Existence of nonnegative solutions for discrete Robin boundary value problems with sign-changing weight

In this paper, we are concerned with the following discrete problem first{−∆2u(t − 1) = λp(t)f(u(t)), t ∈ [1, N − 1]Z,∆u(0) = u(N) = 0,where N > 2 is an integer, λ > 0 is a parameter, p : [1, N −1]Z → R is a sign-changing function, f : [0, +∞) → [0, +∞) isa continuous and nondecreasing function. ∆u(t) = u(t+1)−u(t), ∆2u(t) = ∆(∆u(t)). By using the iterative method andSchauder’s fixed point theorem, we will show the existence of nonnegative solutions to the above problem. Furthermore,we obtain the existence of nonnegative solutions for discrete Robin systems with indefinite weights.

PDF

___

[1] Afrouzi GA, Brown K J. Positive mountain pass solutions for a semilinear elliptic equation with a sign-changing weight function. Nonlinear Analysis 2006; 64(3): 409-416. doi: 10.1016/j.na.2005.06.018
[2] Bai D, Henderson J, Zeng Y. Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities. Boundary Value Problems 2015; 231: 9. doi: 10.1186/s13661-015-0500-8
[3] Chen R, Li X. Positive periodic solutions of the first-order singular discrete systems. Annals of Applied Mathematics 2018; 34 (1): 47-57.
[4] Chen T, Ma R. Existence of positive solutions for difference systems coming from a model for burglary. Turkish Journal of Mathematics 2016; 40 (5): 1049-1057. doi: 10.3906/mat-1506-59
[5] Delgado M, Suárez A. On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem. Proceedings of the American Mathematical Society 2004; 132 (6): 1721-1728. doi: 10.1090/S0002-9939- 04-07233-8
[6] Fan Y, Wang L, Li W. Global behavior of a higher order nonlinear difference equation. Journal of Mathematical Analysis and Applications 2004; 299 (1): 113-126. doi: 10.1016/j.jmaa.2004.06.0 14
[7] Fleming WH. A selection-migration model in population genetics. Journal of Mathematical Biology 1975; 2 (3): 219-233. doi: 10.1007/BF00277151
[8] Guo C, Guo J, Kang S, Li H. Existence and uniqueness of positive solution for nonlinear fractional q-difference equation with integral boundary conditions. The Journal of Applied Analysis and Computation 2020; 10 (1): 153- 164. doi: 10.11948/20190055
[9] Jiang J, Henderson J, Xu J, Fu Z. Positive solutions for a system of Neumann boundary value problems of second-order difference equations involving sign-changing nonlinearities. Journal of Function Spaces 2019; 2019. doi: 10.1155/2019/3203401
[10] Ma R, Gao C, Han X. On linear and nonlinear fourth-order eigenvalue problems with indefinite weight. Nonlinear Analysis 2011; 74 (18): 6965-6969. doi: 10.1016/j.na.2011.07.017
[11] Ma R, Han X. Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function. Applied Mathematics and Computation 2009; 215 (3): 1077-1083. doi: 10.1016/j.amc.2009.06.042
[12] Ma R, Ma H. Global structure of positive solutions for superlinear discrete boundary value problems. Journal of Difference Equations and Applications 2011; 17 (9): 1219-1228. doi: 10.1080/10236190802688352
[13] Ma R, Xu J, Han X. Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight. Nonlinear Analysis 2011; 74 (10): 3379-3385. doi: 10.1016/j.na.2011.02.013