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Solvability of a second order nonlinear neutral delay difference equation. (English) Zbl 1252.39001

Summary: We study the second-order nonlinear neutral delay difference equation \[ \Delta [a_n \Delta (x_n + b_n x_{n-\tau}) + f(n, x_{f_{1n}}, \dots, x_{f_{kn}})] + g(n, x_{g_{1n}}, \dots, x_{g_{kn}}) = c_n,\;n \geq 0. \] By means of the Krasnosel’skii and Schauder fixed point theorems and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34K11 Oscillation theory of functional-differential equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A21 Oscillation theory for difference equations
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