Periodic Solutions Of Differential Equations Research Articles

Bendixon‐Type Theorems with Applications

AbstractBendixson's theorem on the nonexistence of periodic solutions of differential equations is generalized to the N ‐dimensional case. An application of the theorem in formal reaction kinetics is shown. Then a theorem analogous to Schneider's generalization is given. Finally, a theorem on implicit systems is formulated and applied to get a sufficient condition for the absence of periodic solutions in a generalized Liénard‐equation.

Shorter Notes: Better Bounds for Periodic Solutions of Differential Equations in Banach Spaces

Shorter Notes: Better Bounds for Periodic Solutions of Differential Equations in Banach Spaces

Estimates for periodic solutions of differential equations

A class of infinite delay equations which are per- turbations of finitely delayed equations is considered. Asymptotic estimates are obtained for the solutions from which we get the existence of periodic solutions. We review a few technics for fixed point theorems

Better bounds for periodic solutions of differential equations in Banach spaces

Let f f be Lipschitz with constant L L in a Banach space and let x ( t ) x(t) be a P P -periodic solution of x ′ ( t ) = f ( x ( t ) ) x’(t) = f(x(t)) . We show that P ⩾ 6 / L P \geqslant 6/L . An example is given with P = 2 π / L P = 2\pi /L , so the bound is nearly strict. We also give a short proof that P ⩾ 2 π / L P \geqslant 2\pi /L in a Hilbert space.

On the set of periodic solutions of differential equations of Riccati type

The purpose of this paper is to expand upon the results obtained in [4]. We consider the set H of differential equationswhere p and r are continuous real-valued functions of period ω (ω being fixed throughout). The equation (1.1) is denoted by P or (p,r), and we regard H as the set of pairs of continuous functions of period ω.On H we define a norm:

Periodic solutions of differential equations and one-parameter family of operators

A method for computing bounds to periodic solutions of the system y 1=f(t, y), y,f∈R n, f periodic in t with period T, (1) is studied using an integral equation y=Gλy depending on a parameter λ, and whose solutions for each λ are the periodic solutions of (1). λ is then given a value λ 0 so as to determine a set S for which G λ 0 S ⊂ S. Since G λ 0 is proved to satisfy Schauder's conditions, then in S there is a solution of (2), i.e., a periodic solution of (1). The method, which evidently also constitutes an existence proof in S, has the peculiarity of being quite general since it can be used to bound other kinds of solutions of the system y′ = f( t, y), by simply setting up a different integral equation y = G λ y which possesses the solutions to be bounded. This method has then been used to bound periodic solutions of a system arising from the dynamics of two floating bodies, for which G λ S ⊂ S has the form of a system of inequalities.

Weak almost periodic solutions of differential equations

A very important question in many practical situations is to known if there exists a mean value of a bounded solution x of a differential equation x’ =f(t, x), By the mean value we understand the limit lim,,, (l/T) ]lx(f) dr if it exists. Eberlein in [l] has found a class of functions which have the mean value-the weak almost periodic (wap) functions. A bounded continuous function x is weak almost periodic if the set of its translations (x,: s E R} is relatively compact in the weak topology of the space of bounded continuous functions on R. The purpose of this paper is to examine the properties of wap solutions of the differential equation. We demonstrate theorems on the existence of wap solution of the differential equation with the weak almost periodic right-hand side.

Investigations in the theory of bounded and periodic solutions of differential equations

Investigations in the theory of bounded and periodic solutions of differential equations

Periodic solutions of differential equations in Banach spaces

Let X be a Banach space, D⊂X, f: [0,∞)xD→X continuous and ω-periodic. In this paper we consider various conditions on D and f sufficient for existence of an ω-periodic solution of the differential equation u′=f(t,u). In the main, we shall assume that D is closed bounded and convex and f satisfies a boundary condition at δD such that D is flow invariant for u′=f(t,u). The map f is assumed to be either compact or dissipative or a certain perturbation of such maps.

On finding periodic solutions of differential equations

On finding periodic solutions of differential equations

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay. Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f( x) and the fact that the origin belongs to the range of f. This situation is explored in [19].

On the Spectrum of almost periodic solutions of an abstract differential equation

AbstractIf a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

On the existence and uniqueness of periodic solutions for difference equations of second order

On the existence and uniqueness of periodic solutions for difference equations of second order

Bifurcations in nerve membrane dynamics.

The bifurcation theory involves qualitative analysis of variations in singular and certain periodic solutions of differential equations representing a dynamic system. The model proposed by Hodgkin and Huxley for nerve membrane dynamics illustrates certain concepts from this theory. The membrane current, Im is shown to be the bifurcation parameter. At certain values of this parameter, namely at bifurcation values, the character and the stability properties of singular solutions and the other limiting sets such as limit cycles change as predicted by the theory.

Periodic solutions of differential equations in the cylindrical space

Periodic solutions of differential equations in the cylindrical space

Bounds for periodic solutions of differential equations in Banach spaces

Bounds for periodic solutions of differential equations in Banach spaces

The bifurcation of periodic solutions of differential equations having two parameters

The bifurcation of periodic solutions of differential equations having two parameters

Periodic solutions of differential equations with almost constant time lag

Delay-differential equations, or functional differential equations as they are frequently called, have become important in many fields. These equations have been studied by many authors and a moderate sized literature has been developed by Hale and his associates for the case where the equations are linear or slightly perturbed from the linear case. Cooke [l] has pointed out the need for further study of delay-differential equations with perturbed lag, with particular need for investigating the existence of oscillations. Although Perello [2] has established procedures for studying the oscillations of perturbed linear systems, the procedure fails to apply to Cooke’s problem. It is the purpose of this paper to establish that by a slight modification in Perello’s banach spaces, his procedure may be used identically in this case. We will use the notation which Hale and his associates have made standard and will assume that the reader is thoroughly familiar with their theory for linear autonomous functional differential equations. An extensive treatment of this theory may be found in Hale [3] and a brief sketch in Perello [2]. Perello considered equations of the form

Asymptotically Almost Periodic Solutions of Differential Equations

In this paper we examine asymptotically almost periodic solutions of an almost autonomous differential equation. Solutions of this type have been examined previously for plane systems by Wong and Burton [8] and Utz and Waltman [6]. In both of the above papers solutions of the almost autonomous equation spiral to a periodic orbit of the limiting equations and are thus shown to be asymptotically almost periodic. This device is also exploited in the theorems proved here, although the solutions will not be required to approach periodic orbits of the limiting equation in such a uniform manner. Also, conditions will be given so that solutions satisfy a somewhat more standard type of asymptotic almost periodicity than the type discussed in [8] and [6].

On Ryabov’s Asymptotic Characteization of the Solutions of Quasi-Linear Differential Equations with Small Delays

On Ryabov’s Asymptotic Characteization of the Solutions of Quasi-Linear Differential Equations with Small Delays