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     

RBSDE''''s with jumps and the related obstacle problems for integral-partial differential equations

The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.     

Extremal solutions of periodic boundary value problems for first order integro-differential equations of mixed type

This paper investigates the maximal and minimal solutions of periodic boundary value problems for first order nonlinear integro-differential equations of mixed type by establishing a comparison result and using the monotone iterative technique.     

Nonexistence of positive solutions for 2 order integro-differential inequalities

In this paper, we shall show that under suitable conditions on f and K, the inequalities

imply that the integro-differential inequalities

have no positive solutions, respectively.     

RBSDE’s with jumps and the related obstacle problems for integral-partial differential equations

The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Global monotonicity and oscillation for second order differential equation

Oscillatory properties of the second order nonlinear equation

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.     

Periodic solutions for a kind of second order neutral functional differential equations

In this paper, criteria are established for the existence of periodic solutions to a second order sublinear neutral differential equation. Our method is based on careful a priori estimation and continuation theorem, and our sublinear condition is an improvement of the boundedness condition in some recent results.     

A probabilistic approach to second order variational inequalities with bilateral constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of aunique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.     

An iterative splitting approach for linear integro-differential equations

The motivation for this paper is to solve a model based on the dynamics of electrons in a plasma using a simplified Boltzmann equation. Such problems have arisen in active plasma resonance spectroscopy, which is used for plasma diagnostic techniques; see Braithwaite and Franklin (2009) [1]. We propose a modified iterative splitting approach to solve the Boltzmann equations as a system of integro-differential equations. To enable solution by fast and iterative computations, we first transform the integro-differential equations into second order differential equations. Second, we split each second order differential equations into two first order differential equations via a splitting approach. We carry out an error analysis of the higher order iterative approach. Numerical experiments with a simplified Boltzmann equation will be discussed, along with the benefits of computing with this splitting approach.     

Oscillatory solutions of fractional integro-differential equations

We give necessary conditions to get oscillatory solutions of a class of fractional order neutral differential equations with continuously distributed delay by means of the fractional derivative with respect to a given function. In particular, oscillatory solutions of the considered fractional equations with Caputo and Hadamard type of fractional derivatives are established. Some explicit examples are given to illustrate the main results.     

A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators

We prove the equivalence of the three different definitions of the viscosity solution for the integro-differential equation with the Lévy operator. The two of the definitions are known in the preceding works of the author and the others, and the last one is new. A construction of a sequence of the approximating test functions to the subsolution (or the supersolution) is indispensable for the proof, and it is done explicitly in the paper.     

Lp solutions of backward stochastic differential equations with jumps

Given $p\in (1,2)$, we study ${\mathbb{L}}^p$ solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)$-variables. We show that such a BSDEJ with $p$-integrable terminal data admits a unique ${\mathbb{L}}^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.     

Periodic solutions of singular second order differential equations: Upper and lower functions

In this paper, we continue the study of the periodic problem for the second-order equation u^′′+f(u)u^′+g(u)=h(t,u), where h is a Carathéodory function and f,g are continuous functions on (0,+∞) which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation.     

Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.     

Numbers of subnormal solutions for higher order periodic differential equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.     

Nonprincipal solutions of half-linear second order differential equations

We consider the nonoscillatory half-linear second order differential equation

(∗)     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.