TAILIEUCHUNG - Two asymptotic results of solutions for nabla fractional (q, h)-difference equations

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 .

• Toán học
• Nabla fractional difference
• Caputo and Riemann Liouville nabla
• Fractional operators
• Asymptotic behavior
• Quantum calculus