# zbMATH — the first resource for mathematics

On Sturm-Liouville boundary value problems for second-order nonlinear functional finite difference equations. (English) Zbl 1151.39016
Motivated by I. Yaslan Karaca’s results in [ibid. 205, No. 1, 458–468 (2007; Zbl 1127.39028)], the author gives sufficient conditions for the existence of solutions of a Sturm-Liouville boundary value problem for the second-order nonlinear functional finite difference equation
$\Delta^2x(n)=f(n,x(n+1), x(n-\tau_1(n)),\dots,x(n-\tau_m(n))),$
where $$f$$ is allowed to be linear, superlinear or sublinear. Let $$\mathbb{T}$$ be a time scale including $$0$$ and $$T>0$$. Using Schauder’s fixed point theorem and the upper and lower solution method, the author obtains some results on the existence of a positive solution for an $$m$$-point singular $$p$$-Laplacian dynamic equation with a boundary condition.

##### MSC:
 39A12 Discrete version of topics in analysis 39A10 Additive difference equations 34B15 Nonlinear boundary value problems for ordinary differential equations 39A11 Stability of difference equations (MSC2000)
Full Text:
##### References:
 [1] Agarwal, R.P., Focal boundary value problems for differential and difference equations, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0914.34001 [2] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0923.39002 [3] Agarwal, R.P.; Thompson, H.B.; Tisdell, C.C., Three-point boundary value problems for second order discrete equations, Comput. math. appl., 45, 1429-1435, (2003) · Zbl 1055.39024 [4] Avery, R.I.; Peterson, A.C., Multiple positive solutions of a discrete second order conjugate problem, Panamer. math. J., 8, 1-12, (1998) · Zbl 0959.39006 [5] Aykut, N., Existence of positive solutions for boundary value problems of second order functional difference equations, Comput. math. appl., 48, 517-527, (2004) · Zbl 1066.39015 [6] Cabada, A.; Otero-Espinar, V., Fixed sign solutions of second order difference equations with Neumann boundary conditions, Comput. math. appl., 45, 1125-1136, (2003) · Zbl 1055.39001 [7] Deimling, K., Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040 [8] Henderson, J.; Thompson, H.B., Existence of multiple solutions for second order discrete boundary value problems, Comput. math. appl., 43, 1239-1248, (2002) · Zbl 1005.39014 [9] Ji, J.; Yang, B., Eigenvalue comparisons for boundary problems for second order difference equations, J. math. anal. appl., 320, 964-972, (2006) · Zbl 1111.39012 [10] Karaca, I., Discrete third-order three-point boundary value problem, J. comput. appl. math., 205, 458-468, (2007) · Zbl 1127.39028 [11] Liu, Y.; Ge, W., Twin positive solutions of boundary value problems for finite difference equations with $$p$$-Laplacian operator, J. math. anal. appl., 278, 551-561, (2003) · Zbl 1019.39002 [12] Sun, H.; Shi, Y., Eigenvalues of second order difference equations with coupled boundary value conditions, Linear algebra appl., 414, 361-372, (2006) · Zbl 1092.39011 [13] Wong, P.J.Y., Solutions of constant signs of a system of sturm – liouville boundary value problems, Math. comput. modelling, 29, 27-38, (1999) · Zbl 1041.34015 [14] Wong, P.J.Y., Multiple fixed-sign solutions for a system of difference equations with sturm – liouville conditions, J. comput. appl. math., 183, 108-132, (2005) · Zbl 1077.39003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.