一类带Φ-Laplace算子的差分方程的非振荡解问题

On Non-Oscillatory Solutions for a Class of Difference Equation withΦ-Laplace Operator

主要研究带Φ-Laplace算子的差分方程Δ(a nΦ(Δx n))+b n|x n+1|γsgn x n+1=0 n≥1,γ>0的非振荡解问题.在Φ,{a n}和{b n}分别满足一定条件下给出方程的非振荡解是最终严格单调的,并依据非振荡解的极限行为将其分为4类.利用Schauder不动点定理和离散型Lebesgue控制收敛定理证明了方程的4类非振荡解存在.

This paper deals with the problem of non-oscillatory solutions for difference equationΔ(a nΦ(Δx n))+b n|x n+1|γsgn x n+1=0 n≥1,γ>0 involvingΦ-Laplace operator.It gives that all of the non-oscillatory solutions are eventually strongly monotone whenΦand the sequences{a n},{b n}satisfy certain conditions.Then it classifies them into four classes according to the behaviors of the non-oscillatory solutions.Moreover,the paper proves the existence of four types of non-oscillatory solutions to the equation by the Schauder fixed point theorem and the discrete analog of the Lebesgue dominated theorem.