On the qualitative behaviour of solutions to certain second order nonlinear differential equation with delay
In this paper, the authors consider the following two nonlinear differential equation of second order without and with constant delay, respectively \[ x''+a(t)f(x,x')+g(x)=p(t,x,x') \] and \[ x''+a(t)f(x,x')+g(x(t-\tau))=p(t,x,x'), \] where \(\tau\) is a constant delay, \(x\in\mathbb{R}\), \(\mathbb{R}=(-\infty,\infty)\), \(t\in\mathbb{R}^+\), \(\mathbb{R}^+=[0,\infty)\), \(a,f,g\) and \(p\) are continuous functions of the arguments shown explicitly such that the derivatives \(a'(t)=\frac{d}{dt}a(t),\) and \(g'(x)=\frac{dg}{dx}\) exist and are continuous. In addition, it is assumed that the functions \(f,g\) and \(p\) satisfy a Lipschitz condition with respect to their arguments, except \(t\). Hence, the existence and uniqueness of the solutions of the equations considered are guaranteed. In this paper, the Lyapunov's direct method is employed to obtain sufficient conditions for globally asymptotically stability of the zero solution of the first equation and the boundedness solutions and their derivatives of first order for the given first and second differential equations. The obtained results are new and complement existing results in the literature. Two good examples are given to illustrate the main results.
scientific article; zbMATH DE number 6827405
scientific article