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Bounded positive solutions for a third order discrete equation. (English) Zbl 1250.39003

Summary: We study the following third order neutral delay discrete equation \(\Delta (a_n \Delta^2(x_n + p_nx_{n-\tau})) + f(n, x_{n-d_{ln}}, \dots, x_{n-d_{ln}}) = g_n, ~n \geq n_0\), where \(\tau, l \in \mathbb N, ~n_0 \in \mathbb N \cup \{0\}, \{a_n\}_{n \in \mathbb N_{n_{0}}}, \{p_n\}_{n \in \mathbb N_{n_{0}}}, \{g_n\}_{n \in \mathbb N_{n_{0}}}\) are real sequences with \(a_n \neq 0\) for \(n \geq n_0, ~\{d_{in}\}_{n \in \mathbb N_{n_{0}}} \subseteq \mathbb Z\) with \(\lim_{n \rightarrow \infty}(n - d_{in}) = +\infty\) for \(i \in \{1, 2, \dots, l\}\) and \(f \in C(\mathbb N_{n_{0}} \times \mathbb R^l, \mathbb R)\). By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

MSC:

39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A12 Discrete version of topics in analysis
34K40 Neutral functional-differential equations
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