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Classification and existence of positive solutions of fourth-order nonlinear difference equations

Authors:

Institution:

(1) Department of Mathematics and Computer Science, Faculty of Science and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia

Abstract:

We consider a class of fourth-order nonlinear difference equations of the form

$(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad \mathit{\Delta}^2 \left( {p_n \left( {\mathit{\Delta}^2 y_n } \right)^{\alpha } } \right) + q_n y_{n + 3}^{\beta } = 0,\quad n \in \mathbb{N},$

where α and β are the ratios of odd positive integers, and {p _n} and {q _n} are positive real sequences defined for all $n \in \mathbb{N}\left( {n_0 } \right)$ satisfying the condition

$\sum\limits_{{n = n_0 }}^{\infty } {n\left( {\frac{n}{{p_n }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. } \alpha }}} < \infty .}$

We classify the nonoscillatory solutions of (Ω) and establish necessary and/or sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior. Supported by Ministry of Science, Technology and Development of Republic of Serbia – Grant No. 144003.

Keywords:

nonlinear difference equation nonoscillatory solution positive solutions asymptotic behavior classification of solutions oscillation

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