Infinite Systems of Differential Equations

1976 ◽  
Vol 28 (6) ◽  
pp. 1132-1145 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Systems Of Nonlinear Equations ◽  
Infinite Systems ◽  
Systems Of Differential Equations ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Image Position

In an earlier paper [7], we have studied the existence, uniqueness and asymptotic behavior of solutions to certain infinite systems of linear differential equations with constant coefficients. In the present paper we are interested in systems of nonlinear equations whose coefficients are not necessarily constants; more specifically, we are concerned with infinite systems of the form

On Infinite Systems of Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 691-703 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Differential Equations ◽  
Infinite System ◽  
Initial Conditions ◽  
Linear Differential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Infinite Systems ◽  
Linear Differential ◽  
Image Position

Let A = [αtj] (i,j = 1, 2, …) be an infinite matrix with complex entries, and let z = (ζj) (j = 1, 2, …) be a sequence of complex numbers. In this paper we wish to investigate the existence, uniqueness and asymptotic behavior of solutions to the infinite system of linear differential equationswith the initial conditions

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1

1985 ◽  
Vol 100 (3-4) ◽  
pp. 301-326 ◽  
Author(s):  
M. H. Lantsman
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Algebraic Equations ◽  
Order Of Growth ◽  
Constant Coefficients ◽  
Independent Variable ◽  
Linear Differential ◽  
Image Position ◽  
Asymptotic Representations

SynopsisWe consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equationcan be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

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.

Infinite Systems of Differential Equations II

1979 ◽  
Vol 31 (3) ◽  
pp. 596-603 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Differential Equations ◽  
Infinite System ◽  
Nonlinear Differential Equations ◽  
Coefficient Matrix ◽  
Nonnegative Real Number ◽  
Initial Value ◽  
Infinite Systems ◽  
Systems Of Differential Equations ◽  
The Stability ◽  
Image Position

This paper is a continuation of earlier work [6], in which we studied the existence and the stability of solutions to the infinite system of nonlinear differential equations(1.1)i = 1, 2, …. Here s is a nonnegative real number, Rs = {t ∈ R: t ≧ s}, and denotes a sequence-valued function. Conditions on the coefficient matrix A(t) = [aij(t)] and the nonlinear perturbation were established which guarantee that for each initial value c= {ct} ∈ l1, the system (1.1) has a strongly continuous l1valued solution x(t) (i.e., each is continuous and converges uniformly on compact subsets of Rs). A theorem was also given which yields the exponential stability for the nonlinear system (1.1).

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