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On existence of positive solutions for a class of discrete fractional boundary value problems

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Abstract

Motivated by some recent developments in the existence theory of fractional difference equations, in this paper we consider boundary value problem

$$\begin{aligned} -\Delta _{\nu -2}^{\nu }u(t)=&f(t+\nu -1,u(t+\nu -1)),\quad 1<\nu \le 2,\\ u(\nu -2)&=0,\quad \Delta _{\nu -1}^{\nu -1}u(\nu +N)=0, \end{aligned}$$

where \(t\in [0,N+1]_{\mathbb {N}_0} \) and N (\(N\ge 2\)) is an integer. The nonlinear function \(f:[\nu -1,\nu +N]_{\mathbb {N}_{\nu -1}}\times \mathbb {R}\rightarrow \mathbb {R^+}\) is assumed to be continuous. We establish some useful inequalities satisfied by the Green’s function associated with above boundary value problem. Sufficient conditions are developed to ensure the existence and nonexistence of positive solutions for the boundary value problem.

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Acknowledgements

We are very grateful to the reviewer for his useful comments that led to improvement of this manuscript.

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Correspondence to Mujeeb ur Rehman.

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ur Rehman, M., Iqbal, F. & Seemab, A. On existence of positive solutions for a class of discrete fractional boundary value problems. Positivity 21, 1173–1187 (2017). https://doi.org/10.1007/s11117-016-0459-4

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