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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:  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:  Bai, D.: A global result for discrete $\phi$-Laplacian eigenvalue problems.Adv. Difference Equ. 2013 (2013), Article ID 264, 10 pages. MR 3110766 Reference:  Bai, D., Xu, X.: Existence and multiplicity of difference $\phi$-Laplacian boundary value problems.Adv. Difference Equ. 2013 (2013), Article ID 267, 13 pages. MR 3125053 Reference:  Bian, L.-H., Sun, H.-R., Zhang, Q.-G.: Solutions for discrete $p$-Laplacian periodic boundary value problems via critical point theory.J. Difference Equ. Appl. 18 (2012), 345-355. Zbl 1247.39004, MR 2901826, 10.1080/10236198.2010.491825 Reference:  Cabada, A.: Extremal solutions for the difference $\phi$-Laplacian problem with nonlinear functional boundary conditions.Comput. Math. Appl. 42 (2001), 593-601. Zbl 1001.39006, MR 1838016, 10.1016/S0898-1221(01)00179-1 Reference:  Jaroš, J., Kusano, T.: A Picone type identity for second-order half-linear differential equations.Acta Math. Univ. Comen., New Ser. 68 (1999), 137-151. Zbl 0926.34023, MR 1711081 Reference:  Ji, D., Ge, W.: Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with $p$-Laplacian.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68 (2008), 2638-2646. Zbl 1145.34309, MR 2397748, 10.1016/j.na.2007.02.010 Reference:  Kim, C.-G., Shi, J.: Global continuum and multiple positive solutions to a $p$-Laplacian boundary-value problem.Electron. J. Differ. Equ. (electronic only) 2012 (2012), 12 pages. Zbl 1260.34045, MR 2946845 Reference:  Kusano, T., Jaroš, J., Yoshida, N.: A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 40 (2000), 381-395. Zbl 0954.35018, MR 1768900, 10.1016/S0362-546X(00)85023-3 Reference:  Lee, Y.-H., Sim, I.: Existence results of sign-changing solutions for singular one-dimensional $p$-Laplacian problems.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68 (2008), 1195-1209. Zbl 1138.34010, MR 2381665, 10.1016/j.na.2006.12.015 Reference:  Li, Y., Lu, L.: Existence of positive solutions of $p$-Laplacian difference equations.Appl. Math. Lett. 19 (2006), 1019-1023. Zbl 1125.39007, MR 2246169, 10.1016/j.aml.2005.10.020 Reference:  Liu, Y.: Existence results for positive solutions of non-homogeneous BVPs for second order difference equations with one-dimensional $p$-Laplacian.J. Korean Math. Soc. 47 (2010), 135-163. Zbl 1191.39008, MR 2591031, 10.4134/JKMS.2010.47.1.135 Reference:  Řehák, P.: Oscillatory properties of second order half-linear difference equations.Czech. Math. J. 51 (2001), 303-321. Zbl 0982.39004, MR 1844312, 10.1023/A:1013790713905 Reference:  Xia, J., Liu, Y.: Positive solutions of BVPs for infinite difference equations with one-dimensional $p$-Laplacian.Miskolc Math. Notes 13 (2012), 149-176. MR 2970907, 10.18514/MMN.2012.357 Reference:  Yang, Y., Meng, F.: Eigenvalue problem for finite difference equations with $p$-Laplacian.J. Appl. Math. Comput. 40 (2012), 319-340. Zbl 1295.39006, MR 2965334, 10.1007/s12190-012-0559-7 Reference:  Zeidler, E.: Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems.Springer New York (1986). Zbl 0583.47050, MR 0816732 Reference:  Zhang, X., Tang, X.: Existence of solutions for a nonlinear discrete system involving the $p$-Laplacian.Appl. Math., Praha 57 (2012), 11-30. Zbl 1249.39009, MR 2891303, 10.1007/s10492-012-0002-2 .

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