Large-Amplitude Periodic Solutions for Differential Equations with Delayed Monotone Positive Feedback

The aim of this paper is to show that the structure of the global attractor for delayed monotone positive feedback can be more complicated than the union of spindle-like structures between consecutive stable equilibria with respect to the pointwise ordering. Large amplitude periodic orbits—in the sense that they are not between two consecutive stable equilibria—are constructed for nonlinearities close to a step function. For some nonlinearities there are exactly two large amplitude periodic orbits. By describing the unstable sets of these periodic orbits, a complete picture is obtained about the global attractor outside the spindle-like structures.     

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.     

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.     

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

For periodic solutions to the autonomous delay differential equation

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.     

Existence and Multiplicity of Positive Solutions for a Fourth-Order P-Laplace Equations

Ｉｎｔｒｏｄｕｃｔｉｏｎｐ＿Ｌａｐｌａｃｅｅｑｕａｔｉｏｎｓａｒｉｓｅｉｎａｖａｒｉｅｔｙｏｆｉｎｔｅｒｅｓｔｉｎｇａｐｐｌｉｃａｔｉｏｎｓ,ｔｈｅｙａｐｐｅａｒａｓｍｏｄｅｌｓｏｆｇａｓｄｉｆｆｕｓｉｏｎｔｈｒｏｕｇｈｐｏｒｏｕｓｍｅｄｉａ,ａｌｓｏｉｎｔｈｅｔｈｅｏｒｙｏｆｔｈｅｒｍａｌｓｅｌｆ＿ｉｇｎｉｔｉｏｎｏｆａｃｈｅｍｉｃａｌｌｙａｃｔｉｖｅｍｉｘｔｕｒｅｏｆｇａｓｅｓ(ｓｅｅ [1 ]ａｎｄｉｔｓｒｅｆｅｒｅｎｃｅｓ) ,ｒｅｃｅｎｔｌｙ ,ＷＡＮＧＪｕｎ＿ｙｕ (Ｒｅｆ.[1 ] )ｓｔｕｄｉｅｄｔｈ…     

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.      

Periodic Solutions of Delay Differential Equations with a Small Parameter: Existence,Stability and Asymptotic Expansion

We consider a scalar delay differential equation with a small parameter, and employ Walthers method to obtain a result on the existence and stability of a slowly oscillatory periodic solution that represents a refinement of the estimate for the Lipschitz constant of a returning map. We also develop a matching method and obtain asymptotic expansions of the slowly oscillatory periodic solution and its minimal period.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthdayAMS subject classifications: 34K15; 34K20; 34C25.     

Periodic Perturbations of a Class of Functional Differential Equations

Journal of Dynamics and Differential Equations - We study the existence of a connected “branch” of periodic solutions of T-periodic perturbations of a particular class of functional...     

Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations

The occurrence of almost automorphic dynamics for monotone non-autonomous recurrent finite-delay functional differential equations is analyzed. Topological methods are used to ensure its presence in the case of existence of semicontinuous semi-equilibria. When these semi-equilibria are continuous and strong, the presence of almost automorphic extensions is persistent under small perturbations. The above method provides a minimal set isomorphic to the base in the case of a convex semiflow. Some examples show the applicability of these results.     

Periodic Solutions of Second Order Differential Equations with Discontinuous Nonlinearities

We consider the periodic solutions of

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 .     

Center Manifold Theory for Functional Differential Equations of Mixed Type

We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for such equations. We illustrate the application range of our results by discussing an economic life-cycle model that gives rise to functional differential equations of mixed type.     

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 a Boundary-Value Problem of Periodic Type for First-Order Linear Functional Differential Equations

We establish unimprovable sufficient conditions for the unique solvability of the boundary-value problem

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.     

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.