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

### 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*:

**R**→

*B*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

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 55, no. 7, July 2003, pp. 889-00, https://umj.imath.kiev.ua/index.php/umj/article/view/3964.

Issue

Section

Research articles