The asymptotic behavior of Caputo delta fractional equations

Consider the following ν ‐th order Caputo delta fractional equation 0.1Δa ∗ν x ( t ) = c ( t ) x ( t + ν − 1 ) ,t ∈ N a + 1 − ν . The following asymptotic results are obtained. A Theorem Assume 0 < ν < 1 and there exists a constant b 2 such that c ( t )≥ b 2 >0. Then the solutions of the equation (0.1) with x ( a ) > 0 satisfylim t → ∞ x ( t ) = + ∞ .B Theorem Assume 0 < ν < 1 and there exists a constant b 1 such that − ν < c ( t )≤ b 1 <0. Then the solutions of the equation (0.1) with x ( a ) > 0 satisfylim t → ∞ x ( t ) = 0 .This shows that the solutions of the Caputo delta fractional equationΔa ∗ν x ( t ) = cx ( t + ν − 1 ) , 0 < ν < 1 , with x ( a ) > 0 have similar asymptotic behavior with the solutions of the first‐order delta difference equation Δ x ( t ) = c x ( t ), c >− 1. Copyright © 2016 John Wiley & Sons, Ltd.