scholarly journals A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1492
Author(s):  
Mahmoud Abdel-Aty ◽  
Musa E. Kavgaci ◽  
Ioannis P. Stavroulakis ◽  
Nour Zidan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
State Of The Art ◽  
The State ◽  
First Order ◽  
Several Delays

This paper deals with the oscillation of the first-order differential equation with several delay arguments x′t+∑i=1mpitxτit=0,t≥t0, where the functions pi,τi∈Ct0,∞,R+, for every i=1,2,…,m,τit≤t for t≥t0 and limt→∞τit=∞. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.

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.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

scholarly journals Oscillation of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

scholarly journals THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 3601-3627 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
J. NASARRE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Order Differential Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
First Order ◽  
Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

scholarly journals Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 718 ◽  
Author(s):  
Emad R. Attia ◽  
Hassan A. El-Morshedy ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Oscillation Criteria ◽  
Lower Upper ◽  
First Order ◽  
Upper Limit ◽  
First Order Differential Equations

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results.

An Existence Result for Stepanoff Almost-Periodic Differential Equations

1971 ◽  
Vol 14 (4) ◽  
pp. 551-554 ◽  
Author(s):  
S. Zaidman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Existence Result ◽  
Order Differential Equation ◽  
Short Paper ◽  
Almost Periodic ◽  
First Order ◽  
Right Hand

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.

scholarly journals Fractional differential equation with uncertainty

2021 ◽  
Vol 23 (08) ◽  
pp. 181-185
Author(s):  
Karanveer Singh ◽  
◽  
R N Prajapati ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
First Order Differential Equations

We consider a fractional order differential equation with uncertainty and introduce the concept of solution. It goes beyond ordinary first-order differential equations and differential equations with uncertainty.

scholarly journals On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Nabil Sellami ◽  
Romaissa Mellal ◽  
Bahri Belkacem Cherif ◽  
Sahar Ahmed Idris
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Nonlinear Function ◽  
Averaging Theory ◽  
First Order ◽  
Autonomous Differential Equations ◽  
Fifth Order

We study the limit cycles of the fifth-order differential equation x ⋅ ⋅ ⋅ ⋅ ⋅ − e x ⃜ − d x ⃛ − c x ¨ − b x ˙ − a x = ε F x , x ˙ , x ¨ , x ⋯ , x ⃜ with a = λ μ δ , b = − λ μ + λ δ + μ δ , c = λ + μ + δ + λ μ δ , d = − 1 + λ μ + λ δ + μ δ , e = λ + μ + δ , where ε is a small enough real parameter, λ , μ , and δ are real parameters, and F ∈ C 2 is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.

Stability Analysis and Application for Equilibrium Point of Differential Equation

2014 ◽  
Vol 631-632 ◽  
pp. 261-264
Author(s):  
Dan Li
Keyword(s):  
Social Sciences ◽  
Differential Equation ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Species Coexistence ◽  
Order Differential Equation ◽  
Stable Equilibrium ◽  
Stable Equilibrium Point ◽  
First Order ◽  
The Social

A large number of problems in natural sciences (physics, chemistry, biology, astronomy, etc.) and the social sciences (engineering, economic, military, etc.) can be described by using differential equations. In this paper, we study two kinds of differential equations, i.e., the first order differential equation and the second order differential equation, and give the definitions of equilibrium point and stable equilibrium point. Moreover, we discuss the judgment methods of stable equilibrium point, and give the application of stable equilibrium point in the species coexistence problem.

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