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**Unbounded positive solutions and Mann iterative schemes of a second-order nonlinear neutral delay difference equation.**
*(English)*
Zbl 1266.65203

Summary: This paper is concerned with the solvability of the second-order nonlinear neutral delay difference equation \(\Delta^2(x_n + a_nx_{n-\tau}) + \Delta h(n, x_{h_{1n}}, x_{h_{2n}}, \dots, x_{h_{kn}}) + f(n, x_{f_{1n}}, x_{f_{2n}}, \dots, x_{f_{kn}}) = b_n\)0 \(\forall n \geq n_0\). Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

### MSC:

65Q10 | Numerical methods for difference equations |

39A12 | Discrete version of topics in analysis |

34K40 | Neutral functional-differential equations |

39A22 | Growth, boundedness, comparison of solutions to difference equations |

### Keywords:

numerical examples; second-order nonlinear neutral delay difference equation; uncountably many unbounded positive solutions; Mann-type iterative schemes; error estimates
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\textit{Z. Liu} et al., Abstr. Appl. Anal. 2013, Article ID 245012, 12 p. (2013; Zbl 1266.65203)

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