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.


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)


Zbl 1127.39028
Full Text: DOI


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