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Positive solutions for nonlinear discrete periodic boundary value problems. (English) Zbl 1197.39006

The article deals with the following boundary value problem

$-{\Delta }\left[p\left(t-1\right){\Delta }u\left(t-1\right)\right]+q\left(t\right)u\left(t\right)=rg\left(t\right)f\left(u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in {\left[1,T\right]}_{ℤ},$
$u\left(0\right)=u\left(T\right),\phantom{\rule{1.em}{0ex}}p\left(0\right){\Delta }u\left(0\right)=p\left(T\right){\Delta }u\left(T\right),$

where $r$ is a positive parameter, $T>2$, $f\in C\left(ℝ,ℝ\right)$, $sf\left(s\right)>0$ for $s\ne 0$ and there exist the limits

${f}_{0}=\underset{|s|\to 0}{lim}\phantom{\rule{4pt}{0ex}}\frac{f\left(s\right)}{s},\phantom{\rule{2.em}{0ex}}{f}_{\infty }=\underset{|s|\to \infty }{lim}\phantom{\rule{4pt}{0ex}}\frac{f\left(s\right)}{s},$

$p,g:\phantom{\rule{4pt}{0ex}}ℤ\to \left(0,\infty \right)$ and $q:\phantom{\rule{4pt}{0ex}}ℤ\to \left[0,\infty \right)$, $q¬\equiv 0$, are $T$-periodic. The main result is the following:

The boundary value problem under consideration has two $T$-periodic solutions ${u}^{+}$ and ${u}^{-}$, ${u}^{+}\left(t\right)>0$ and ${u}^{-}\left(t\right)<0$ for $t\in \left(0,T\right)$, provided that either ${\lambda }_{1}/{f}_{\infty } or ${\lambda }_{1}/{f}_{0}, where ${\lambda }_{1}$ is the first eigenvalue of the linear eigenvalue problem

$-{\Delta }\left[p\left(t-1\right){\Delta }u\left(t-1\right)\right]+q\left(t\right)u\left(t\right)=rg\left(t\right)u\left(t\right),\phantom{\rule{1.em}{0ex}}t\in {\left[1,T\right]}_{ℤ},$
$u\left(0\right)=u\left(T\right),\phantom{\rule{1.em}{0ex}}p\left(0\right){\Delta }u\left(0\right)=p\left(T\right){\Delta }u\left(T\right)·$

##### MSC:
 39A23 Periodic solutions (difference equations) 39A12 Discrete version of topics in analysis 34B15 Nonlinear boundary value problems for ODE 34C25 Periodic solutions of ODE 39A22 Growth, boundedness, comparison of solutions (difference equations) 34L05 General spectral theory for OD operators
##### References:
 [1] M.A. Krasnosel’skii, Positive solutions of operator equations. Translated from the Russian by Richard E. Flaherty, Groningen 1964 · Zbl 0121.10604 [2] Gustafson, G. B.; Schmitt, K.: Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations, J. differential equations 12, 129-147 (1972) · Zbl 0227.34017 · doi:10.1016/0022-0396(72)90009-5 [3] Nussbaum, R. D.: Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. differential equations 14, 360-394 (1973) · Zbl 0311.34087 · doi:10.1016/0022-0396(73)90053-3 [4] Atici, F. M.; Guseinov, G. Sh.: On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. comput. Appl. math. 132, No. 2, 341-356 (2001) · Zbl 0993.34022 · doi:10.1016/S0377-0427(00)00438-6 [5] Jiang, D.; Chu, J.; O’regan, D.; Agarwal, R.: Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. math. Anal. appl. 286, No. 2, 563-576 (2003) · Zbl 1042.34047 · doi:10.1016/S0022-247X(03)00493-1 [6] O’regan, D.; Wang, H.: Positive periodic solutions of systems of second order ordinary differential equations, Positivity 10, 285-298 (2006) · Zbl 1103.34027 · doi:10.1007/s11117-005-0021-2 [7] Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations 190, No. 2, 643-662 (2003) · Zbl 1032.34040 · doi:10.1016/S0022-0396(02)00152-3 [8] Zhang, Z.; Wang, J.: Positive solutions to a second order three-point boundary value problem, J. math. Anal. appl. 285, No. 1, 237-249 (2003) · Zbl 1035.34011 · doi:10.1016/S0022-247X(03)00396-2 [9] Graef, J. R.: Lingju Kong, haiyan Wang, existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. differential equations 245, No. 5, 1185-1197 (2008) · Zbl 1203.34028 · doi:10.1016/j.jde.2008.06.012 [10] Atici, F. M.; Guseinov, G. Sh.: Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. Anal. appl. 232, 166-182 (1999) · Zbl 0923.39010 · doi:10.1006/jmaa.1998.6257 [11] Fonda, A.; Habets, P.: Periodic solutions of asymptotically positively homogeneous differential equations, J. differential equations 81, No. 1, 68-97 (1989) · Zbl 0692.34041 · doi:10.1016/0022-0396(89)90178-2 [12] Rachunková, I.; Tvrdý, M.; Vrkoč, I.: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. differential equations 176, No. 2, 445-469 (2001) · Zbl 1004.34008 · doi:10.1006/jdeq.2000.3995 [13] Wang, Haiyan: Positive periodic solutions of functional differential equations, J. differential equations 202, No. 2, 354-366 (2004) · Zbl 1064.34052 · doi:10.1016/j.jde.2004.02.018 [14] Chu, J.; Torres, P. J.; Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems, J. differential equations 239, No. 1, 196-212 (2007) · Zbl 1127.34023 · doi:10.1016/j.jde.2007.05.007 [15] Yu, J.; Guo, Z.: Multiplicity results for periodic solutions to delay differential equations via critical point theory, J. differential equations 218, No. 1, 15-35 (2005) · Zbl 1095.34043 · doi:10.1016/j.jde.2005.08.007 [16] Ding, Tongren; Zanolin, F.: Time-maps for the solvability of periodically perturbed nonlinear Duffing equations, Nonlinear anal. TMA 17, No. 7, 635-653 (1991) · Zbl 0777.34030 · doi:10.1016/0362-546X(91)90111-D [17] Ma, Ruyun: Nonlinear discrete Sturm–Liouville problems at resonance, Nonlinear anal. 67, No. 11, 3050-3057 (2007) · Zbl 1129.39006 · doi:10.1016/j.na.2006.09.058 [18] Ma, Ruyun; Ma, Huili: Unbounded perturbations of nonlinear discrete periodic problem at resonance, Nonlinear anal. TMA 70, No. 7, 2602-2613 (2009) · Zbl 1166.39008 · doi:10.1016/j.na.2008.03.047 [19] Jr, J. R. Ward: Global bifurcation of periodic solutions to ordinary differential equations, J. differential equations 142, No. 1, 1-16 (1998) · Zbl 0903.34033 · doi:10.1006/jdeq.1997.3336 [20] Ma, R.: Bifurcation from infinity and multiple solutions for periodic boundary value problems, Nonlinear anal. 42, No. 1, 27-39 (2000) · Zbl 0966.34015 · doi:10.1016/S0362-546X(98)00327-7 [21] Zeidler, E.: Nonlinear functional analysis and its applications, (1986) · Zbl 0583.47050 [22] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9 [23] Wang, Yi; Shi, Yuming: Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, J. math. Anal. appl. 309, No. 1, 56-69 (2005) · Zbl 1083.39019 · doi:10.1016/j.jmaa.2004.12.010