Stability of Third-Order 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 7 in Theory of Third-Order Differential Equations, 2014, pp 455-502 from Springer

Abstract: Abstract This chapter deals with the stability and asymptotic stability of solutions of the unperturbed and the perturbed third-order nonlinear differential equations $$x^{\prime\prime\prime} + \psi\bigl(x,x^{\prime}\bigr)x^{\prime\prime} + f \bigl(x,x^{\prime}\bigr) =0 $$ and $$x^{\prime\prime\prime} + \psi\bigl(x,x^{\prime}\bigr)x^{\prime\prime} + f \bigl(x,x^{\prime}\bigr) = p(t), $$ where ψ, f, ψ x , f x ∈C(R×R,R) and p∈C([0,∞),R). Stability of solutions of equations of the form $$\begin{aligned} x^{\prime\prime\prime} + \psi\bigl(x,x^{\prime},x^{\prime\prime} \bigr)x^{\prime \prime} + f\bigl(x,x^{\prime}\bigr) = p\bigl(t,x,x^{\prime},x^{\prime\prime} \bigr) \end{aligned}$$ has also been considered in this chapter. On the way, we provide some new results on the stability of zero solutions of the autonomous equation $$\begin{aligned} &x^{\prime\prime\prime}(t) = p_1 x^{\prime\prime}(t) + p_2x^{\prime\prime}(t- \tau) + q_1 x^{\prime}(t) + q_2 x^{\prime}(t- \tau) + r_1 x(t) + r_2 x(t-\tau),\\ &\quad t \geq0 \end{aligned}$$ with the initial condition $$x(t) = \phi(t),\quad t\in[-\tau,0], $$ where p 1, p 2, q 1, q 2, r 1 and r 2 are real constants, τ>0 is a real number and ϕ∈C([−τ,0),R) is an initial function.

Keywords: Third-order Linear Differential Equation; Global Asymptotic Stability; Second-order Nonlinear Differential Equation; Delay Differential Equations System; Equivalent Initial Value Problem (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_7

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DOI: 10.1007/978-81-322-1614-8_7

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