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Vietnam Journal of Mathematics 38:1 (2010) 79-87  

The Solvability of a Higher-order Nonlinear Neutral Delay Difference Equation

Zhenyu Guo

Abstract.  This paper studies the solvability of the following higher-order nonlinear neutral delay difference equation $$ \aligned \Delta\Big(a_{kn}\cdots\Delta\big(a_{2n}\Delta(a_{1n}\Delta(x_n+b_nx_{n-d}))\big)\Big) +\sum_{j=1}^sp_{jn}f_j(x_{n-r_{jn}})=q_n, \quad n\ge n_0, \endaligned $$ where $n_0\ge0,n\ge0,d>0,k>0,j>0,s>0$ are integers, $\{a_{in}\}_{n\ge n_0}(i=1,2,\cdots,k)$, $\{b_n\}_{n\ge n_0}$, $\{p_{jn}\}_{n\ge n_0}(j=1,2,\cdots,s)$ and $\{q_n\}_{n\ge n_0}$ are real sequences, , fj C(R,R) and xfj(x) 0 for any x 0 (j = 1,2,, s). Some sufficient conditions for existence of nonoscillatory solutions of this equation are established and expatiated through five theorems according as the range of value of the sequence bn.

2000 Mathematics Subject Classification: 34K15, 34C10.

Keywords: Nonoscillatory solution, neutral delay difference equation, contraction mapping.

 

 

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