An integral criterion for oscillation of linear differential equations of second order

It is proved that if for some n>2 the function x^1–nA_n(x), where A_n(x) is the n-th primitive ofa(x), is not bounded above, then the equation y +a(x)y = 0 oscillates.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 249–251, February, 1978.In conclusion, I thank R. S. Ismagilov for useful discussions about the problem of osillation.     

On periodic solutions of non-autonomous second order hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system

On periodic solutions of nonlinear second order differential systems

The paper deals with the second order differential systems with periodic boundary conditions. In the first part of the paper four different methods are employed to find the unique solution of the linear systems. One of the methods is a shooting type which converts the periodic boundary value problem into its equivalent initial value problem. In the second part of the paper several sufficient conditions are provided for the existence and uniqueness for the nonlinear systems. The technique developed for the linear systems to convert into its equivalent initial value problem is used in an iterative way for the nonlinear systems. It is shown that the iterations converge quadratically.     

Non-collision periodic solutions of second order singular dynamical systems

We study the existence of non-collision periodic solutions for second order singular dynamical systems. The repulsive case and the attractive case are dealt with using a unified topological approach. The proof is based on a well-known fixed point theorem for completely continuous operators, involving a new type of cone. We do not need to consider so-called strong force conditions. Moreover, for the repulsive case, the critical case can be covered. Recent results in the literature, even in the scalar case, are complemented, generalized and improved.     

Stability criterion for second order linear impulsive differential equations with periodic coefficients

In this paper we obtain instability and stability criteria for second order linear impulsive differential equations with periodic coefficients. Further, a Lyapunov type inequality is also established. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotating periodic solutions for super-linear second order Hamiltonian systems

In this paper we consider a class of super-linear second order Hamiltonian systems. We use Morse theory to obtain the existence and multiplicity of rotating periodic solutions, which might be periodic, subharmonic or quasi-periodic ones.     

A stability criterion for periodic solutions of two-dimensional discontinuous dynamical systems

We establish stability conditions for periodic solutions of two-dimensional systems of ordinary differential equations with pulse influence. We study the properties of the jump operator for such systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1262–1266, September, 1999.     

Notes on periodic solutions of subquadratic second order systems

Some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory.     

New existence of periodic solutions for second order non-autonomous Hamiltonian systems

By using mountain pass theorem and local link theorem, some existence theorems are obtained for periodic solutions of second order non-autonomous Hamiltonian systems under local superquadratic condition and other suitable conditions.     

Multiple periodic solutions for second order systems with changing sign potential

This paper deals with the multiplicity of solutions of a second order nonautonomous system. We extend a previous result of the author relaxing the assumptions on the sign of the potential.     

Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

In this paper, we consider the second order Hamiltonian system $\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= ?d ²/dt ² ?A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.