On the iteration of a continuous mapping of a compact space into itself

1950 ◽  
Vol 46 (1) ◽  
pp. 46-56 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Continuous Mapping ◽  
Transformation Theory ◽  
Dependent Variables ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Mechanics ◽  
Image Position ◽  
System 1 ◽  
Lipschitz Conditions ◽  
Elementary Topology

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the formand the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).

On the asymptotic behaviour of the solutions of a class of non-linear differential equations

1950 ◽  
Vol 46 (3) ◽  
pp. 406-418
Author(s):  
F. G. Friedlander
Keyword(s):  
Existence And Uniqueness ◽  
Lipschitz Constant ◽  
Asymptotic Properties ◽  
Physical Significance ◽  
Right Hand ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Mechanics ◽  
Image Position ◽  
The Right

1. This paper is concerned with certain asymptotic properties of the solutions of the differential equationwhere dots indicate differentiation with respect to t, k is a small parameter, and f(x, ẋ, t) satisfies certain conditions which will be formulated below. Equations of this type occur frequently in non-linear mechanics; for k = 0 a system satisfying (1·1) behaves as a harmonic oscillator. To ensure the existence and uniqueness of the solutions of (1·1) it must be assumed that the right-hand side is bounded and satisfies a Lipschitz condition, at least for finite x, ẋ and say all t ≥ 0. The parameter k may be considered as a measure of the ‘smallness’ of the upper bound, and of the Lipschitz constant, of the right-hand side, and need not have any intrinsic physical significance.

Solutions of a non-linear differential equation. I

1967 ◽  
Vol 63 (3) ◽  
pp. 743-754 ◽  
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

We consider in this paper solutions of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

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 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

Characterization of Non-Linear Transformations Possessing Kernels

1970 ◽  
Vol 22 (3) ◽  
pp. 449-471 ◽  
Author(s):  
Victor J. Mizel
Keyword(s):  
Random Processes ◽  
Fading Memory ◽  
Linear Functionals ◽  
Linear Transformations ◽  
Main Motivation ◽  
Carathéodory Conditions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

Recently, in collaboration with Martin [10] and Sundaresan [11], I obtained a characterization of certain classes of non-linear functionals defined on spaces of measurable functions (see also [12]). The functionals in question had the form(1.1)with a continuous “kernel” φ: R → R,or(1.2)with a separately continuous kernel φ: R2 → R. There are direct applications of this work to the theory of generalized random processes in probability (see [8]) and to the theory of fading memory in continuum mechanics [3]. However, the main motivation for these studies was an interest in possible application to the functional analytic study of non-linear differential equations. From the standpoint of this latter application it would also be desirable to characterize the broader class of functionals having the form(1.3)where the kernel φ: R × T → R satisfies “Carathéodory conditions”.

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.

Commuting flows and conservation laws for Lax equations

1979 ◽  
Vol 86 (1) ◽  
pp. 131-143 ◽  
Author(s):  
George Wilson
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Kdv Equation ◽  
Lax Representation ◽  
Linear Evolution ◽  
Commuting Flows ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In recent years there has been great progress in the study of certain systems of non-linear partial differential equations, namely those that have a ‘Lax representation’Here P and L are linear differential operators in one variable x, whose coefficients are l × l matrices of functions of x and t. Thus L has the formwhere each ui is a matrix of functions ui,αβ(x, t), 1 ≤ α, β ≤ l. The symbol Lt means that we differentiate each coefficient of L, and as usual [P, L] = PL − LP. The coefficients of P are supposed to be polynomials in the ui, αβ and their x-derivatives, so that (1·1) is equivalent to a system of non-linear ‘evolution equations’ for the variables ui, αβ. The simplest example is the Korteweg–de Vries (KdV) equationwhich has a Lax representation with(Here l = 1, and there is only one coefficient ui, αβ. ) The connexion between the KdV equation and this ‘Schrodinger operator’ L was discovered by Gardner, Greene, Kruskal and Miura(6), but it was P. Lax (13) who first pointed out explicitly what we call the Lax representation given by (1·1) and (1·2). The notation (P, L), due to Gel'fand and Dikii(7), reflects this fact.

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).

Analytical theory of non-linear oscillations

1974 ◽  
Vol 76 (1) ◽  
pp. 285-296 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Existence Theorem ◽  
Wide Class ◽  
Periodic Oscillations ◽  
Real Domain ◽  
Non Linear ◽  
General Existence ◽  
Linear Differential ◽  
Image Position

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equationwhere g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

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