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 Title: Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems (English) Author: Bai, Dingyong Author: Chen, Yuming Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 60 Issue: 4 Year: 2015 Pages: 343-353 Summary lang: English . Category: math . Summary: We discuss the discrete $p$-Laplacian eigenvalue problem, $\begin {cases} \Delta (\phi _p(\Delta u(k-1)))+\lambda a(k)g(u(k))=0,\quad k\in \{1,2, \ldots , T\},\\ u(0)=u(T+1)=0, \end {cases}$ where $T>1$ is a given positive integer and $\phi _p(x):=|x|^{p-2}x$, $p > 1$. First, the existence of an unbounded continuum $\mathcal {C}$ of positive solutions emanating from $(\lambda , u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal {C}$ is a monotone continuous curve globally defined for all $\lambda >0$. (English) Keyword: discrete $p$-Laplacian eigenvalue problem Keyword: positive solution Keyword: continuum Keyword: Picone-type identity Keyword: lower and upper solutions method MSC: 34B09 MSC: 39A10 MSC: 39A12 idZBL: Zbl 06486915 idMR: MR3396469 DOI: 10.1007/s10492-015-0100-z . Date available: 2015-06-30T11:58:28Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/144311 . Reference: [1] Agarwal, R. P., Perera, K., O'Regan, D.: Multiple positive solutions of singular discrete $p$-Laplacian problems via variational methods.Adv. Difference Equ. 2005 (2005), 93-99. Zbl 1098.39001, MR 2197124 Reference: [2] Bai, D.: A global result for discrete $\phi$-Laplacian eigenvalue problems.Adv. Difference Equ. 2013 (2013), Article ID 264, 10 pages. 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