Oscillatory and Asymptotic Behaviour of Solutions of Third-Order Delay 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 6 in Theory of Third-Order Differential Equations, 2014, pp 335-453 from Springer

Abstract: Abstract This chapter is concerned with the asymptotic behaviour of solutions of $$x^{\prime\prime\prime}(t)+a(t)x^{\prime\prime}(t)+b(t)x^{\prime }(t)+c(t)x \bigl(g(t)\bigr)=0, $$ where a,b,c and g∈C([σ,∞),R),σ∈R,g(t)≤t and g(t)→∞ as t→∞. In the process, we present some recent results on the properties of nonoscillatory solutions of the nonlinear equations $$\begin{aligned} x^{\prime\prime\prime}(t)+q(t)x^{\prime}(t)+p(t)f\bigl(x\bigl(g(t)\bigr)\bigr) & =0, \\ \bigl(r(t) \bigl(x^{\prime\prime}(t)\bigr)^{\gamma}\bigr)^{\prime}+p(t)f \bigl(x\bigl(g(t)\bigr)\bigr) & =0, \\ \bigl(r(t) \bigl(x^{\prime\prime}(t)\bigr)^{\gamma}\bigr)^{\prime}+p(t)x^{\gamma} \bigl(g(t)\bigr) & =0 \end{aligned}$$ and $$r_{3}(t) \bigl(r_{2}(t) \bigl(r_{1}(t)x^{\prime}(t) \bigr)^{\prime}\bigr)^{\prime }+q(t)x^{\prime}(t)+p(t)f\bigl(x \bigl(g(t)\bigr)\bigr)=0, $$ where r and r i ,i=0,1,2,3 are defined as earlier, f:R→R and γ>0 is a ratio of odd positive integers. Third-order delay differential equations with distributed deviating arguments have also been considered. In addition to the above, some interesting results have been given on the oscillation of solutions of the nonlinear nonhomogeneous third-order delay differential equations.

Keywords: Delay Differential Equations; Nonoscillatory Solution; Second-order Linear Differential Equation; Third-order Ordinary Differential Equation; Finding Sufficient Conditions (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-81-322-1614-8_6

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DOI: 10.1007/978-81-322-1614-8_6

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