In this paper, we present two asymptotic results on linear nabla fractional difference equations originating from the recent papers as well as their new extensions on the (q, h)-time scale. | Turk J Math (2018) 42: 2214 – 2242 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Two asymptotic results of solutions for nabla fractional (q, h)-difference equations Feifei DU1,∗,, Lynn ERBE2 ,, Baoguo JIA1 ,, Allan PETERSON2 , 1 School of Mathematics, Sun Yat-Sen University, Guangzhou, . China 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska, USA Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we study the Caputo and Riemann–Liouville nabla (q, h) -fractional difference equation and obtain the following two main results: ˜ σ(a) . Then any solution, Theorem A Assume 0 0 satisfies lim x(t) = 0. t→∞ σ 2 (a) ˜ Theorem B Assume 0 0 . Then x(t) > 0 , t ∈ T (q,h) and lim x(t) = 0. t→∞ Theorem A and Theorem B extend the results in other recent works of the authors. Key words: Nabla fractional difference, (q, h) -calculus, monotonicity, asymptotic behavior 1. Introduction In recent years, fractional calculus has attracted increasing interest. Although several results of fractional differential/difference equations are already published [1–6, 13–16, 22, 23, 25–30, 32, 33, 35, 40–44, 46, 47], the development of a qualitative theory for fractional difference equations is still in its beginning due to the memory effects of fractional operators. ∗Correspondence: qinjin65@ 2010 AMS Mathematics Subject Classification: 39A12, 39A70 This work was supported by the National Natural Science Foundation of China (No. 11271380) and the Guangdong Province Key Laboratory of Computational Science. 2214 This work is licensed under a Creative Commons Attribution International License. DU et al./Turk J Math The extension of the basic notions of discrete fractional calculus to the (q, h)-calculus setting appear in [10, 12]. The (q, h)-calculus reduces to discrete nabla-calculus (see [18, Chapter .
