Optimal Existence Theory for Single and Multiple Positive Periodic Solutions of Functional Differential Equations

This paper deals with a new optimal existence theory for single and multiple positive periodic solutions of functional differential equations using a fixed-point theorem in cones. We illustrate our theory by examining several biomathematical models. The paper improves and extends previous results in the literature.     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

Unique Periodic Orbits for Delayed Positive Feedback and the Global Attractor

The delay differential equation, (t)=–x(t)+f(x(t–1)), with >0 and a real function f satisfying f(0)=0 and f>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C=C([–1, 0], ) of initial data for solutions of the equation. In this paper, for a set of and f which include examples from neural network theory, we show that this three-dimensional set is the global attractor, i.e., the compact invariant set which attracts all bounded subsets of C. The proof involves, among others, results on uniqueness and absence of periodic orbits.     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

Positive Solutions of Linear Impulsive Differential Equations

The paper deals with the existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to the investigation of a similar problem for higher-order linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of the equations. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 291–297, July–September, 2005.     

Periodic Solutions of Second Order Differential Equations with Discontinuous Nonlinearities

We consider the periodic solutions of

Heteroclinic Connection of Periodic Solutions of Delay Differential Equations

For a certain class of delay equations with piecewise constant nonlinearities we prove the existence of a rapidly oscillating stable periodic solution and a rapidly oscillating unstable periodic solution. Introducing an appropriate Poincaré map, the dynamics of the system may essentially be reduced to a two dimensional map, the periodic solutions being represented by a stable and a hyperbolic fixed point. We show that the two dimensional map admits a one dimensional invariant manifold containing the two fixed points. It follows that the delay equations under consideration admit a one parameter family of rapidly oscillating heteroclinic solutions connecting the rapidly oscillating unstable periodic solution with the rapidly oscillating stable periodic solution.      

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Positive Solutions to Singular Second and Third Order Differential Equations for Quantum Fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system for quantum fluids with positive constant temperature in bounded domains for arbitrary large data. The momentum equation can be written as a dispersive third-order equation for the particle density where viscous effects are incorporated. The phenomena that admit positivity of the solutions are studied. The cases, one space dimensional dispersive or non-dispersive, viscous or non-viscous, are thoroughly analyzed with respect to positivity and existence or non-existence of solutions, all depending on the constitutive relation for the pressure law. We distinguish between isothermal (linear) and isentropic (power law) pressure functions of the density. It is proved that in the dispersive, non-viscous model, a classical positive solution only exists for “small” (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist for “large” current densities. The dispersive, viscous problem admits a classical positive solution for all current densities, both for the isentropic and isothermal case, with an “ultra-diffusion” condition. The proofs are based on a reformulation of the equations as a singular elliptic second-order problem and on a variant of the Stampacchia truncation technique. Some of the results are extended to general third-order equations in any space dimension. Accepted July 1, 2000?Published online February 14, 2001     

On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

For periodic solutions to the autonomous delay differential equation

Global Continua of Rapidly Oscillating Periodic Solutions of State-Dependent Delay Differential Equations

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

Delay Equations with Rapidly Oscillating Stable Periodic Solutions

We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently large.     

Existence and Asymptotic Behavior of Positive Solutions of Neutral Impulsive Differential Equations

In this article, we consider first-order neutral impulsive differential equations with constant coefficients and constant delays. We study the asymptotic behavior of eventually positive solutions of these equations and establish necessary and sufficient conditions for the existence of such solutions. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 304–318, July–September, 2005.     

Periodic or Bounded Solutions of Carathéodory Systems of Ordinary Differential Equations

In this paper we extend the guiding function approach to show that there are periodic or bounded solutions for first order systems of ordinary differential equations of the form x′=f(t,x), a.e. t∈[a,b], where f satisfies the Carathéodory conditions. Our results generalize recent ones of Mawhin and Ward.     

Construction of Periodic Solutions of Linear Systems of Partial Differential Equations with Linear Deviations of Arguments

We prove a theorem on the existence of a solution of a system of partial differential equations with linearly transformed arguments that is continuously differentiable and bounded on ².__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 462–467, October–December, 2004.     

Metastable Periodic Patterns in Singularly Perturbed Delayed Equations

We consider the scalar delayed differential equation e[(x)\dot](t)=-x(t) +f(x(t-1)){\epsilon\dot x(t)=-x(t)\,+f(x(t-1))}, where ${\epsilon\,{>}\,0}${\epsilon\,{>}\,0} and f verifies either df/dx > 0 or df/dx < 0 and some other conditions. We present theorems indicating that a generic initial condition with sign changes generates a solution with a transient time of order exp(c/e){{\rm exp}(c{/}\epsilon)}, for some c > 0. We call it a metastable solution. During this transient a finite time span of the solution looks like that of a periodic function. It is remarkable that if df/dx > 0 then f must be odd or present some other very special symmetry in order to support metastable solutions, while this condition is absent in the case df/dx < 0. Explicit e{\epsilon}-asymptotics for the motion of zeroes of a solution and for the transient time regime are presented.     

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.