Ulam‐Hyers stability of Caputo fractional difference equations

Abstract

We study the Ulam-Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam-Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results. KEYWORDS Caputo operator, nabla difference equation, Ulam-Hyers stability MSC CLASSIFICATION 39A70; 39A12 INTRODUCTIONUnder what conditions one can still say that the result of a theorem holds or is approximately true when altering slightly the assumption? This question was originally raised by Ulam during a talk at the University of Wisconsin in 1940. 1,2 Then, after a year, Hyers answered this question for a special case. 3 After that, considering Cauchy differences, Rassias 4 generalized the work.Since then, the concept of Ulam stability has been investigated as well as extended in several different directions. There are many papers on this issue that have been published so far. Jung et al [5][6][7]8,9 for example, have considered the Ulam-Hyers stability for differential equations. Besides, Ulam-Hyers stabilities for fractional cases have been studied, one can see in previous studies. [10][11][12][13][14][15][16] Moreover, some works have been done for difference equations. Jung and Nam, 17-20 for example, have investigated Ulam-Hyers stability for first-order difference equations, first-order matrix difference equations, and first-order inhomogeneous matrix difference equations. Besides, Ulam-Hyers stability for nonhomogeneous linear difference equations with constant stepsize has been studied by Onitsuka. 21 For the study of fractional difference equations, one can see Bas and Ozarslan, Chen et al, There are only a few papers which consider the Ulam-Hyers stability for fractional difference equations. Recently, Jagan Mohan Jonnalagadda 25 has formulated and obtained Ulam-Hyers stability for Riemann-Liouville fractional nabla difference equations. His approach was using the N-transform. To obtain the Ulam-Hyers stability result, see Jagan Mohan. 25, Theorem 8 and 10 He used the estimation of nabla Mittag-Leffler functions shown in a previous study. 25, Lemma 4 Unfortunately, it is important to mention that this argument has not been proven rigorously since it is not trivial. Because of the fact that we do not know whether the estimations F (− , t ) ≤ 1 and F , (− , t ) ≤ 1 Γ( ) are correct or not, we cannot get 25, (24), (31) easily. Thus, the proofs of Jagan Mohan 25, theorem 8 and 10 are incomplete. However, in our paper, we use a Math Meth Appl Sci. 2019;42:7461-7470.wileyonlinelibrary.com/journal/mma

Ulam-Hyers stability results for a novel nonlinear Nabla Caputo fractional variable-order difference system

“…In this paper, we are concerned with a class of nonlinear variable-order Nabla Caputo fractional difference system, which is quite different from the related references discussed in the literature [5,9,10,16,18,19,23].…”

“…Assume that f (t, x) = sin x(t) Ax(t) = 9 100 e −t x(t) , t ∈ N12 0 , [t 0, T ] = [0, 3]∪[3, 6]∪[6, 9]∪[9,12],x(0) = 1 , ν k = 1 , t kl = 3 * k , k = 0, 1, 2, 3 , then N = 4 , L = M 1 = 0.100 , M A = 0.090. And by Mathematica software, we know N k=2 H ν (k−2) t (k−1)l , t (k−2)l = 4.476,andsup t∈N T a H ν (N −1) t, t (N −1)l = 1.096, t ∈ N 12 9 ,then M = 4.476 + 1.096 = 5.572 satisfying 0 < M < M −1 A , and we can have(M A + L) N k=2 H ν (k−2) t (k−1)l , t (k−2)l = 0.850admitting the inequality (3.5).…”

Monotonicity results for nabla fractional h‐difference operators

“…Discrete fractional calculus and its applications have become an attractive topic in recent years since Miller and Ross 1 initiated the discrete fractional calculus in 1988. For the basic theory of discrete fractional calculus, we could refer to previous works [2][3][4][5][6][7][8][9][10][11][12][13] and the references therein. It is well-known that monotonicity results play an important role in the study of discrete fractional calculus, and numerous monotonicity results about fractional calculus have been published.…”

Discrete fractional order two-point boundary value problem with some relevant physical applications

“…The corresponding discrete counter part, fractional order difference equations (FODEs), have appeared as a new research area for mathematicians and scientists. The study of discrete fractional calculus was initiated by Miller and Ross [26] and then developed by several other researchers [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. In the meantime, researchers have adopted the fact that dealing with FODEs provides a more accurate description than FDEs and the use of FODEs facilitates applications that require computational and simulation analysis.…”