Stability properties of linear volterra equations

Stability Properties of Linear Volterra Integrodifferential Equations in a Banach Space

“…In fact, in case the dimension X is finite, the problem is solved a‰rmatively, and consequently the UAS property for the zero solution of (E) can be characterized in terms of the integrability of the resolvent operator Rðt; sÞ of (E), together with the uniform boundedness of Rðt; sÞ; for the details, see [10] and [24]. In case the dimension X is infinite, however, the aspect is quite di¤erent.…”

“…We emphasize that the method employed in [9,10] for the equations with dim X < l is not directly applicable to the analysis for the equations with infinite dimensional X. Indeed, if one proceeds the way in [9,10] for the analysis in case of dim X ¼ l, there will arise a di‰culty in the establishment of the integrability of the resolvent. To overcome this, in this paper we will focus attention on a property for the resolvent which is weaker than the integrability, and deduce that the total stability property in (E) is equivalent to the weaker property for the resolvent (Theorem 3.1).…”

“…Furthermore, they obtained a result which characterizes the uniform asymptotic stability property in terms of the integrability of the resolvent for the equations, and applied it to ensure the existence of bounded solutions for the nonhomogeneous equation with bounded forcing terms. The purpose of this paper is to extend several results in [9,10] to the equations with infinite dimensional X. We emphasize that the method employed in [9,10] for the equations with dim X < l is not directly applicable to the analysis for the equations with infinite dimensional X.…”

“…The purpose of this paper is to extend several results in [9,10] to the equations with infinite dimensional X. We emphasize that the method employed in [9,10] for the equations with dim X < l is not directly applicable to the analysis for the equations with infinite dimensional X. Indeed, if one proceeds the way in [9,10] for the analysis in case of dim X ¼ l, there will arise a di‰culty in the establishment of the integrability of the resolvent.…”

“…Volterra integrodi¤erential equations have been treated in several works; see, e.g., [4,5,6,7,9,10,12,13,16,17,21,24] and the references therein. In case of dim X < l, the authors in [9,10] treated linear Volterra integrodi¤erential equations, and established some equivalent relationships among several stability properties in Volterra integrodi¤erential equations (E) and ðE l Þ. Furthermore, they obtained a result which characterizes the uniform asymptotic stability property in terms of the integrability of the resolvent for the equations, and applied it to ensure the existence of bounded solutions for the nonhomogeneous equation with bounded forcing terms.…”