On the existence of periodic solutions of Rayleigh equations

In this paper, we study the existence of periodic solutions of Rayleigh equation

Unbounded solutions of differential equations of second order

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

We establish existence of asymptotically almost periodic mild solutions for a class of semi-linear second-order abstract retarded functional differential equations with infinite delay. Research supported in part by FONDECYT, grant 1050314.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation where p > 0 is a constant. These results are extensions of the earlier results of Hille, Wintner, Opial, Yan for second order linear differential equations and include the recent results of Li and Yeh, Kusano and Yoshida, Yang and Lo for half-linear differential equations. Authors’ address: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China     

On periodic solutions of second-order differential equations with attractive-repulsive singularities

Sufficient conditions for the existence of a solution to the problem

     

On the periodic boundary-value problem for systems of second-order nonlinear ordinary differential equations

The periodic boundary value problem for systems of secondorder ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.     

Multiple periodic solutions of scalar second order differential equations

We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987) [1] and Del Pino, Manasevich and Murua (1992) [2]. The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré-Birkhoff Fixed Point Theorem.     

Pairs of positive solutions for the periodic scalar p-Laplacian

We study a nonlinear periodic problem driven by the scalar p-Laplacian and having a nonsmooth potential (hemivariational inequality). Using a combination of variational techniques and degree-theoretic methods based on a degree map for certain multivalued perturbations of (S)₊operators, we establish the existence of two positive solutions.     

On general algebraic mechanisms for producing centers in polynomial differential systems

We classify nondegenerate centers of systems of the form

On the periodic solutions of certain class of seventh-order differential equations

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).     

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

A global solution curve for a class of periodic problems, including the pendulum equation

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Similarity solutions of differential equations for boundary layer approximations in porous media

This paper is concerned with the ordinary differential equation

On the existence and stability of periodic orbits in non ideal problems: General results

In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E. , where , are T periodic functions of t and there is a ₀ ∈ Ω such that f ( a ₀) = 0 and f ′( a ₀) is a nilpotent matrix. When n = 3 and f (x) = (0, q (x ₃) , 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system: the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld Effect as a bifurcation of periodic orbits.