The Criteria for Oscillations of Second Order Linear Functional and Difference Equations With Delay - Sharadha, P. (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, \(u’’(t) + r(t) u ( τ(t)) = 0\), for \(t t_0\). Here \(r ([t_0, ∞), R^+)\), \(τ ([t_0, ∞), R)\), where \(r\) and \(τ\) are continuous functions of non-negative real number and \(τ(t)\) is an increasing function and it is also delay operator then \(τ(t) t\), for \(t_0 t\), and \(_t→ ∞ τ(t) = ∞\). The second order functional equations of oscillatory theory is \(u(h(t)) = A(t)u(t) + B(t)u(h(h(t)))\), for \(t t_0\), where \(A\) and \(B\) are continuous funtions and \(A, B ([t_0, ∞), R)\), \(h ([t_0, ∞), R^+)\) 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 \(δ^2u(m) + r(m)u(τ(m)) = 0\), where \(δ^2 = δ(δ), δ u(m) = u(m + 1) - u(m)\), \(_n→∞ τ(m) = +∞\), \(τ : Z^+ → Z^+\), \(r : Z^+ → R^+\), \(τ(m) m - 1\).