Jia Baoguo

Convexity for nabla and delta fractional differences

In this paper we demonstrate that Theorem A If satisfies , for each , with , then , for . Theorem B If satisfies , for each , with , and . Then , for . This demonstrates that, in some sense, the positivity of the th order fractional difference has a strong connection to the convexity of . In Section 3, by means of an example, we show that a recent result in {C. Goodrich, A convexity result for fractional differences, Appl. Math. Lett. 35 (2014), pp. 58–62.} is incorrect as stated.

A Wong-type oscillation theorem for second order linear dynamic equations on time scales

We obtain Wong-type oscillation theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by oscillation results due to Wong [15]. As a particular application of our results, we show that the difference equation is oscillatory iff |b|>1.

Oscillation of a family of q-difference equations

Abstract We obtain the complete classification of oscillation and nonoscillation for the q -difference equation x Δ Δ ( t ) + b ( − 1 ) n t c x ( q t ) = 0 , b ≠ 0 , where t = q n ∈ T = q N 0 , q > 1 , c , b ∈ R . In particular we prove that this q -difference equation is nonoscillatory, if c > 2 and is oscillatory, if c 2 . In the critical case c = 2 we show that it is oscillatory, if | b | > 1 q ( q − 1 ) , and is nonoscillatory, if | b | ≤ 1 q ( q − 1 ) .