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The existence of positive solutions for a fourth-order difference equation with sum form boundary conditions. (English) Zbl 1474.39011

Summary: We consider the fourth-order difference equation: \(\Delta(z(k + 1) \Delta^3 u(k - 1)) = w(k) f(k, u(k))\), \(k \in \{1,2, \ldots, n - 1 \}\) subject to the boundary conditions: \(u(0) = u(n + 2) = \sum_{i = 1}^{n + 1} g(i) u(i)\), \(a \Delta^2 u(0) - b z(2) \Delta^3 u(0) = \sum_{i = 3}^{n + 1} h(i) \Delta^2 u(i - 2)\), \(a \Delta^2 u(n) - b z(n + 1) \Delta^3 u(n - 1) = \sum_{i = 3}^{n + 1} h(i) \Delta^2 u(i - 2)\), where \(a, b > 0\) and \(\Delta u(k) = u(k + 1) - u(k)\) for \(k \in \{0,1, \ldots, n - 1 \}\), \(f : \{0,1, \ldots, n \} \times [0, + \infty) \rightarrow [0, + \infty)\) is continuous. \(h(i)\) is nonnegative \(i \in \{2,3, \ldots, n + 2 \}\); \(g(i)\) is nonnegative for \(i \in \{0,1, \ldots, n \}\). Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.

MSC:

39A12 Discrete version of topics in analysis
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Wang, J.; Yu, C.; Guo, Y., Triple positive solutions of a nonlocal boundary value problem for singular differential equations with \(p\)-Laplacian, Abstract and Applied Analysis, 2013 (2013) · Zbl 1278.34021 · doi:10.1155/2013/613672
[2] Feng, M., Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions, Applied Mathematics Letters, 24, 8, 1419-1427 (2011) · Zbl 1221.34062 · doi:10.1016/j.aml.2011.03.023
[3] Hao, X.; Liu, L.; Wu, Y., Positive solutions for nonlinear nth-order singular nonlocal boundary value problems, Boundary Value Problems, 2007 (2007) · Zbl 1148.34015 · doi:10.1155/2007/74517
[4] Hao, X.; Liu, L.; Wu, Y.; Sun, Q., Positive solutions for nonlinear \(n\) th-order singular eigenvalue problem with nonlocal conditions, Nonlinear Analysis: Theory, Methods and Applications, 73, 6, 1653-1662 (2010) · Zbl 1202.34038 · doi:10.1016/j.na.2010.04.074
[5] Webb, J. R. L.; Infante, G., Positive solutions of nonlocal boundary value problems: a unified approach, Journal of the London Mathematical Society, 74, 3, 673-693 (2006) · Zbl 1115.34028 · doi:10.1112/S0024610706023179
[6] Ma, R.; Xu, L., Existence of positive solutions of a nonlinear fourth-order boundary value problem, Applied Mathematics Letters, 23, 5, 537-543 (2010) · Zbl 1195.34037 · doi:10.1016/j.aml.2010.01.007
[7] Cabada, A.; Enguica, R. R., Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 74, 10, 3112-3122 (2011) · Zbl 1221.34061 · doi:10.1016/j.na.2011.01.027
[8] Xu, F., Three symmetric positive solutions of fourth-order nonlocal boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 96, 1-11 (2011) · Zbl 1340.34084
[9] Ma, R.; Chen, T., Existence of positive solutions of fourth-order problems with integral boundary conditions, Boundary Value Problems, 2011, 1-17 (2011) · Zbl 1208.34016 · doi:10.1155/2011/297578
[10] Bai, Z., Positive solutions of some nonlocal fourth-order boundary value problem, Applied Mathematics and Computation, 215, 12, 4191-4197 (2010) · Zbl 1191.34019 · doi:10.1016/j.amc.2009.12.040
[11] Webb, J. R. L.; Infante, G.; Franco, D., Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proceedings of the Royal Society of Edinburgh A: Mathematics, 138, 2, 427-446 (2008) · Zbl 1167.34004 · doi:10.1017/S0308210506001041
[12] Zhang, B.; Kong, L.; Sun, Y.; Deng, X., Existence of positive solutions for {BVP}s of fourth-order difference equations, Applied Mathematics and Computation, 131, 2-3, 583-591 (2002) · Zbl 1025.39006 · doi:10.1016/S0096-3003(01)00171-0
[13] Kong, L.; Kong, Q.; Zhang, B., Positive solutions of boundary value problems for third-order functional difference equations, Computers & Mathematics with Applications, 44, 3-4, 481-489 (2002) · Zbl 1057.39014 · doi:10.1016/S0898-1221(02)00170-0
[14] Zhang, X.; Feng, M.; Ge, W., Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications, 69, 10, 3310-3321 (2008) · Zbl 1159.34020 · doi:10.1016/j.na.2007.09.020
[15] Guo, Y.; Yang, F.; Liang, Y., Positive solutions for nonlocal fourth-order boundary value problems with all order derivatives, Boundary Value Problems, 2012, article 29 (2012) · Zbl 1279.34041
[16] Wang, Q.; Guo, Y.; Ji, Y., Positive solutions for fourth-order nonlinear differential equation with integral boundary conditions, Discrete Dynamics in Nature and Society, 2013 (2013) · Zbl 1270.34037 · doi:10.1155/2013/684962
[17] Zhang, X.; Ge, W., Symmetric positive solutions of boundary value problems with integral boundary conditions, Applied Mathematics and Computation, 219, 8, 3553-3564 (2012) · Zbl 1311.34054 · doi:10.1016/j.amc.2012.09.037
[18] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones. Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5 (1988), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0661.47045
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