Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Existence of periodic solutions for first order differential equations

In this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities, namely the method of upper and lower solutions, and the alternative method for nonlinear problems at resonance. The results obtained are in terms of the behavior of the nonlinear part at infinity.     

Existence of positive periodic solutions of periodic boundary value problem for second order ordinary differential equations

In this paper the existence of positive solutions are obtained for a class of second order differential equations. The proof is based on the fixed point index theory in cones.     

Twist periodic solutions of second order singular differential equations

We study the twist periodic solutions of second order singular differential equations. Such twist periodic solutions are stable in the sense of Lyapunov and present much interesting dynamical features around them. The proof is based on the third-order approximation method. The estimates of periodic solutions of Ermakov-Pinney equations and the estimates on rotation numbers of Hill equations play an important role in the analysis.     

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.     

EXISTENCE OF SOLUTIONS FOR SECONDORDER IMPULSIVE DIFFERENTIAL EQUATIONS

In this paper, we apply the coincidence degree theory to study nonlinear second orderimPulsive periodic boundary value problems (PBVP), and show some sufficient conditions for the existence of solutions.     

Existence of almost periodic solutions of second order neutral delay differential equations with piecewise constant argument

By constructing almost periodic sequence solutions to difference equations, the existence of almost periodic solutions of neutral delay differential equations with piecewise constant argument

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded.     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded. This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006).     

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 the existence of almost periodic solutions of some dissipative second order differential equations

Summary The search for almost periodic solutions of any dissipative equation of the form(1.1), in which p(t) is an almost periodic function, has come to be closely linked up with a number of standard ? convergence ? restrictions on f, g′, g″ and k (see, for example,[2] and[3]). The object of the present paper is to show that as far as the existence, alone, of an almost periodic solution of(1.1) is concerned these ? convergence ? restrictions on f, g′, g″ and k are quite unnecessary. The first result (Theorem 1) shows in fact that the conditions(1.2) alone are quite sufficient for the existence of an almost periodic solution of(1.1); and Theorem 2 extends this result (though under stronger conditions on f and g) to the case in which the forcing function depends on x and x as well. Partially supported by N.S.F. research project G-57 at The University of Michigan.     

Existence of periodic solutions of second order nonlinear p-Laplacian difference equations

By using the critical point theory, the existence of periodic solutions to second order nonlinear p-Laplacian difference equations is obtained. The main approach used is a variational technique and the saddle point theorem. The problem is to solve the existence of periodic solutions of second order nonlinear p-Laplacian difference equations.     

Existence of nontrivial periodic solutions for first order functional differential equations

In this paper, in the case of not requiring the nonlinear terms to be non-negative the existence of nontrivial periodic solutions for the first order functional differential equations is considered by using the partial ordering theory.     

Existence of solutions for second order impulsive differential equations with deviating arguments

This paper deals with impulsive second order differential equations with deviating arguments. We investigate the existence of solutions of such problems with nonlinear boundary conditions. To obtain corresponding results we discuss also second order impulsive differential inequalities with deviating arguments.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Existence of oscillatory solutions of forced second order delay differential equations

In this paper, under weaker hypothesis, the global existence of oscillatory solutions is established for a forced second order nonlinear delay differential equation. The results of this paper generalize and improve some known results.     

Multiple periodic solutions for a class of second order differential equations

By using the coincidence degree method, the existence of multiple periodic solutions for a class of second order differential equations is obtained under the existence of upper and lower solutions.