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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)
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