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Asymptotic behavior of solutions of fractional nabla q-difference equations

Authors
  • Jia, Baoguo
  • Erbe, Lynn
  • Peterson, Allan
Type
Published Article
Journal
Georgian Mathematical Journal
Publisher
De Gruyter
Publication Date
Jun 23, 2017
Volume
26
Issue
1
Pages
21–28
Identifiers
DOI: 10.1515/gmj-2017-0030
Source
De Gruyter
Keywords
License
Yellow

Abstract

We are concerned with the ν-th order nabla fractional q-difference equation (quantum equation) ∇ q , ρ ⁢ ( 1 ) ν ⁡ x ⁢ ( t ) = c ⁢ ( t ) ⁢ x ⁢ ( t ) , t ∈ q ℕ 1 \nabla^{\nu}_{{q,\rho(1)}}x(t)=c(t)x(t),t\in q^{\mathbb{N}_{1}} , where q > 1 {q>1} , ℕ 1 = { 1 , 2 , … } {\mathbb{N}_{1}=\{1,2,\dots\}} , ρ ⁢ ( 1 ) = q - 1 {\rho(1)=q^{-1}} . We prove that for 0 < ν < 1 {0<\nu<1} and c ⁢ ( t ) ≤ 0 {c(t)\leq 0} , t ∈ ℕ 1 {t\in\mathbb{N}_{1}} , any solution of the q-difference equation with x ⁢ ( 1 ) > 0 {x(1)>0} satisfies lim t → ∞ ⁡ x ⁢ ( t ) = 0 \lim_{t\to\infty}x(t)=0 . This asymptotic result shows that the solutions of the nabla fractional q-difference equation ∇ q , ρ ⁢ ( 1 ) ν ⁡ x ⁢ ( t ) = c ⁢ x ⁢ ( t ) {{\nabla^{\nu}_{{q,\rho(1)}}}x(t)=cx(t)} , 0 < ν < 1 {0<\nu<1} , c < 0 {c<0} , have asymptotic behavior similar to that of the solutions of the first order nabla q-difference equation ∇ q ⁡ x ⁢ ( t ) = c ⁢ x ⁢ ( t ) {\nabla_{q}x(t)=cx(t)} , c < 0 {c<0} , t ∈ q ℕ 1 {t\in q^{\mathbb{N}_{1}}} .

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