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Bound of solutions to third-order nonlinear differential equations with bounded delay. (English) Zbl 1193.34139

Consider the delay differential equation

x ''' +f(x,x ' ,x '' )+g(x(t-r(t)),x ' (t)-r(t))+h(x(t-r(t)))=p(t,x,x ' ,x(t-r(t)),x ' (t-r(t)),x '' )(*)

under the assumption that the delay satisfies

0r(t)α,r ' (t)β,α>0,0<β<1·

The author gives additional conditions on f, g, h, p such that the solution of the Cauchy problem of (*) is uniformly bounded including its first and second derivative. The proof is based on the construction of a Lyapunov functional.

MSC:
34K12Growth, boundedness, comparison of solutions of functional-differential equations
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