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2014 | Buch

Theory of Third-Order Differential Equations

verfasst von: Seshadev Padhi, Smita Pati

Verlag: Springer India

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book discusses the theory of third-order differential equations. Most of the results are derived from the results obtained for third-order linear homogeneous differential equations with constant coefficients. M. Gregus, in his book written in 1987, only deals with third-order linear differential equations. These findings are old, and new techniques have since been developed and new results obtained.

Chapter 1 introduces the results for oscillation and non-oscillation of solutions of third-order linear differential equations with constant coefficients, and a brief introduction to delay differential equations is given. The oscillation and asymptotic behavior of non-oscillatory solutions of homogeneous third-order linear differential equations with variable coefficients are discussed in Ch. 2. The results are extended to third-order linear non-homogeneous equations in Ch. 3, while Ch. 4 explains the oscillation and non-oscillation results for homogeneous third-order nonlinear differential equations. Chapter 5 deals with the z-type oscillation and non-oscillation of third-order nonlinear and non-homogeneous differential equations. Chapter 6 is devoted to the study of third-order delay differential equations. Chapter 7 explains the stability of solutions of third-order equations. Some knowledge of differential equations, analysis and algebra is desirable, but not essential, in order to study the topic.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this chapter, we consider the third-order linear differential equation with constant coefficients of the form

$$ x^{\prime\prime\prime}+ax^{\prime\prime}+bx^{\prime}+cx=0,\quad t \geq \sigma, $$

(1.1)

where a, b, and c are real constants. We have considered eight different cases for the constants a, b, and c while studying the structures of spaces of oscillatory and nonoscillatory solutions of (7.5). Further, it is observed that different structures of solution spaces of (7.5) appear for the eight different cases on a, b, and c. Two comparison theorems are also given in this chapter on the oscillation theory of the nonhomogeneous equation

$$ x^{\prime\prime\prime}+ax^{\prime\prime}+bx^{\prime}+cx=f, $$

(1.2)

where a, b, c, and f are constants. Introductions to third-order delay differential equations and third-order canonical differential equations are also given. The results obtained in this chapter provide some basic ideas on the oscillation and nonoscillation of third-order linear and nonlinear differential equations with variable coefficients.

Seshadev Padhi, Smita Pati

Chapter 2. Behaviour of Solutions of Linear Homogeneous Differential Equations of Third Order

Abstract

This chapter is concerned with the oscillation and nonoscillation of solutions and their asymptotic behaviour of the linear homogeneous equation with variable coefficients of the form

$$x^{\prime\prime\prime}+a(t)x^{\prime\prime}+b(t)x^{\prime}+c(t)x=0, $$

where a,b∈C ¹([σ,∞),R) and c∈C([σ,∞),R). Many interesting and nontrivial results are given on the oscillation of solutions of the equation in the following six different cases: (i) a(t)≥0, b(t)≤0, c(t)>0, (ii) a(t)≤0, b(t)≤0, c(t)>0, (iii) a(t)≤0, b(t)≤0, c(t)<0, (iv) a(t)≥0, b(t)≤0, c(t)<0, (v) a(t)≥0, b(t)≥0, c(t)>0, and (vi) a(t)≤0, b(t)≥0, c(t)<0. Open problems are incorporated at the end of the chapter for future work.

Seshadev Padhi, Smita Pati

Chapter 3. Oscillation of Solutions of Linear Nonhomogeneous Differential Equations of Third Order

Abstract

Linear nonhomogeneous differential equations of the form

$$ x^{\prime\prime\prime}+a(t)x^{\prime\prime}+b(t)x^{\prime}+c(t)x=f(t), $$

(3.1)

where a,b∈C ¹([σ,∞),R) and c∈C([σ,∞),R), has been considered in Chap. 3. Existence of oscillatory solutions of the equation has been given in this chapter for the cases (i) a(t)≥0, b(t)≤0, c(t)>0, (ii) a(t)≤0, b(t)≤0, c(t)>0, and (iii) a(t)≥0, b(t)≥0, c(t)>0. Rest cases have been left as an open problem. Further, Green’s function method has been given in this chapter so that all oscillatory solutions of a linear nonhomogeneous differential equation of the form (7.65) tends to zero eventually.

Seshadev Padhi, Smita Pati

Chapter 4. Oscillation and Nonoscillation of Homogeneous Third-Order Nonlinear Differential Equations

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.

Seshadev Padhi, Smita Pati

Chapter 5. Oscillation and Nonoscillation of Nonlinear Nonhomogeneous Differential Equations of Third Order

Abstract

This chapter is concerned with the study of oscillatory and asymptotic behaviour of nonoscillatory solutions of nonhomogeneous third-order differential equations of the form

$$x^{\prime\prime\prime} + a(t) x^{\prime\prime} + b(t)x^{\prime} + c(t) x = f \bigl(t,x,x^{\prime},x^{\prime\prime}\bigr), $$

where a, b and c∈C([σ,∞),R) and f:[σ,∞)×R ³→R, σ∈R. z-type oscillation criteria has been used in this chapter to study the nonoscillation of solutions of the considered equation. As an application, some sufficient conditions have been given for the nonoscillation of different mathematical models in Engineering.

Seshadev Padhi, Smita Pati

Chapter 6. Oscillatory and Asymptotic Behaviour of Solutions of Third-Order Delay Differential Equations

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.

Seshadev Padhi, Smita Pati

Chapter 7. Stability of Third-Order Differential Equations

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.

Seshadev Padhi, Smita Pati

Backmatter

Titel: Theory of Third-Order Differential Equations
verfasst von: Seshadev Padhi
Smita Pati
Verlag: Springer India
Electronic ISBN: 978-81-322-1614-8
Print ISBN: 978-81-322-1613-1
DOI: https://doi.org/10.1007/978-81-322-1614-8