On positive solutions to nonlocal fractional and integer-order difference equations. (English) Zbl 1289.39008

The paper deals with the following discrete boundary value problem for a fractional difference equation: \[ \begin{gathered} -\Delta^\nu y(t)=f(t+\nu-1, y(t+\nu-1)), \\ y(\nu-2)=\psi(y),\quad y(\nu+b)=\phi(y), \end{gathered} \] where \(1<\nu\leq 2\). The author investigates the case when both functionals \(\psi\) and \(\phi\) are linear, but not necessarily nonnegative, generalizing in that way some recent results in this field. Under these assumptions on functionals \(\psi\) and \(\phi\) and certain additional conditions, the main result states the existence of a positive solution to the above discrete fractional boundary value problem. Moreover, since the integer case \(\nu=2\) is included in the consideration, the paper provides a new result also in the “classical” (i.e., integer-order) theory of boundary value problems for difference equations. The results are illustrated by two examples.


39A12 Discrete version of topics in analysis
39A05 General theory of difference equations
26A33 Fractional derivatives and integrals
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