Chuanxi Qian

A three point boundary value problem for nonlinear fourth order differential equations

Abstract In this paper, the authors consider the nonlinear fourth order ordinary differential equation (E) u″″(t)=λg(t)f(u), 0 with the boundary conditions (B) u(0)=u′(1)=u″(0)=u″(p)−u″(1)=0. Some results on the existence and nonexistence of positive solutions to problem (E)–(B) are obtained. Results on the existence of infinitely many positive solutions are also presented. Examples are included to demonstrate that the results are sharp.

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem (E) x (2m) (t) + (-1) m+1 f(x(t)) = 0, 0 < t < 1, (B) x (2i) (0) = x (2i) (1) = 0, i = 0, 1, 2, …, m - 1, and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

Asymptotic behavior of a third-order nonlinear differential equation

Abstract Consider the third-order nonlinear differential equation x‴+ψ(x,x′)x″+f(x,x′)=p(t), where ψ,f,fx∈C(R×R,R) and p∈C([0,∞),R). We obtain sufficient conditions for every solution of the equation to be bounded; we also establish criteria for every solution of the equation to converge to zero.

Global stability in a nonlinear difference equation

Consider the nonlinear difference equation where By using Liapunovs method, we establish some sufficient conditions for Eq. (0.1) to have a globally asymptotically stable positive equilibrium. Our results can be applied to some difference equations derived from mathematical biology.

A General Comparison Result for Higher Order Nonlinear Difference Equations With Deviating Arguments

The authors consider m -th order nonlinear difference equations of the form D m p x n + i h j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m S 1, n ] N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i j ( D i m 1 p x n ), i =1,2,…, m , j x n = x n +1 m x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m L 1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n ) M X as n M X , and h j : N 0 2 R M R is continuous with uh j ( n , u )>0 for u p 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included.

Convergence of a Difference Equation and Its Applications

In this paper, we establish a sufficient condition for every solution of the forced difference equation

Formulas for Liapunov functions for systems of linear difference equations

x_{n+1} - x_n + p_nx_{n-k} = r_n, \ \ n=0, 1, \ldots