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.


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
Full Text: DOI


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