Existence and stability of almost-periodic solutions of quasi-linear differential equations with deviating argument.
For the equation \[ x'(t)= A(t)x(t)+ f(t,x(t),\dots,x(t-\tau_k(t))), \] the existence of an almost-periodic solution is shown under the following assumptions: \(A(t)\) is an \(n\times n\)-matrix, almost-periodic in \(t\); \(\tau_j\) almost-periodic; \(f(t,u_0,\dots,u_k)\) is almost-periodic in \(t\) locally uniformly in the \(u_j\) and Lipschitzian in the \(u_j\), with sufficiently small Lipschitz constant; \(x'= A(t)x\) satisfies an exponential dichotomy. Under some additional assumptions and with \(\tau_j\geq 0\), such a solution is shown to be exponentially stable.
scientific article; zbMATH DE number 2140801