ON POSITIVE SOLUTIONS TO NONLOCAL FRACTIONAL AND INTEGER-ORDER DIFFERENCE EQUATIONS

By Christopher S. Goodrich. 2011

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In this paper, we consider a discrete fractional boundary value problem of the form -Δυy(t) = f(t+υ - 1; y(t + υ - 1)), y(υ - 2) = ψ(y), y(υ + b) = ф(y), where t Є [0,b]N0, f : [υ – 1,... , υ + b - 1]Nυ-2 x R → [0,+∞) is continuous, ψ, ф : C([υ - 2; υ + b]Nυ-2 ) → R are given functionals, and 1 < υ ≤ 2. We show that provided that both ψ and ф are linear functionals, then under certain conditions the fractional boundary value problem will have at least one positive solution even if neither ψ nor ф is nonnegative for all y ≥ 0. This provides new results not only for the fractional boundary value problem but also in the case when υ = 2. Our results also generalize some recent work on the conjugate fractional boundary value problem. We conclude with two examples to illustrate our results.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/cf61a63f8d72074a77e6943ae9fe3ba3dda5d4b8

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In Applied Mathematics

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In Numerical Analysis

Citations in the corpus

Exponential functions of discrete fractional calculus (2013, methodology): “We refer the reader to [2, 7, 12, 13, 14, 15, 16], for example, and more recently [8, 10, 11].”
On a discrete fractional three-point boundary value problem (2012, methodology): “The present author has produced a couple of recent works on discrete non-local FBVPs (see [19,20]), but using techniques very different from the ones employed here.”