General functional differential systems with asymptotically constant solutions (Q805857)

The note contains existence results for solutions of systems of functional differential equations \(X'(t)=F(t,X(\cdot))\) on (- \(\infty,\infty)\) that satisfy final value properties of the form \(\lim_{t\to \infty}X(t)=c\) and are constant on \((-\infty,t_ 0]\), where c is a given vector and \(t_ 0\) is given or arbitrary. Typical assumptions include the following: F is continuous with respect to locally uniform convergence and satisfies an estimate \(| F_ i(t,X)| \leq \omega (t,\rho_ 1,...,\rho_ n)\) \((-\infty <t<\infty)\) whenever \(| X_ i(s)| \leq \rho_ i\) for \(i=1,...,r\), \(| X_ i(s)| \geq \rho_ i\) for \(i=r+1,...,n\), for all \(-\infty <s<\infty\), and \(\int^{\infty}_{0}\omega (t,\rho_ 1,...,\rho_ n)dt<\infty.\) The proofs use the Schauder-Tikhonov fixed point theorem.