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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.
\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.