Boundedness of solutions and asymptotic properties of some systems of differential equations

1994 ◽  
Vol 46 (7) ◽  
pp. 1035-1038
Author(s):  
V. G. Gorodetskii
Keyword(s):  
Differential Equations ◽  
Asymptotic Properties ◽  
Boundedness Of Solutions ◽  
Systems Of Differential Equations

scholarly journals On time transformations for differential equations with state-dependent delay

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Alexander Rezounenko
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
The State ◽  
Delay Systems ◽  
Constant Delay ◽  
Systems Of Differential Equations ◽  
State Dependent ◽  
State Dependent Delay ◽  
Time Transformations

AbstractSystems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

scholarly journals Investigation of asymptotic behavior of solutions of one class of systems of differential equations with deviation of an argument

2019 ◽  
Author(s):  
N. V. Varekh ◽  
N. L. Kozakova ◽  
A. O. Lavrentieva
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Mathematical Problem ◽  
Asymptotic Properties ◽  
Theoretical Physics ◽  
General Nature ◽  
Asymptotic Behavior Of Solutions ◽  
Time Interval ◽  
Systems Of Differential Equations ◽  
Derivatives Of

In this paper, we study the asymptotic behavior of solutions at an infinite time interval of one class of systems of differential equations with the deviation of an argument, which are a generalization of the Emden-Fowler equation in the sublinear case. Conditions were found under which each solution either oscillates strongly or all its components monotonically end to zero at infinity. Two theorems under different constraints on the deviation of an argument are proved. Equation d(n)y(t)/dtn + δ p(t)f(y(t)) = 0, f(u) = uα, δ = -1 or 1, has been the object of much research. Some cases of this equation are models of processes in theoretical physics (Emden, Fowler, Fermi equations). After that, this physical problem becomes a mathematical problem at an infinite interval. It is found that the asymptotic properties of the solutions depend on the sign δ, type of nonlinearity f(u) (f(u) = uα), (0< α <1 – sublinear case, α = 1 – linear case, α >1 – superlinear), n – even or odd. For this equation, conditions have already been found under which, when δ = 1 and n are even, all solutions oscillates; if n is odd, then each solution either oscillates or monotonically goes to zero indefinitely. If δ = -1, n is even, then each solution oscillates either monotonically to zero or to infinity when t → ∞ together with the derivatives of order (n -1). If δ = -1, n is odd, then each solution oscillates or is monotonically infinite for t → ∞ together with the derivatives of order (n -1). Then, the following results were obtained for differential systems and equations with the general nature of the argument rejection (differential-functional equations). The next stage of the study is to summarize the results for such systems. This article investigates the system of differential equations with the deviation of the argument for the case δ = 1, n = 3. The obtained results are refined and the results obtained earlier are generalized. Two theorems with different assumptions about rejection of the argument by analytical methods are proved. These theorems have different applications. The results of the study are a generalization of the sublinear case for odd n.

scholarly journals Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times

2021 ◽  
Vol 104 (4) ◽  
pp. 142-150
Author(s):  
O.N. Stanzhytskyi ◽  
◽  
A.T. Assanova ◽  
M.A. Mukash ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Averaging Method ◽  
Nonlinear Dynamical Systems ◽  
Asymptotic Properties ◽  
Stable Solution ◽  
Impulsive Systems ◽  
Nonlinear Dynamical ◽  
Systems Of Differential Equations ◽  
Averaged Systems ◽  
Fixed Times

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.

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