Abstract
This paper discusses qualitative properties of the two-term linear fractional difference equation where , , and is the α th order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions. MSC:39A30, 39A12, 26A33.
Highlights
1 Introduction This paper studies stability and asymptotic properties of the linear fractional difference equation
Where < α
In Section, we present an alternative expression of ( . ) in the form of a Volterra difference equation of convolution type
Summary
This paper studies stability and asymptotic properties of the linear fractional difference equation. ). where < α < , λ = are real scalars, y(n) is the function defined on the set of positive integers N , and the symbol ∇α is the αth order Riemann-Liouville difference operator introduced as follows. For any ν > , we define the νth fractional sum of y : N → R at n by ∇–ν y(n) =. Where is the Euler Gamma function and (s)(μ) = (s + μ) , (s)
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