Existence and Stability of Almost Periodic Solutions for Quasilinear Delay Systems and the Halanay Inequality

Abstract

We present some easily verifiable conditions for the existence and global asymptotical stability of almost periodic solutions to systems of delay differential equations in the form u′(t)=−Au(t)+Wg(t,ut)+f(t), where A,W∈L(Rn) and f(t),g(t,ϕ) are almost periodic in t uniformly on ϕ from bounded subsets of C([−h,0],Rn). With this purpose, we use the exponential dichotomy theory together with usual positivity arguments. In particular, the semigroup version of the so-called Perron–Frobenius theorem is applied to study the generalized Halanay inequality. Finally, systems with maxima are studied in detail.