The boundedness to nonlinear differential equations of fourth order with delay

Abstract

In this paper, we study the boundedness of solutions to fourth order nonlinear delay differential equation \begin{equation*} \begin{array}{c} x^{(4)}(t)+f_{1}(x^{\prime \prime }(t))x^{\prime \prime \prime }(t)+f_{2}(x^{\prime }(t),x^{\prime \prime }(t))x^{\prime \prime }(t)+g(x^{\prime }(t-r))+h(x(t-r)) \\ =p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t),x^{\prime \prime }(t-r),x^{\prime \prime \prime }(t)), \end{array} \end{equation*} where $r>0$ is a constant delay. In proving our main result, we make use of the Lyapunov's second method by constructing a Lyapunov functional.