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

Krisztin, T.

Funkc. Ekvacioj, Ser. Int. 29, 1-10 (1986).

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.

MSC:

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

Keywords:

rate of convergence of solutions; functional differential equations; Lyapunov-Razumikhin type technique; unbounded delays; non-exponential estimates