Anti-periodic solutions for a class of \(2n\)th-order nonlinear differential equations with delays (Q2892240)

The following differential delay equation is considered NEWLINE\[NEWLINE u^{(2n)}+\sum_{k=1}^{2n-1}(-1)^k a_k(u^{(k-1)}(t))u^{(k)}(t)-g(t,u(t-\tau(t)))=e(t),\quad n\geq2, \tag{1}NEWLINE\]NEWLINE where all involved functions are continuous in their variables. Additionally, the functions \(a_k\) are all even, \(g(t,u)\) has the property \(g(t+\frac{T}2,-u)=g(t,u)\;\forall (t,u)\in\mathbb R^2\) (an ``anti-periodicity'' property), \(\tau(t)\) is periodic with period \(\frac{T}2\), and \(e(t)\) is anti-periodic with \(e(t+\frac{T}2)=-e(t)\;\forall t\in\mathbb R\).NEWLINENEWLINEA real-valued function \(u(t)\) is defined to be anti-periodic if \(u(t+\omega)=-u(t)\) for some \(\omega>0\) and all \(t\in\mathbb R\). Clearly, \(u(t)\) is also periodic with period \(T=2\omega\).NEWLINENEWLINEImposing a set of additional hypotheses, a sufficient condition is derived for the existence of a \(\frac{T}2\)-anti-periodic solution of equation (1). Two examples illustrating the applicability of the main result are given.