The Criteria for Oscillations of Second Order Linear Functional and Difference Equations With Delay

Abstract

We establish some conditions for all the solutions of the oscillatory theory of second-order linear differential equations with delay, \(u''(t) + r(t) u ( \tau(t)) = 0\), for \(t \ge t_0\). Here \(r \in ([t_0, \infty), R^+)\), \(\tau \in ([t_0, \infty), R)\), where \(r\) and \(\tau\) are continuous functions of non-negative real number and \(\tau(t)\) is an increasing function and it is also delay operator then \(\tau(t) \le t\), for \(t_0 \le t\), and \(\lim\limits_{t\to \infty} \tau(t) = \infty\). The second order functional equations of oscillatory theory is \(u(h(t)) = A(t)u(t) + B(t)u(h(h(t)))\), for \(t \ge t_0\), where \(A\) and \(B\) are continuous funtions and \(A, B \in ([t_0, \infty), R)\), \(h \in ([t_0, \infty), R^+)\) are continuous real valued functions and \(u\) is also a real valued function with unknown variables. The second-order linear difference equation of oscillatory theory is \(\delta^2u(m) + r(m)u(\tau(m)) = 0\), where \(\delta^2 = \delta(\delta), \delta u(m) =
u(m + 1) - u(m)\), \(\lim\limits_{n\to\infty} \tau(m) = +\infty\), \(\tau : Z^+ \to Z^+\), \(r : Z^+ \to R^+\), \(\tau(m) \le m - 1\).