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.     

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 positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equation

u^′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),     

Existence of positive periodic solutions of first‐order neutral differential equations

This paper presents the existence of positive periodic solutions for first‐order neutral differential equation with distributed deviating arguments. We apply Krasnoselskii's fixed point theorem to obtain our results. An example is given to support the theory. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

     

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE$u^{″}+\lambda a(t)g(u)=0,\lambda >0,$ where $g(x)$ is a positive function on ${\mathbb{R}}^{+}$, superlinear at zero and sublinear at infinity, and $a(t)$ is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities $g(x)$ when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.     

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.     

On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations

The existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations. The arguments are based only upon the positivity of the Green's functions and the Krasnosel'skii fixed point theorem. As an example, a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity.     

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.     

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.     

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.     

Existence of positive periodic solutions of first order neutral differential equations with variable coefficients

This work deals with the existence of positive $\omega$-periodic solutions for the first order neutral differential equation. The results are established using Krasnoselskii’s fixed point theorem. An example is given to support the theory.     

Existence of positive solutions of nonlinear fractional differential equations

Existence of positive solutions for the nonlinear fractional differential equation D^su(x)=f(x,u(x)), 0<s<1, has been studied (S. Zhang, J. Math. Anal. Appl. 252 (2000) 804-812), where D^s denotes Riemann-Liouville fractional derivative. In the present work we study existence of positive solutions in case of the nonlinear fractional differential equation:

L(D)u=f(x,u),u(0)=0,0<x<1,