Existence and Uniqueness Theorem of the Solution to a Class of Nonlinear Nabla Fractional Difference System with a Time Delay

Chen, Churong; Jia, Baoguo; Liu, Xiang; Erbe, Lynn

doi:10.1007/s00009-018-1258-x

Cited by 23 publications

(11 citation statements)

References 14 publications

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“…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].…”

Section: Resultsmentioning

confidence: 99%

“…In [18,23,35], the authors studied fractional difference equations, and the existence of solutions were established by employing Schauder's fixed point theorem. In [10,19], Luo and Chen investigated the uniqueness results for a class of nonlinear fractional difference system with time delay and gave the proof by contradiction and generalized Gronwall inequality. He et al gave existence results for fractional discrete equations by means of topological degree methods in [15], and many other conclusions can see [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Luo¹,

Abdeljawad²,

Luo³

2021

Turk J Math

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Luo¹,

Abdeljawad²,

Luo³

2021

Turk J Math

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“…Remark 3.7 It is noted that for a nonnegative function f (t), the fractional integral t 0 (ts) α 1 -α 2 -1 f (s) ds may be monotonically increasing or decreasing with respect to t for 0 < α 1α 2 < 1 (see [3,9,11,30]). To prove that the integral term t 0 (ts) α 1 -α 2 -1 f (s) ds is monotonically increasing for f (t) ≥ 0, there is an alternative approach found in [10] (Lemma 5).…”

Section: Corollary 36mentioning

confidence: 99%

Finite-time stability of multiterm fractional nonlinear systems with multistate time delay

Arthi¹,

Brindha²,

2021

Adv Differ Equ

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This work is mainly concentrated on finite-time stability of multiterm fractional system for $0 < \alpha _{2} \leq 1 < \alpha _{1} \leq 2$ 0 < α 2 ≤ 1 < α 1 ≤ 2 with multistate time delay. Considering the Caputo derivative and generalized Gronwall inequality, we formulate the novel sufficient conditions such that the multiterm nonlinear fractional system is finite time stable. Further, we extend the result of stability in the finite range of time to the multiterm fractional integro-differential system with multistate time delay for the same order by obtaining some inequality using the Gronwall approach. Finally, from the examples, the advantage of presented scheme can guarantee the stability in the finite range of time of considered systems.

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“…Besides, Ulam‐Hyers stability for nonhomogeneous linear difference equations with constant stepsize has been studied by Onitsuka . For the study of fractional difference equations, one can see Bas and Ozarslan, Chen et al, and Xu et al There are only a few papers which consider the Ulam‐Hyers stability for fractional difference equations. Recently, Jagan Mohan Jonnalagadda has formulated and obtained Ulam–Hyers stability for Riemann–Liouville fractional nabla difference equations.…”

Section: Introductionmentioning

confidence: 99%

Ulam‐Hyers stability of Caputo fractional difference equations

Chen

Böhner

Jia

2019

Math Methods in App Sciences

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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

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Existence and Uniqueness Theorem of the Solution to a Class of Nonlinear Nabla Fractional Difference System with a Time Delay

Cited by 23 publications

References 14 publications

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

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

Finite-time stability of multiterm fractional nonlinear systems with multistate time delay

Ulam‐Hyers stability of Caputo fractional difference equations

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