Bound of solutions to third-order nonlinear differential equations with bounded delay
Consider the delay differential equation \[ \begin{multlined} 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'')\end{multlined}\tag{\(*\)} \] under the assumption that the delay satisfies \[ 0\leq r(t)\leq\alpha,\quad r'(t)\leq\beta,\quad \alpha> 0,\quad 0<\beta< 1. \] 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.
scientific article; zbMATH DE number 5693306
scientific article