponential dichotomy (in theorems 4.3 and 4.5) of the evolution family \({\mathcal U}\) on the half-line \(\mathbb{R}_+\) in terms of the infinitesimal generator of \({\mathcal T}\).


34G10 Linear differential equations in abstract spaces
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
47H20 Semigroups of nonlinear operators
Full Text: DOI


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