Almost periodic solutions for undamped nonhomogeneous delay-differential equations (Q2782629)

The main purpose of the presented paper is to show that if a time-delay term is introduced into a second-order scalar differential equation of the form NEWLINE\[NEWLINEx''+ x-\varepsilon vx+ \varepsilon^3 x^3= f(t),NEWLINE\]NEWLINE where \(\varepsilon\) and \(v\) are positive constants and \(f\) is almost-periodic, the same result will hold provided the magnitude of this term is sufficiently small, and will follow from a more general \(n\)-dimensional equation of the form NEWLINE\[NEWLINEx'= (A+\varepsilon C(t))x+\varepsilon g(x,\varepsilon)+ \varepsilon\alpha h(x_t)+ \varepsilon p(t),NEWLINE\]NEWLINE where \(A\) and \(C(t)\) are real \(n\times n\)-matrices with \(A\) similar to a diagonal matrix with pure imaginary entries, the entries of \(C(t)\) are almost-periodic, \(g\) and its first partial derivatives with respect to the components of \(x\) are continuous in \((x,\varepsilon)\), and \(p(t)\) is an \(\mathbb{R}^n\)-valued almost-periodic function. \(\varepsilon\) and \(\alpha\) are positive constants and \(h: C_r\to \mathbb{R}^n\) is a function on \(C_r\), the set of functions \(\phi(\theta)\) continuous on \([-r,\theta]\) to \(\mathbb{R}^n\).