On the Impulsive Implicit $\Psi$--Hilfer Fractional Differential Equations with Delay

Jyoti P. Kharade; Kishor D. Kucche

REVIEW 3 major objections 6 minor 25 references

On the Impulsive Implicit $\Psi$--Hilfer Fractional Differential Equations with Delay

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves existence, uniqueness, and Ulam–Hyers–Mittag–Leffler stability for impulsive implicit $\Psi$-Hilfer fractional differential equations with time delay.

desk verdict The existence/uniqueness half is a sound routine extension; the UHML stability proof has a load-bearing gap and the paper should be accepted only after a rewrite. read the letter →

arxiv 1908.07793 v1 pith:BKKVNIMB submitted 2019-08-21 math.DS math.AP

Jyoti P. Kharade , Kishor D. Kucche This is my paper · ORCID

classification math.DSmath.AP MSC 34A0834D2034A3735A23

keywords Ψ-HilferfractionalderivativeimpulsivedifferentialequationstimedelayUlam-Hyers-Mittag-LefflerstabilityexistenceanduniquenessGronwallinequalityPicardoperatorintegralequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that a broad class of fractional differential equations—implicit, with impulses, a time delay, and a $\Psi$-Hilfer fractional derivative—has exactly one solution and is stable in a strong sense: any approximate solution stays close to the true solution, with the error controlled by a Mittag-Leffler function. The class unifies many familiar fractional derivatives, so a positive result would transfer to Caputo, Riemann-Liouville, and Hilfer settings as special cases. The authors prove a contraction on a weighted Banach space of piecewise-continuous functions, then use an extended Gronwall inequality and Picard operator theory to control the distance between nearby solutions. They also show that ordinary Ulam-Hyers and generalized Ulam-Hyers stability are particular cases of the Mittag-Leffler stability they obtain.

What carries the argument

The load-bearing object is the $\Psi$-Hilfer fractional derivative, order $\alpha \in (0,1)$ and type $\beta \in [0,1]$, together with its companion $\Psi$-Riemann-Liouville fractional integral $I^{\alpha;\Psi}_{0+}$; the parameter $\rho=\alpha+\beta-\alpha\beta$ sets the weight $(\Psi(t)-\Psi(0))^{1-\rho}$ used in the solution space and in all bounds. The argument rides on two devices: the equivalence Theorem 3.2, which turns the impulsive implicit delay problem into the fixed-point equation (3.2), and the operator $T$ from (4.1), whose contraction property under (H1)-(H3) yields the unique solution. For stability, a second operator $Q$ is built on the same data, its unique fixed point $z^*$ is used as a comparison envelope, and the extended Gronwall inequality (Lemma 2.4) converts the envelope's integral inequality into the explicit Mittag-Leffler estimate. Picard operator theory and the abstract Gronwall lemma tie the comparison step together: for $z \le Qz$, the fixed point $z^*$ is an upper bound.

What would settle it

A concrete check would be to compute $z^*(t_2)-z^*(t_1)$ from the fixed-point equation (4.14) with $\Psi(t)=t$, $\alpha=1/2$, and a nonnegative $z$ supported near the origin: the Riemann-Liouville integral $I^{1/2}_{0+}z(t)$ decreases on an interval (for $z=1_{[0,1]}$, it equals $2(\sqrt{t}-\sqrt{t-1})/\sqrt{\pi}$ for $t>1$), so the lower bound used to prove $z^*$ increasing breaks down; if the actual fixed point is nevertheless increasing, the stability claim would survive.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 4.1: under the Lipschitz conditions (H1)-(H3), the implicit impulsive $\Psi$-Hilfer delay problem (1.1)-(1.4) has a unique solution in the weighted space $X_{C,\rho,\Psi}$, and the underlying equation (1.1)-(1.2) is Ulam-Hyers-Mittag-Leffler stable. The proof first converts the problem to the equivalent fractional integral equation (3.2) via Theorem 3.2, so that the solution is a fixed point of the operator $T$ defined in (4.1). $T$ is shown to be a contraction, giving existence and uniqueness. For stability, an approximate solution $v$ is compared with the true solution $u$ through a bound of the form $(\Psi(t)-\Psi(0))^{1-\rho}|v(t)-u(t)| \le \epsilon C_{p,E_\alpha} E_\alpha(\zeta_{f,\Psi}(\Psi(t)-\Psi(0))^\alpha)$, and the constants are made explicit. The final section applies the result when $\Psi(t)=t$ to obtain delay versions of Caputo and Riemann-Liouville problems.

Load-bearing premise

The proof's stability conclusion depends on the assertion that the comparison function z* is increasing, so that z*(h(t)) ≤ z*(t) whenever the delay satisfies h(t) ≤ t; this monotonicity is used to drop the delayed term from the Gronwall estimate, and it is not automatic for fractional integrals of order α < 1, so the conclusion would not follow from the written argument if it fails.

Editorial extensions

If this is right

If Theorem 4.1 is correct, every system in this class has exactly one solution, so modelling with implicit Ψ-Hilfer equations is mathematically safe rather than heuristic.
The Ulam-Hyers-Mittag-Leffler bound means that errors in an approximate solution propagate with the explicit envelope ε C_{p,E_α} E_α(ζ_{f,Ψ}(Ψ(t)-Ψ(0))^α), not just qualitatively.
Because ordinary Ulam-Hyers and generalized Ulam-Hyers stability are shown to be special cases, the same theorem covers the standard stability notions for Caputo and Riemann-Liouville limits (β=1 and β=0 with Ψ(t)=t).
The contraction constant in (H3) gives a checkable condition; in the worked examples with Ψ(t)=t it is about 0.1912 for the Caputo case and 0.1013 for the Riemann-Liouville case.
The extended Gronwall inequality is the technical bridge from local estimates to global bounds, so any future extension of the class will need an analogue of that lemma.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the monotonicity step for the comparison fixed point used in the stability estimate is repaired or replaced by a delay-adjusted Gronwall argument, the same template should cover state-dependent delays, since the delay only enters through the z(h(s)) term.
I would expect the contraction-plus-Gronwall framework to transfer to systems of such equations or to several delays, because the sum of finitely many delayed terms preserves the structure of the extended Gronwall inequality.
A testable extension is to weaken the global Lipschitz condition (H1) to local Lipschitz conditions: the contraction argument would then suggest local well-posedness, although the paper only states global conditions.
Because the stability constants are explicit, an independent numerical experiment on the paper's Caputo example could check whether the predicted envelope is tight or merely sufficient.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The existence/uniqueness part is a competent, standard contraction argument in the weighted space. The class is new in the sense that nobody in the cited line put implicit nonlinearity, impulses, Psi-Hilfer derivative, and delay together, but the method is entirely routine. The equivalence and the fixed-point estimates line up; I'd accept that half.

The UHML stability proof, however, is not correct as written. The step "from (4.10) we have z <= Q(z)" is false. In (4.10) the delayed term inside the fractional integral is (Psi(s)-Psi(0))^{1-rho}|v(h(s))-u(h(s))|, while Q uses z(h(s)) with the same weight evaluated at h(s). Since h(s) <= s and Psi is increasing, the actual term is bigger by a factor >=1, so the right-hand side of (4.10) is Qz plus an extra nonnegative integral. You cannot conclude z <= Qz, and Lemma 2.3 drops out. Separately, the proof that the fixed point z* is increasing asserts monotonicity of a fractional integral of order alpha<1, which is false in general; the displayed lower bound with M and the Abel kernels does not establish it. The impulse sum written as sum_{0

There are also smaller, real defects: m in (4.9) is undefined; the Gamma denominator in the series reduction is wrong (Gamma((n+1)alpha) instead of Gamma((n+1)alpha+1), with an extra 1/Gamma(alpha)); and the final stability constant zeta_{f,Psi} omits the factor K, which the example silently inherits. These are not just typos because the advertised bound depends on them.

On the citation pattern: the authors lean on their own prior work [23], [24], but those are background lemmas, not circular dependencies. Fine.

Net: the uniqueness/existence result is probably salvageable and is a modest contribution to the Psi-Hilfer literature. The UHML stability theorem, as the headline result, is unproven. I would send it to a referee rather than desk-reject, because the first half is solid and the stability gap, while serious, might be repairable with a better comparison operator. But I would not accept the current version.

Referee Report

3 major / 6 minor

Summary. The paper investigates existence, uniqueness, and Ulam-Hyers-Mittag-Leffler (UHML) stability for an impulsive implicit Psi-Hilfer fractional differential equation with time delay. It proves an equivalent integral representation (Theorem 3.2), then uses the Banach contraction principle to obtain a unique solution in a weighted piecewise-continuous space (Theorem 4.1(1)). The main advertised result is Theorem 4.1(2), which claims UHML stability via a comparison argument involving a Picard operator Q and an extended Gronwall lemma; Ulam-Hyers and generalized Ulam-Hyers stability are derived as corollaries in Remark 4.2. A worked example illustrates the constants.

Significance. If correct, the stability theorem would extend UHML stability theory to a broad class of implicit impulsive fractional equations with delay, unifying and generalizing earlier results such as [12]–[16]. The existence-uniqueness part is a routine but clean contraction argument with an explicit condition (H3). However, the stability proof contains a false comparison step and an unjustified monotonicity assertion; since stability is the central advertised contribution, the paper's main result is not established as written. The paper does not contain machine-checked proofs or reproducible code, but the symbolic derivations are conventional for the field.

major comments (3)

[Section 4, Eqs. (4.10)–(4.11)] The comparison step stated as “for z(t) = (Ψ(t)-Ψ(0))^{1-ρ}|v(t)-u(t)| from (4.10) we have z ≤ Q(z)” is invalid. In (4.10), the delayed term in the integrand is (Ψ(s)-Ψ(0))^{1-ρ}|v(h(s))-u(h(s))|, which equals ((Ψ(s)-Ψ(0))/(Ψ(h(s))-Ψ(0)))^{1-ρ} z(h(s)) when h(s)>0, not z(h(s)). Since h(s) ≤ s and Ψ is increasing, the factor is at least 1, so the right-hand side of (4.10) is Qz(t) plus a nonnegative extra integral. From z(t) ≤ RHS and RHS ≥ Qz(t) one cannot infer z(t) ≤ Qz(t). Therefore Lemma 2.3 cannot be applied, and the UHML bound (4.16) does not follow from the written argument.
[Section 4, after Eq. (4.14)] The proof that the fixed point z* is increasing is also flawed. In the computation of z*(t2)-z*(t1), the displayed lower bound replaces the factor (z*(s)+z*(h(s))) by a global minimum M inside the difference of two Abel integrals. For 0 < α < 1 the kernel Ψ'(s)(Ψ(t2)-Ψ(s))^{α-1} is smaller than Ψ'(s)(Ψ(t1)-Ψ(s))^{α-1} on [0,t1], so the contribution from the interval [0,t1] to the difference is nonpositive; it cannot be bounded below by M times the positive total kernel difference. Consequently the assertion z*(h(t)) ≤ z*(t) is unproved, and the reduction of (4.14) to the Gronwall inequality in the subsequent lines is unsupported.
[Theorem 4.1(2) and Remark 4.2] Because the two errors above occur in the load-bearing comparison and monotonicity steps, the claimed UHML stability of (1.1)–(1.2) is not established. Remark 4.2, which derives Ulam-Hyers and generalized Ulam-Hyers stability from (4.16), and the stability assertions in Example 5.1 inherit this gap. The existence-uniqueness part, Theorem 4.1(1), appears sound, but the paper's main advertised contribution is the stability result, and a new comparison principle or a different Gronwall argument would be needed to repair the proof.

minor comments (6)

[Section 4, Eq. (4.9)] In the substitution step, the differential should be dθ rather than ds, and the term (Ψ(s)-Ψ(0)^{nα}) is missing a closing parenthesis; also the constant m is used before being defined—presumably m equals the number of impulses p.
[Section 4, Eq. (4.11)] The operator Q is defined on B = C([-r,b], R+), but the formula uses w(t_k^-) in the impulse sum and z(h(s)) for arguments that may lie in [-r,0]; the notation should be z throughout, and the extension of z to [-r,0] should be specified.
[Section 4, after Eq. (4.14)] The phrase “z* is increasing operator” should read “z* is an increasing function”; the word operator is inappropriate for a real-valued function.
[Section 4, opening sentence] The sentence “This section deals with the In this section, we derive...” contains a duplicated fragment and should be rewritten.
[Example 5.1, inequality (5.3)] The display in (5.3) uses |u|, |v|, and |w| inside the definition of f, which conflicts with the solution variable v; the test inequality should be written entirely in terms of v and its derivative.
[References] Reference [17] to Sousa, Kucche, and Capelas de Oliveira has a garbled title (“-Hilfer impulsive fractional differential equations”) and should be checked against the published version in Applied Mathematics Letters.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a contraction-plus-Gronwall argument; the cited self-lemma is background and not equivalent to the target theorem.

full rationale

No circular step is identifiable in the paper's derivation chain. Theorem 4.1 proves existence, uniqueness, and UHML stability from hypotheses (H1)-(H3) by passing to the equivalent integral equation (3.2), applying the Banach contraction principle to the operator T, and then using external Gronwall-type lemmas (Lemma 2.3 from Rus, Lemma 2.4 from Sousa et al.). The equivalence in Theorem 3.2 is imported from the authors' own prior work, reference [24], so there is a self-citation; however, that cited lemma is a standard representation result and is not equivalent to the target existence, uniqueness, or UHML stability conclusion. It is used as a tool, not as an assumed version of the theorem. The UHML stability argument does not fit parameters to data, does not rename a known empirical pattern, and does not import a uniqueness theorem from the authors' own work. The step 'from (4.10) we have z ≤ Q(z)' is mathematically suspect because the delayed term in (4.10) carries the weight (Psi(s)-Psi(0))^{1-rho} while Q contains z(h(s)) with weight (Psi(h(s))-Psi(0))^{1-rho}; the later assertion that z* is increasing is also false for fractional integrals of order alpha < 1. These are correctness gaps in the written proof, not circular reductions: the conclusion is not assumed as an input, and no equation is equivalent to the output by definition. Under the hard rule requiring a quoted reduction for any circularity finding, the honest result is a non-finding with score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the positive constants K, Lf, LJk are hypotheses of the Lipschitz conditions, and the stability constants Cp,Ealpha and zeta_f,Psi are outputs of the proof rather than inputs. The central derivation relies on standard fractional calculus identities, a cited solution formula from the authors' earlier work, and two comparison lemmas. No invented entities are introduced.

assumptions (5)

domain assumption Psi-Riemann-Liouville fractional integral and Psi-Hilfer derivative satisfy the identities in Theorem 2.1 (I^alpha H D^alpha f = ... and H D^alpha I^alpha f = f).
Invoked throughout Section 3 and 4 without proof; cited to [19]. These identities are standard in the Psi-Hilfer calculus but are not derived in this paper.
domain assumption Lemma 3.1: the solution representation for a non-impulsive Psi-Hilfer equation is exactly formula (3.1).
Used to build the impulsive integral equation in Theorem 3.2. Cited to [24], a related preprint by two of the present authors; the lemma is not reproven here.
domain assumption Extended Gronwall inequality with impulses (Lemma 2.4): U(t) <= V(t) + g(t) integral A U ds + sum beta_k U(t_k^-) implies the product bound.
This is the key comparison tool for the stability estimate; cited to [23]. Its validity is not demonstrated in the paper.
standard math Abstract Gronwall lemma for ordered metric spaces (Lemma 2.3): x <= T(x) implies x <= x_T^* for increasing Picard operators.
Used to compare z with the fixed point z* in the stability proof; cited to [22].
standard math Banach contraction principle is applicable in the weighted spaces X_{C,rho,Psi} and B.
Used to obtain the unique fixed point of T and Q. The paper checks the contraction constants but does not restate the theorem.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the Impulsive Implicit $\Psi$--Hilfer Fractional Differential Equations with Delay." pith.science (2026). https://pith.science/paper/BKKVNIMB

@misc{pith2026190807793,
  author       = {Pith},
  title        = {Pith review of: On the Impulsive Implicit $\Psi$--Hilfer Fractional Differential Equations with Delay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BKKVNIMB}},
  note         = {Machine review of arXiv:1908.07793}
}

abstract

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $\Psi$--Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam--Hyers and generalized Ulam--Hyers stability are the specific cases of Ulam--Hyers--Mittag--Leffler stability. Extended version of Gronwall inequality, abstract Gronwall lemma and Picard operator theory are the primary devices in our investigation. We give an example to illustrate the obtained results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 23 canonical work pages

[23]

Vanterler da C

J. Vanterler da C. Sousa, D.S. Oliveira, E. Capelas de Ol iveira, A note on the mild solutions of Hilfer impulsive fractional diﬀerential equations, arXiv: 1811.09256 (2018)

arXiv 2018
[24]

K. D. Kucche, J. P. Kharade, J. Vanterler da C. Sousa, On t he Nonlinear Impulsive ΨHilfer Fractional Diﬀerential Equations, arXiv:1901.01814 (2019 )

arXiv 1901
[12]

K. Shah, A. Ali, S. Bushnaq, Hyers-Ulam stability analy sis to implicit Cauchy problem of frac- tional diﬀerential equations with impulsive conditions, Ma th. Meth. Appl. Sci. 2018 (41) (2018) 8329-8343

work page 2018
[16]

K. Liu, J. Wang, D. ORegan, Ulam-Hyers-Mittag–Leﬄer st ability for Ψ-Hilfer fractional-order delay diﬀerential equations, Advances in Diﬀerence Equation s (2019) 2019:50

work page 2019
[1]

J. Wang, M. Fekan, Y. Zhou, A survey on impulsive fraction al diﬀerential equations, Fractional Calculus and Applied Analysis 19 (4) (2016) 806–831

work page 2016
[2]

J. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fracti onal diﬀerential equations, Applied Mathematics and Computation 242 (2014) 649–657

work page 2014
[3]

Fe˘ckan, Y

J.Wang, M. Fe˘ckan, Y. Zhou, Ulams type stability of impulsive ordinary diﬀe rential equations, Journal of Mathematical Analysis and Applications 395 (201 2) 258–264. 14

work page
[4]

J. Wang, Y. Zhou, M. Fe˘ckan, On recent developments in the theory of boundary value p roblems for impulsive fractional diﬀerential equations, Computers and Mathematics with Applications 64 (2012) 3008–3020

work page 2012

Show all 25 references

[5]

Feckan, Y

M. Feckan, Y. Zhou, J. Wang, On the concept and existence o f solution for impulsive fractional diﬀerential equations, Commun Nonlinear Sci Numer Simulat 1 7 (7) (2012) 3050–3060

2012
[6]

Benchohra, B

M. Benchohra, B. Slimani, Existence and uniqueness of so lutions to impulsive fractional diﬀeren- tial equations, Electronic Journal of Diﬀerential Equation s 2009 (10) (2009) 1–11

2009
[7]

G. M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional diﬀerential equations, Nonlinear Analysis 72 (2010) 1604–1615

2010
[8]

Nieto , Abelghani Ouahab , Venktesh Venktesh, Imp licit fractional diﬀerential equations via Liouville -Caputo derivative, Mathematics 3 (2015) 398 -411

Juan J. Nieto , Abelghani Ouahab , Venktesh Venktesh, Imp licit fractional diﬀerential equations via Liouville -Caputo derivative, Mathematics 3 (2015) 398 -411

2015
[9]

Benchohra, J

M. Benchohra, J. Lazreg, On stability for nonlinear impl icit fractional diﬀerential equations, Le Matematiche Vol. LXX (2015) Fasc. II, 49-61

2015
[10]

Benchohra, M

M. Benchohra, M. Said Souid, Integrable solutions for i mplicit fractional order diﬀerential equa- tions, Nonlinear Analysis 69 (2008) 2677-2682

2008
[11]

K. D. Kucche, J. J. Nieto, V. Venktesh, Theory of nonline ar implicit fractional diﬀerential equa- tions, Diﬀer. Equ. Dyn. Syst. DOI 10.1007/s12591-016-0297- 7

doi
[13]

Eghbali, V

N. Eghbali, V. Kalvandi, J. M. Rassias, A ﬁxed point appr oach to the Mittag-Leﬄer-Hyers-Ulam stability of a fractional integral equation, Open Math. 14 ( 2016) 237-246

2016
[14]

J. R. Wang, Y. Zhang, Ulam-Hyers-Mittag-Leﬄer stabili ty of fractional-order delay diﬀerential equations, Optimization 63 (2014) 1181-1190

2014
[15]

A. U. K. Niazi, J. Wei, M. U. Rehman , P. Denghao, Ulam-Hye rs-Mittag-Leﬄer stability for nonlinear fractional neutral diﬀerential equations, Sborn ik: Mathematics 209 (9) (2018) 1337- 1350

2018
[17]

Vanterler da C

J. Vanterler da C. Sousa, K. D. Kucche , E. Capelas de Oliv eira, Stability of -Hilfer impulsive fractional diﬀerential equations, Appl. Math. Lett. 88 (201 9) 73–80
[18]

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory an d applications of fractional diﬀerential equations, Elsevier. Science, B.V. Amsterdam (2006)

2006
[19]

Vanterler da C

J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the ψ–Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simulat. 60 (2018) 72-91

2018
[20]

Z. Bai, X. Dong, C. Yin, Existence results for impulsive nonlinear fractional diﬀerential equation with mixed boundary conditions, Boundary Value Problems (2 016) 2016:63. 15

2016
[21]

World scientiﬁc (2014)

Zhou Y., Basic theory of fractional diﬀerential equatio ns. World scientiﬁc (2014)

2014
[22]

I.A.Rus, Growall lemmas: ten open problems, Sci. Math. Jpn. 70 (2009) 221–228

2009
[25]

Vanterler da C

J. Vanterler da C. Sousa, E. Capelas Oliveira, A Gronwal l inequality and the Cauchy-type problem by means of ψ-Hilfer operator, Diﬀ. Equ. & Appl. 11 (1) (2019) 87–106

2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [23]

Vanterler da C

J. Vanterler da C. Sousa, D.S. Oliveira, E. Capelas de Ol iveira, A note on the mild solutions of Hilfer impulsive fractional diﬀerential equations, arXiv: 1811.09256 (2018)

arXiv 2018

[2] [24]

K. D. Kucche, J. P. Kharade, J. Vanterler da C. Sousa, On t he Nonlinear Impulsive ΨHilfer Fractional Diﬀerential Equations, arXiv:1901.01814 (2019 )

arXiv 1901

[3] [12]

K. Shah, A. Ali, S. Bushnaq, Hyers-Ulam stability analy sis to implicit Cauchy problem of frac- tional diﬀerential equations with impulsive conditions, Ma th. Meth. Appl. Sci. 2018 (41) (2018) 8329-8343

work page 2018

[4] [16]

K. Liu, J. Wang, D. ORegan, Ulam-Hyers-Mittag–Leﬄer st ability for Ψ-Hilfer fractional-order delay diﬀerential equations, Advances in Diﬀerence Equation s (2019) 2019:50

work page 2019

[5] [1]

J. Wang, M. Fekan, Y. Zhou, A survey on impulsive fraction al diﬀerential equations, Fractional Calculus and Applied Analysis 19 (4) (2016) 806–831

work page 2016

[6] [2]

J. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fracti onal diﬀerential equations, Applied Mathematics and Computation 242 (2014) 649–657

work page 2014

[7] [3]

Fe˘ckan, Y

J.Wang, M. Fe˘ckan, Y. Zhou, Ulams type stability of impulsive ordinary diﬀe rential equations, Journal of Mathematical Analysis and Applications 395 (201 2) 258–264. 14

work page

[8] [4]

J. Wang, Y. Zhou, M. Fe˘ckan, On recent developments in the theory of boundary value p roblems for impulsive fractional diﬀerential equations, Computers and Mathematics with Applications 64 (2012) 3008–3020

work page 2012