Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

Second order linear differential equations with almost periodic solutions

In this paper, a characterization of almost periodic strongly continuous Sine operator function is given, and in a Hilbert space, it is shown that the almost periodicity of a Sine operator function implies that of the corresponding Cosine operator function.     

Almost periodic solutions of linear partial differential equations

Let L be an arbitrary linear partial differential operator and let f be an almost periodic function for t in R^m. In this paper we present sufficient conditions that a bounded solution u of Lu = f be almost periodic. Our work generalizes the theorem of Sibuya [5] for Poisson's equation and the theorems of Favard [3] and Bochner [1] for ordinary differential equations.     

On periodic solutions of linear differential equations with pulsed influence

We study periodic solutions of ordinary linear second-order differential equations with publsed influence at fixed and nonfixed times. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 141–148, January, 1997.     

Symmetric linear multistep methods for second-order differential equations with periodic solutions

Special symmetric linear multistep methods for second-order differential equations without first derivatives are proposed. The methods can be tuned to a possibly a priori knowledge of the user on the location of the frequencies, that are dominant in the exact solution. On the basis of such extra information the truncation error can considerably be reduced in magnitude. Numerical results are compared with results produced by the symmetric methods of Lambert and Watson and the method of Gautschi.     

On subnormal solutions of second order linear periodic differential equations

In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.     

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

Abel-like differential equations with no periodic solutions

We present various criteria for the non-existence of positive periodic solutions of generalized Abel differential equations with periodic coefficients that can change sign. As an application, we obtain some families of planar vector fields without limit cycles.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

Stability of solutions of a system of linear differential equations with periodic stochastic coefficients

Systems of equations for first and second moments are investigated and transformed. Stability of solutions of a first-order linear differential equations is analyzed. Stability of solutions of the stochastic Mathieu equation is investigated and the boundaries of the instability region are determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 119–126, 1986.     

Existence results of periodic solutions for non-autonomous differential delay equations with asymptotically linear properties

In this paper, we study a class of non-autonomous differential delay equations which can be changed to Hamiltonian systems. By estimating Maslov-type index of the related Hamiltonian systems at infinity and at origin, we establish the existence of periodic solutions of the differential delay equations.     

Certain differential equations have only isolated periodic orbits

Summary Some results of Poincaré and Dulac concerning non-isolated periodic orbits and singular cycles in the plane are here extended to certain classes of autonomous analytic ordinary differential equations of higher dimension. The equations in these classes are then shown to have only isolated periodic orbits provided that all their critical points satisfy a simple condition. A further condition at infinity can ensure that the equation has only finitely many periodic orbits.     

First order ordinary differential equations with several periodic solutions

The method of upper and lower solutions and convexity arguments are used to prove sharp results for the existence and multiplicity of periodic solutions for first order ordinary differential equations depending upon a parameter.Dedicated to Professor H. W. Knobloch for his sixtieth birthday