On the rate of convergence of solutions of functional differential equations. (English) Zbl 0601.34046

The goal of this paper is to estimate the rate of convergence of solutions of functional differential equations, where the right-hand side of the equation contains an ordinary and a functional part of the same order. For these equations asymptotic stability is impossible (as in the case where any constant is a solution). We use Lyapunov-Razumikhin type technique. As an application, the scalar equation (*) \(x'(t)=a(t,x(t),x(t-r(t)))\) is studied under the conditions: a,r are continuous functions \(0\leq r(t)\leq r_ 0\), a(t,x,.) is non-decreasing, for a fixed compact interval I there exists a function \(b\in C(R^+,R^+)\) such that x,y\(\in I\), \(x\geq y\) imply \(a(t,x,y)\leq b(t)(y- x),\) x,y\(\in I\), \(x\geq y\) imply \(a(t,x,y)\geq b(t)(y-x),\) \(B=\sup_{t\geq 0}\int_{t}^{t+r_ 0}b(s)ds<\infty.\) It is shown that if \(\phi\) (s)\(\in I\) for \(-r_ 0\leq s\leq 0\), then there is a c depending on \(\phi\) such that for the solution x(0,\(\phi)\) of (*) we have \(| x(0,\phi)(t)-c| \leq Ke^{-\beta t}\) for \(t\geq 0\), where \(K=I(l- e^{-B})^{-1},\beta =-(1/(2r_ 0))\log (l-e^{-B}).\) The case of certain unbounded delays is also considered and non-exponential estimates are also obtained.


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34D20 Stability of solutions to ordinary differential equations