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

On existence of positive solutions for a class of discrete fractional boundary value problems 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} - Δ ν - 2 ν u ( t ) = f ( t + ν - 1 , u ( t + ν - 1 ) ) , 1 < ν ≤ 2 , u ( ν - 2 ) = 0 , Δ ν - 1 ν - 1 u ( ν + N ) = 0 , where $$t\in [0,N+1]_{\mathbb {N}_0}$$ t ∈ [ 0 , N + 1 ] N 0 and N ( $$N\ge 2$$ N ≥ 2 ) is an integer. The nonlinear function $$f:[\nu -1,\nu +N]_{\mathbb {N}_{\nu -1}}\times \mathbb {R}\rightarrow \mathbb {R^+}$$ f : [ ν - 1 , ν + N ] N ν - 1 × R → 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 21 (3) – Nov 29, 2016
15 pages

/lp/springer_journal/on-existence-of-positive-solutions-for-a-class-of-discrete-fractional-wzCvcPP31h
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0459-4
Publisher site
See Article on Publisher Site

### 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} - Δ ν - 2 ν u ( t ) = f ( t + ν - 1 , u ( t + ν - 1 ) ) , 1 < ν ≤ 2 , u ( ν - 2 ) = 0 , Δ ν - 1 ν - 1 u ( ν + N ) = 0 , where $$t\in [0,N+1]_{\mathbb {N}_0}$$ t ∈ [ 0 , N + 1 ] N 0 and N ( $$N\ge 2$$ N ≥ 2 ) is an integer. The nonlinear function $$f:[\nu -1,\nu +N]_{\mathbb {N}_{\nu -1}}\times \mathbb {R}\rightarrow \mathbb {R^+}$$ f : [ ν - 1 , ν + N ] N ν - 1 × R → 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.

### Journal

PositivitySpringer Journals

Published: Nov 29, 2016

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