A generalization of a boundedness theorem for a certain third-order differential equation

1967 ◽  
Vol 63 (3) ◽  
pp. 735-742 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Third Order ◽  
Boundedness Theorem ◽  
The Real ◽  
Third Order Differential Equation ◽  
Image Position

1. This paper investigates the boundedness, as t → ∞, of the solutions of the real differential equationwhere α is a constant and φ2, φ3, ψ depend on the arguments shown with φ2, φ′3, ψ continuous.

On the existence of periodic solutions of a certain third-order differential equation

1960 ◽  
Vol 56 (4) ◽  
pp. 381-389 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Continuous Periodic Function ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Image Position

In this paper we shall be concerned with the differential equationin which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:under which every solution x(t) of (1) satisfieswhere t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),or, in the case (3),In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

Further results for the solutions of a third-order differential equation

1963 ◽  
Vol 59 (1) ◽  
pp. 111-116 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Primary Object ◽  
Image Position

1. The equation considered here is of the formwhere a, b are constants, h(x) is differentiable and h′(x), p(t) are continuous in x, t respectively. The primary object of the paper is to prove the followingTheorem 1. Suppose that(i) a > 0,b > 0;(ii) h(0) = 0, h(x)/x ≥ δ > 0 (x ≠ 0);(iii) h′(x) ≤ c for all x where ab > c > 0.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

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