高阶非线性时滞差分方程解的振动性和渐近稳定性

Oscillation and asymptotic behavior of solutions for the higher order nonlinear delay difference equations

研究一类高阶非线性时滞差分方程Δd+1xn-1+pnf(xn-τ)+qng(xn-σ)=0的解的振动性和差分方程Δd+1xn-1+pnf(xn-τn)+qng(xn-σn)=0解的渐近稳定性,其中d为偶数,pn,qn≥0.τ,σ>0.τn,σn都是整数,f,g是非减函数,当x≠0时xf(x)-xg(x)>0.在文献[1-4]的基础上,给出其振动的充要条件,指出非振动解当n→+∞时渐近趋于零或趋于非零有限值时的充分条件.改进和推广了[5-6]相应的结果,且举出两例说明定理的应用.

Consider the oscillation of solutions of the higher order nonlinear difference equation with variable delay for △^（d＋1） x（n-1）＋pnf（x（n-τ））＋qng（x（n-σ））=0 and asymptotic behavior of △^（d＋1）x（n-1）＋pnf（x（n-τn））＋qng（x（n-σn））=0 where d is enven, Pn,qn≥0.τ,σ〉0.and τn,σn are integer number, f, g are non -decreasing function such that x≠0 xf（x）-xg（z）〉0 . Based on [1] - [4], A necessary and sufficient condition under which solutions are conditions. Some sufficient conditions under which the non - oscillatory solution of the equation tends to zero or finite value as n→∞ are obtained. Some results in [5 ] and [6] are extended and improved. Applications of main results are shown by two examples.