On bounded solutions of nonlinear differential equations at resonance (Q698874)

The author deals with the differential equation \[ x'-Ax = f(t,x), \] where \(A\) is a linear bounded selfadjoint operator on a Hilbert space \(H = C(\mathbb{R})\) such that \(0\) is an isolated point of the spectrum of \(A\) and \( f: \mathbb{R} \times H \to H\) is a continuous mapping. Under suitable boundedness and monotonicity type conditions on \(f\) -- using fixed-point theorems for completely continuous maps and homotopy invariance of Leray-Schauder degree -- the author proves the existence of a bounded solution on \(\mathbb{R}\).