Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments

1978 ◽  
Vol 81 (3-4) ◽  
pp. 195-210 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Unknown Function ◽  
Continuous Functions ◽  
Linear Differential Equations ◽  
Deviating Arguments ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined bywhere r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

scholarly journals Boundedness of Solutions for Second Order Differential Equations

2020 ◽  
Vol 12 (4) ◽  
pp. 58
Author(s):  
Daniel C. Biles
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equations ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

We present new theorems which specify sufficient conditions for the boundedness of all solutions for second order non-linear differential equations. Unboundedness of solutions is also considered.

Periodic Solutions by Picard's Approximations

1968 ◽  
Vol 11 (5) ◽  
pp. 743-745 ◽  
Author(s):  
T.A. Burton
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
General Framework ◽  
Floquet Theory ◽  
Linear Differential Equations ◽  
All Solutions ◽  
Private Correspondence ◽  
Linear Differential ◽  
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In [1] Demidovic considered a system of linear differential equationswith A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

scholarly journals On the non-existence of L2 solutions of a class of non-linear differential equations

1965 ◽  
Vol 14 (4) ◽  
pp. 257-268 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Half Line ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equationdoes not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

Periodic solutions of non-linear differential equations of the second order. V

1951 ◽  
Vol 47 (4) ◽  
pp. 752-755 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

1·1. Let van der Pol's equation be taken in the formwhere ε1, ε2, k1 and k2 are small, and ω ≠ 0 is a constant, rational or irrational, independent of them.

Differential inequalities and extension of Lyapunov's method

1964 ◽  
Vol 60 (4) ◽  
pp. 891-895 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Differential Equations ◽  
Comparison Principle ◽  
Direct Method ◽  
Linear Differential Equations ◽  
Maximal Solution ◽  
Differential Inequalities ◽  
Non Linear ◽  
Linear Differential ◽  
Lyapunov’S Method ◽  
Image Position

One of the most important techniques in the theory of non-linear differential equations is the direct method of Lyapunov and its extensions. It depends basically on the fact that a function satisfying the inequalityis majorized by the maximal solution of the equationUsing this comparison principle and the concept of Lyapunov's function various properties of solutions of differential equations have been considered (1–11).

On the integrability of solutions of perturbed non-linear differential equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 309-318 ◽  
Author(s):  
Paul W. Spikes
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

SynopsisSufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

Periodic solutions of non-linear differential equations of the second order. IV

1951 ◽  
Vol 47 (4) ◽  
pp. 741-751 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Theoretical Investigation ◽  
Linear Differential Equations ◽  
The Real ◽  
Subharmonic Solutions ◽  
Real Domain ◽  
Non Linear ◽  
Linear Differential ◽  
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1.1. This paper is a theoretical investigation in the real domain of the existence of subharmonic solutions of non-linear differential equations of the formwhere F is analytic and of least period 2π/ω in t; ε = (ε1, …, εn) is small; and F(x, ẋ, 0, t) is not linear in x and ẋ.

scholarly journals Linear differential equations with unbounded delays and a forcing term

2004 ◽  
Vol 2004 (4) ◽  
pp. 337-345 ◽  
Author(s):  
Jan Čermák ◽  
Petr Kundrát
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Equations ◽  
Continuous Functions ◽  
Positive Continuous Function ◽  
All Solutions ◽  
Linear Differential

The paper discusses the asymptotic behaviour of all solutions of the differential equationy˙(t)=−a(t)y(t)+∑i=1nbi(t)y(τi(t))+f(t),t∈I=[t0,∞), with a positive continuous functiona, continuous functionsbi,f, andncontinuously differentiable unbounded lags. We establish conditions under which any solutionyof this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

On the relation between distinct particular solutions of equation

1993 ◽  
Vol 123 (5) ◽  
pp. 917-926
Author(s):  
Guan Ke-ying ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Particular Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThere exists a relation (1.5) between any n + 2 distinct particular solutions of the differential equationIn this paper, we show that when and only when n = 0, 1 and 2, this relation can be represented by the following form:provided the form of this relation function Φn depends only on n and is independent of the coefficients of the equation. This result reveals interesting properties of these non-linear differential equations.

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