Asymptotic Solutions of nth Order Dynamic Equation and Oscillations
نویسنده
چکیده
We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson’s asymptotic theorem.Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theoremwe prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.
منابع مشابه
Asymptotic behavior of n-th order dynamic equations
We are concerned with the asymptotic behavior of solutions of an nth order linear dynamic equation on a time scale in terms of Taylor monomials. In particular, we describe the asymptotic behavior of the so-called (first) principal solution in terms of the Taylor monomial of degree n−1. Several interesting properties of the Taylor monomials are established so that we can prove our main results.
متن کاملON THE PERIODIC SOLUTIONS OF A CLASS OF nTH ORDER NONLINEAR DIFFERENTIAL EQUATIONS *
The nth order differential equation x + c (t )x + ƒ( t,x) = e(t),n>3 is considered. Using the Leray-Schauder principle, it is shown that under certain conditions on the functions involved, this equation possesses a periodic solution.
متن کاملNonlinear Fuzzy Volterra Integro-differential Equation of N-th Order: Analytic Solution and Existence and Uniqueness of Solution
This paper focuses on the fuzzy Volterra integro-differential equation of nth order of the second-kind with nonlinear fuzzy kernel and initial values. The derived integral equations are solvable, the solutions of which are unique under certain conditions. The existence and uniqueness of the solutions are investigated in a theorem and an upper boundary is found for solutions. Comparison of the e...
متن کاملAsymptotic representation of low-frequency dynamic tides in close binaries
An asymptotic representation of low-frequency dynamic tides in close binaries is developed. The dynamic tides are treated as low-frequency, linear, isentropic, forced oscillations of a non-rotating spherically symmetric star. The asymptotic representation is developed to the second order in the forcing frequency. For the sake of simplification, the star is assumed to be everywhere in radiative ...
متن کاملExtension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems
The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...
متن کامل