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).

scholarly journals Existence of Periodic Solutions for Nonlinear Fully Third-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yunfei Zhang ◽  
Minghe Pei
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Existence Results ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Growth Restrictions

In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.

scholarly journals Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations

2003 ◽  
Vol 2003 (4) ◽  
pp. 209-228 ◽  
Author(s):  
O. Rabiei Motlagh ◽  
Z. Afsharnezhad
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Order Equation ◽  
Second Order ◽  
Third Order ◽  
Bifurcation Theorem ◽  
Existence Of Periodic Solutions

The existence of periodic solutions for the third-order differential equationx¨˙+ω2x˙=μF(x,x˙,x¨)is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions ifF(x,x˙,x¨)has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).

scholarly journals On the limit cycles for a class of eighth-order differential equations

2020 ◽  
Vol 6 (1) ◽  
pp. 53-61
Author(s):  
Chems Eddine Berrehail ◽  
Zineb Bouslah ◽  
Amar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Rational Numbers ◽  
Eighth Order ◽  
Existence Of Periodic Solutions

AbstractIn this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p2λ2 + p2µ2 + λ2µ2 + p2 + λ2 + µ2, B = p2 λ2 + p2µ2 + λ2µ2 + p2λ2µ2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

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.

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.

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