Oscillation and Nonoscillation of Homogeneous Third-Order Nonlinear Differential Equations

Seshadev Padhi and Smita Pati
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Seshadev Padhi: Birla Institute of Technology, Mesra, Department of Applied Mathematics
Smita Pati: Birla Institute of Technology, Mesra, Department of Applied Mathematics

Chapter Chapter 4 in Theory of Third-Order Differential Equations, 2014, pp 193-303 from Springer

Abstract: Abstract This chapter deals with third-order linear and nonlinear homogeneous differential equations of the form $$x^{\prime\prime\prime} + a(t)x^{\prime\prime} + b(t)x^{\prime} + c(t) x^\alpha = 0 $$ and $$x^{\prime\prime\prime} + a(t)x^{\prime\prime} + b(t)x^{\prime} + c(t) f(x) = 0 $$ where a, b and c∈C([σ,∞),R), α>0 is a ratio of odd integers, f∈C(R,R) such that $\frac{f(x)}{x} \geq\beta >0$ for x≠0. The necessary and sufficient conditions have been given in terms of the coefficient functions for the oscillation and nonoscillation of solutions of the considered equations for the following cases: (i) a(t)≥0, b(t)≤0 and c(t)>0; and (ii) a(t)≤0, b(t)≤0 and c(t)>0.

Keywords: Third-order Nonlinear Differential Equation; Kneser Solution; Nonoscillatory Solution; Integral Average Condition; Oscillatory Solutions (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-81-322-1614-8_4

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