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Zbl 1193.34139
Tunç, Cemil
Bound of solutions to third-order nonlinear differential equations with bounded delay.
(English)
[J] J. Franklin Inst. 347, No. 2, 415-425 (2010). ISSN 0016-0032; ISSN 1879-2693/e

Consider the delay differential equation $$\multline x'''+ f(x,x',x'')+ g(x(t- r(t)), x'(t)- r(t))\\ +h(x(t- r(t)))= p(t,x,x',x(t- r(t)), x'(t- r(t)), x'')\endmultline\tag{*}$$ under the assumption that the delay satisfies $$0\le r(t)\le\alpha,\quad r'(t)\le\beta,\quad \alpha> 0,\quad 0<\beta< 1.$$ 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.
[Klaus R. Schneider (Berlin)]
MSC 2000:
*34K12 Properties of solutions of functional-differential equations

Keywords: Lyapunov functional; third-order delay differential equation

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