On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

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.

     

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.     

Anti-periodic solutions for second order differential equations

By using the bilinear form and the Leray-Schauder principle, we prove the new anti-periodic existence results for second order differential equations.     

A remark on the existence of slowly growing positive solutions to second order super-linear ordinary differential equations

In this paper second order super-linear ordinary differential equations are considered, and a sufficient condition for the existence of a slowly growing positive solution is given.     

Unbounded solutions of differential equations of second order

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

Nonprincipal solutions of half-linear second order differential equations

We consider the nonoscillatory half-linear second order differential equation

(∗)     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Nonoscillatory solutions of forced differential equations of the second order

We prove that under certain assumptions if lu + b²u = 0, b > 0, where l is an operator Schrödinger equation defined in section 2 and u exp(?br) → 0 as r → ∞ then u?L². This result shows in particular that the resonant states behave exactly as O(r^?1, exp(ikr)) as r → ∞. It also provides dichotomy theorems in a general situation.     

Growth of solutions of second order linear differential equations

We consider the differential equation , where and are entire functions. Provided and as outside a set of finite logarithmic measure, we prove that all nonconstant solutions of this equation are of infinite order.

     

Asymptotic positivity of solutions of second order differential equations

A body in a damped oscillator eventually stops at the origin. Can we drag the body to the positive side by giving a positive driving force? Unfortunately, due to the oscillatory motion of the body, it is not true in general. In this paper, we give a sufficient condition on the driving force guaranteeing the asymptotic positivity of the position of the body, which means the negative part of the position vanishes in time. Also the result will be extended to a wider class of differential equations including the damped oscillator.     

Growth of solutions of second order linear differential equations

This paper is devoted to studying the growth of solutions of equations of type f^″+h(z)eazf^′+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.