Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

<p style='text-indent:20px;'>The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $\end{document}</tex-math></inline-formula> is an unknown solution, <inline-formula><tex-math id="M2">\begin{document}$ \Delta^2 u(k) $\end{document}</tex-math></inline-formula> is its second-order forward difference, <inline-formula><tex-math id="M3">\begin{document}$ k_0 $\end{document}</tex-math></inline-formula> is a fixed integer and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> are real numbers, <inline-formula><tex-math id="M6">\begin{document}$ m\not = 0, 1 $\end{document}</tex-math></inline-formula>.</p>