On bounded solutions of nonlinear differential equations in Banach spaces. (English) Zbl 0593.34062

Dawidowski, Marian; Rzepecki, Bogdan

Demonstr. Math. 18, 91-102 (1985).

The authors’ purpose is to prove the existence of bounded solutions of the differential equation: (1) \(x'(t)=A(t)x(t)+F(t,x(t))\) under the assumption that the differential linear equation: (2) \(x'(t)=A(t)x(t)\) admits a regular exponential dichotomy and F satisfies some regularity condition expressed in terms of the measure of noncompactness \(\alpha\).

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A34	Nonlinear ordinary differential equations and systems
34C11	Growth and boundedness of solutions to ordinary differential equations
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

first order differential equation; bounded solutions; regular exponential dichotomy