Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem. (English) Zbl 1084.47522

Burton, T. A.

Nonlinear Stud. 9, No. 2, 181-190 (2002).

The author uses a modification of Krasnosel’skiĭ’s fixed point theorem [T. A. Burton, Proc. Am. Math. Soc. 124, No. 8, 2383–2390 (1996; Zbl 0873.45003)] to give sufficient conditions for the asymptotic stability of the zero solution of the functional-differential equation \(x'=-a(t)x^3(t)+b(t)x^3(t- r(t))\), where \(r(t)\) need neither be bounded nor differentiable, while \(a\) and \(b\) can be unbounded.

Reviewer: L. Hatvani (MR 2003e:34133)

MSC:

47H10	Fixed-point theorems
34K20	Stability theory of functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

asymptotic stability; zero solution; functional-differential equation with unbounded coefficientsl Ljapunov functional; Krasnoselskij fixed point theorem

Citations:

Zbl 0873.45003