Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

  • M. F. Gorodnii

Abstract

For a sectorial operator A with spectrum σ(A) that acts in a complex Banach space B, we prove that the condition σ(A) ∩ i R = Ø is sufficient for the differential equation \(\varepsilon x_\varepsilon^\prime\prime(t)+x_\varepsilon^\prime(t)=Ax_\varepsilon(t)+f(t), t \in R,\) where ε is a small positive parameter, to have a unique bounded solution x ε for an arbitrary bounded function f: RB that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as ε → 0+ to the unique bounded solution of the differential equation x′(t) = Ax(t) + f(t).
Published
25.07.2003
How to Cite
GorodniiM. F. “Stability of Bounded Solutions of Differential Equations With Small Parameter in a Banach Space”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 55, no. 7, July 2003, pp. 889-00, https://umj.imath.kiev.ua/index.php/umj/article/view/3964.
Section
Research articles