### A note on bounded solutions of second order differential equations at resonance

DOI: http://dx.doi.org/10.12775/TMNA.1999.040

#### Abstract

In the paper we study the existence of bounded solutions for differential

equations of the form:

$ x''-Ax= f(t,x)$, where $A\in L(H)$, $f: {\mathbb R}\times H \to H$

($H$ -- a Hilbert space) is a continuous mapping.

Using a perturbation of the equation, the Leray-Schauder topological degree

and fixed point theory, we overcome the difficulty that the linear

problem is non-Fredholm in any resonable Banach space.

equations of the form:

$ x''-Ax= f(t,x)$, where $A\in L(H)$, $f: {\mathbb R}\times H \to H$

($H$ -- a Hilbert space) is a continuous mapping.

Using a perturbation of the equation, the Leray-Schauder topological degree

and fixed point theory, we overcome the difficulty that the linear

problem is non-Fredholm in any resonable Banach space.

#### Keywords

Differential equations; resonance; boundedness; unbounded domain

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