The Criteria for Oscillations of Second Order Linear Functional and Difference Equations With Delay (Journal of Informatics and Mathematical Sciences, 2024)
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 \(\delta^2 = \delta(\delta), \delta u(m) =u(m + 1) - u(m)\), \(\lim\limits_{n\to\infty} \tau(m) = +\infty\), \(\tau : Z^+ \to Z^+\), \(r : Z^+ \to R^+\), \(\tau(m) \le m - 1\).