Boundedness of third-order delay differential equations in which \(h\) is not necessarily differentiable (Q2817432)

In this paper, the author considers the following nonlinear differential equation of third order with constant delay NEWLINE\[NEWLINE{x}''' + a{x}'' + b{x}' + h(x(t - r)) = p(t,x,{x}',{x}''),\tag{1}NEWLINE\]NEWLINENEWLINENEWLINE\noindent where \(a, b\) and \(r\) are some positive constants \(x \in \mathbb R,\mathbb R = ( - \infty ,\infty ),t \in \mathbb R _ + ,\mathbb R _ + = [0,\infty ),\) and the functions \(h\) and \(p\) are continuous functions for the arguments displayed explicitly in equation (1). The author gives new sufficient conditions guaranteeing the uniform boundedness and uniform ultimate boundedness of solutions of equation (1). A new theorem is proved on that subject. The obtained result improves earlier works in the literature and the proof is based on the method of Lyapunov functional. No example is given for illustrations.