On representation formulas for solutions of linear differential equations with Caputo fractional derivatives

Mikhail Gomoyunov

REVIEW 3 major objections 4 minor 24 references

On representation formulas for solutions of linear differential equations with Caputo fractional derivatives

T0 review · 3 major / 4 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 explicit representation formulas for solutions of linear Caputo fractional equations with variable coefficients when the initial condition is prescribed on an interval, not at a single point.

desk verdict A useful and likely correct extension of representation formulas to intermediate-point initial data; two proof gaps need filling, but neither looks fatal. read the letter →

arxiv 1908.08319 v1 pith:MPFA7WQ5 submitted 2019-08-22 math.OC math.CAmath.DS

Mikhail Gomoyunov This is my paper · ORCID

classification math.OCmath.CAmath.DS MSC 26A3334A0834A30

keywords fractionaldifferentialequationsCaputoderivativefundamentalsolutionmatrixrepresentationformulavariationofconstantsintermediateinitialconditionHöldercontinuityCauchyproblem

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

The paper aims to prove that solutions of a variable-coefficient linear system with a Caputo fractional derivative can be written in explicit closed form even when the initial condition is not a single value but a prescribed history on an interval $[t_0,t_*]$. Such nonlocal initial data arise naturally in control problems and differential games for fractional systems, where explicit representation formulas are the standard building block for feedback construction and numerical methods. The main result, Theorem 4.2, gives a Duhamel-type representation in which the whole pre-history affects the future only through a modified forcing term $b_*$ built from $w_*$. Along the way the paper develops a careful theory of the regularized fundamental solution matrix $F$: existence, boundedness, H\"older continuity in both arguments, and a dual definition.

What carries the argument

The central object is the regularized fundamental solution matrix $F(t,s)$, defined as the continuous solution of integral equation (3.4) and related to the Riemann\u2013Liouville fundamental matrix $Z$ by $F(t,s)=(t-s)^{1-\alpha}Z(t,s)$; it is bounded and H\"older continuous in both variables on $\Omega=\{(t,s): t_0\le s\le t\le\vartheta\}$, and it satisfies the dual equation $G=F$ (Proposition 3.3). The step that makes the general intermediate-point case work is Lemma 4.1's integral identity, which turns the left-sided fractional integral of the initial history into a right-sided integral over $[t_*,t]$: $\int_{t_0}^{t_*}\phi(\tau)/(t-\tau)^{1-\alpha}\,d\tau=\int_{t_*}^{t}\psi(\tau)/((t-\tau)^{1-\alpha}(\tau-t_*)^\alpha)\,d\tau$, with $\psi$ defined by (4.6). This identity converts the nonlocal memory of $w_*$ into the singular but manageable forcing term $b_*$ and is what the representation formula (4.8) rests on.

What would settle it

Take $t_0=0$, $t_*=1$, $\alpha=1/2$, $\phi(\tau)=1$ and evaluate the identity in Lemma 4.1 numerically at several $t>1$; both sides must agree. A disagreement at any one $t$, or failure of the resulting $b_*$ from (4.9) to be locally integrable on $(t_*,\vartheta]$, would falsify the proof of Theorem 4.2.

Watch

Extended reading notes

Core claim

At the center is Theorem 4.2. For the Cauchy problem $({}^C\!D^\alpha_{t_0+}x)(t)=A(t)x(t)+b(t)$, $x(t)=w_*(t)$ on $[t_0,t_*]$, the solution is claimed to be $$x(t)=\Bigl(\operatorname{Id}+\int_{t_*}^{t}\frac{F(t,\tau)A(\tau)}{(t-\tau)^{1-\$\alpha$}}\,d\tau\Bigr)w_*(t_*)+\int_{t_*}^{t}\frac{F(t,\tau)b_*(\tau)}{(t-\tau)^{1-\$\alpha$}}\,d\tau,\quad t\in[t_*,\vartheta],$$ where $F$ is the regularized fundamental solution matrix defined by (3.4) and $b_*$ is the forcing term (4.9) that carries the memory of $w_*$. The proof rewrites the fractional integral of the initial history over $[t_0,t_*]$ as an integral over $[t_*,t]$ with a singular kernel (Lemma 4.1), so the nonlocal data become part of the forcing term and the standard Duhamel argument applies. Corollary 4.1 gives an equivalent form with the history appearing as an explicit double integral.

Load-bearing premise

The load-bearing premise is Lemma 4.1's integral identity, whose proof is delegated to a cited theorem rather than carried out; if that identity fails, or if the resulting $b_*$ is not integrable, formula (4.8) collapses.

Editorial extensions

If this is right

For $t_*=t_0$, formula (4.8) reduces to Theorem 4.1 and recovers the known Duhamel representation for standard Caputo initial conditions.
The whole pre-history of the solution enters the future only through $b_*$; once $w_*(t_*)$ and $b_*$ are known, the solution on $[t_*,\vartheta]$ is determined by the same convolution as in the pointwise-initial-value case.
The fundamental matrix $F$ is bounded and H\"older continuous in both variables and satisfies the dual identity $G=F$, so the same object describes forward and backward propagation on $\Omega$.
The alternative formula (4.15) is simpler in shape but has a second term that need not vanish as $t\downarrow t_*$; this is an explicit warning that numerical schemes based on (4.15) must treat the neighbourhood of $t_*$ specially.

Reading between the lines

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

Because Lemma 4.1 only needs the fractional integral of the history to make sense, the memory-to-forcing conversion should extend to initial histories rougher than $AC^\alpha$; testing this would require proving an $L^p$ or distributional version of the identity.
The singular $(\tau-t_*)^{-\alpha}$ factor in $b_*$ means any numerical discretization of (4.8) must resolve a weak endpoint singularity; the paper flags the analogous difficulty for (4.15) but does not propose a scheme, leaving adaptive or product-integration methods as a natural next step.
Since $F$ satisfies a dual equation, the same formulas should transpose to adjoint or backward-in-time problems in dynamic programming for fractional systems, though the paper does not develop that direction.

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

The paper extends Duhamel-type representation formulas for linear Caputo fractional ODEs with variable coefficients to the case of nonlocal initial data on an interval [t0,t*]. As far as I can tell, that case is genuinely new; Bourdin and the standard texts only cover t*=t0. The main formula (4.8), with the modified forcing term b* in (4.9), is the real contribution, and the derivation is mostly detailed and standard. The fundamental solution matrix F is handled carefully, with Hölder continuity and a dual equation, and the comparison with Bourdin is fair. The reliance on the author's earlier existence results is fine; there's no circularity.

The soft spots are two. Lemma 4.1 is the hinge—it converts the nonlocal memory from the initial interval into the forcing term b*. Its proof is one sentence delegating to Samko-Kilbas-Marichev. That's thin for a lemma that carries the new idea. A referee should ask for a full derivation, or at least a precise statement of the theorem being used and a check of its hypotheses. This is fixable, not fatal.

The second concern is the Fubini-Tonelli interchange in Theorem 4.2. The paper notes that b* is not essentially bounded and says (4.12) and continuity of ψ suffice. The stress-test note worries that continuity alone gives no bound. I checked: continuity on a compact interval makes ψ bounded, and then (4.12) gives |b*(τ)| ≤ C (τ-t*)^(-α) + |b(τ)|. That's enough to make the repeated integral absolutely convergent, because the kernel (τ-ξ)^(α-1) is integrable against that singularity. So the stress-test's demand for a Hölder estimate is overstated. Still, the paper's sketch is too terse; the estimate should be written out.

Overall the central argument holds up. This paper is for readers in fractional differential equations and control who need explicit representations for nonlocal initial data. It deserves a serious referee. I'd send it to review with a request for a full proof of Lemma 4.1 and a clear justification of the interchange.

Referee Report

3 major / 4 minor

Summary. The paper studies a linear fractional differential equation with a Caputo derivative of order α∈(0,1) and variable coefficients, subject to a Cauchy condition specified on an initial interval [t0,t*] rather than at a single point. The authors introduce a nonsingular fundamental matrix F(t,s) through a regularized Volterra-type integral equation, establish its Hölder continuity with respect to both variables, prove a dual characterization, and then derive two representation formulas for the solution. The main result (Theorem 4.2, Eq. (4.8)) expresses the solution for t≥t* as an affine function of the terminal value w*(t*) plus a forced integral involving an explicitly defined function b* that encodes the memory of the initial segment. A second formula (Corollary 4.1, Eq. (4.15)) rewrites this in terms of w*(t0) and a double integral. The central derivation follows a contraction-mapping argument for F and a verification that the proposed formulas satisfy the equivalent integral equation from Proposition 2.1.

Significance. If the representation formulas are correct, they extend the variation-of-constants formula to nonlocal Caputo initial conditions, a setting relevant for control problems and differential games with fractional dynamics. The paper provides a self-contained study of the fundamental solution matrix with explicit constants (MF, HF in Propositions 3.1 and 3.2), which is a useful contribution in its own right. The main strategy is sound: the proofs of the fundamental matrix properties use standard contraction and Bellman-Gronwall techniques, and the representation formulas are verified by substitution into the integral equation. The chief unresolved points are two technical justifications in Section 4: the integral identity in Lemma 4.1 is delegated to an external source, and the Fubini-Tonelli interchange involving the singular function b* is only sketched. Both are load-bearing for the claimed representation, but they appear fixable with additional estimates.

major comments (3)

[Section 4, Lemma 4.1] Lemma 4.1 is the hinge of Theorem 4.2: its first equality in (4.7) converts the nonlocal memory term over [t0,t*] into a forcing term over [t*,t], and its second equality defines the singular part of b* in (4.9). The proof, however, consists of a single sentence referring to 'the scheme from [22, Theorem 13.10]' without stating the theorem or verifying its hypotheses. The exact result in [22] should be quoted with its conditions, or a complete proof should be given, because without (4.7) the representation formula (4.8) does not follow.
[Section 4, Theorem 4.2] In the proof of Theorem 4.2, the author claims that the equality analogous to (4.3) is proved by the same steps as in Theorem 4.1, noting only that 'the function b* is not essentially bounded in general' and that 'equality (4.12) and the inclusion ψ*(·)∈C([t*,θ],Rn) should be used' when applying Fubini-Tonelli. This is insufficient. A rigorous proof must establish the absolute integrability of the double integral containing b*. For example, from (4.12) one obtains |b*(τ)| ≤ C(τ−t*)^{−α} + |b(τ)|, and the inner kernel integral ∫_{r}^{t}(t−τ)^{α−1}(τ−r)^{α−1}dτ equals B(α,α)(t−r)^{2α−1}, which is integrable against this bound. Alternatively, the explicit expression (4.11) for ψ* can be used. The authors should include such an estimate so that the interchange of the order of integration in the proof of (4.14) is fully justified.
[Section 4, Eq. (4.13)] The assertion that the function y defined by (4.13) is continuous on [t0,θ] is not proven. The formula for y involves b*, which near t* is a difference of two singular terms (see (4.9)); continuity at t* requires showing that these singularities cancel and that the resulting limit equals w*(t*). The paper states only that this follows from the inclusions and Proposition 3.2. A verification of the cancellation, or an alternative argument, is needed before applying Proposition 2.1 to conclude that y coincides with the solution.

minor comments (4)

[Section 4, Lemma 4.1] In the second equality of (4.7), the factor (t−t*)^α is not clearly separated in the typeset display; a cleaner presentation would avoid possible confusion about which terms carry the singular factor.
[Section 4, Theorem 4.2] The verification of the identity used in (4.10), namely ψ*(t*) = (w*(t*)−w*(t0))/Γ(1−α), relies on the reflection formula for the gamma function and the definition of the Riemann-Liouville integral; it would be helpful to mention this explicitly, as it is a key step.
[Section 4, Corollary 4.1] In the proof of equality (4.16), the beta integral identity ∫_{t*}^{t}(t−τ)^{α−1}(τ−t*)^{−α}dτ = Γ(α)Γ(1−α) is used implicitly; citing this standard identity would improve readability.
[References] The paper relies on Proposition 2.1 from the author's earlier work [11] for existence and uniqueness. This is acceptable, but since it is central to the method, a brief indication of the proof or a more precise theorem statement would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the representation formulas are derived constructively from the integral-equation characterization and the fundamental-matrix definition; the flagged gaps are omitted proofs and a terse Fubini justification, not reductions to inputs.

full rationale

The central results, Theorems 4.1 and 4.2, are proved constructively: a candidate y is defined by the claimed representation formula and is then shown to satisfy the equivalent integral equation (2.4), so uniqueness from Proposition 2.1 yields equality. No fitted parameter is introduced and no target formula is assumed as an input. The load-bearing auxiliary facts are Proposition 2.1 (existence, uniqueness, and integral-equation equivalence), delegated to the author's earlier work [11]/[9], and Lemma 4.1's integral identity, delegated to the textbook scheme [22, Theorem 13.10]. These are independent supports: Proposition 2.1 has stated assumptions that do not include the representation formulas, and [22] is a standard external reference. The self-citations to [9]-[11] are prior theorems, not unverified re-statements of the target result, so they do not raise the circularity score. The skeptical objections about the unproved identity in Lemma 4.1 and the Fubini–Tonelli step for the unbounded b* are correctness or rigor concerns, not instances in which a prediction is equivalent to its input by construction. Hence no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. Its only analytic device is the singular effective forcing term b* (4.9), which is a function constructed from the initial data, not a new quantity with independent evidence. The fundamental solution matrix F is inherited from [3]. No free parameters are fitted to data.

assumptions (6)

domain assumption A(.) in L^infinity([t0,theta], R^{n x n}) and b(.) in L^infinity([t0,theta], R^n), see (2.1)
The theory is developed for essentially bounded coefficients and forcing terms; this excludes singular or unbounded A and b.
domain assumption The initial function w*(.) belongs to AC^alpha([t0,t*], R^n), see (2.3)
This space ensures the Caputo derivative of w* exists and the integral equation (2.4) is well posed.
domain assumption Proposition 2.1: existence, uniqueness, and equivalence with the integral equation (2.4), quoted from [11, Proposition 2] and [9, Theorem 3.1]
The representation formulas are derived from the integral equation; if this existence/uniqueness result were false, the formulas would not describe the solution.
ad hoc to paper Lemma 4.1: the identity (4.7) converting the memory term over [t0,t*] into a singular forcing term; the proof is delegated to 'the scheme from [22, Theorem 13.10]'
This is a newly stated technical lemma used only for the intermediate-point initial condition; it is load-bearing for Theorem 4.2.
standard math Standard continuity and inversion properties of Riemann-Liouville integrals and Caputo derivatives (Propositions 1.1 and 1.2, from [9])
These are classical results used throughout; they are not proved in the paper.
standard math The Bellman-Gronwall lemma (Lemma 3.1), cited as a corollary of [7, Lemma 6.19] and [10, Proposition 2]
Used in the proof of Holder continuity with respect to the second variable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On representation formulas for solutions of linear differential equations with Caputo fractional derivatives." pith.science (2026). https://pith.science/paper/MPFA7WQ5

@misc{pith2026190808319,
  author       = {Pith},
  title        = {Pith review of: On representation formulas for solutions of linear differential equations with Caputo fractional derivatives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPFA7WQ5}},
  note         = {Machine review of arXiv:1908.08319}
}

read the original abstract

In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the H\"{o}lder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[22]

Samko, A.A

S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications . Gordon and Breach Science Pub- lishers, Yverdon (1993)

work page 1993
[1]

Atanackovic, D

T. Atanackovic, D. Dolicanin, S. Pilipovic, and B. Stank ovic, Cauchy problems for some classes of linear fractional diﬀerential e qua- tions. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1039–1059; DOI: 10.2478/s13540-014-0213-1

work page doi:10.2478/s13540-014-0213-1 2014
[2]

Bonilla, M

B. Bonilla, M. Rivero, and J.J. Trujillo, On systems of li near fractional diﬀerential equations with constant coeﬃcients. Appl. Math. Comput. 187, No 1 (2007), 68–78; DOI: 10.1016/j.amc.2006.08.104

work page doi:10.1016/j.amc.2006.08.104 2007
[3]

Bourdin, Cauchy–Lipschitz theory for fractional mul ti-order dy- namics: State-transition matrices, Duhamel formulas and d ual- ity theorems

L. Bourdin, Cauchy–Lipschitz theory for fractional mul ti-order dy- namics: State-transition matrices, Duhamel formulas and d ual- ity theorems. Diﬀer. Integral Equ. 31, No 7-8 (2018), 559–594; https://projecteuclid.org/euclid.die/1526004031

work page arXiv 2018
[4]

L. Bourdin, Weighted H¨ older continuity of Riemann–Lio uville frac- tional integrals — Application to regularity of solutions t o fractional Cauchy problems with Carath´ eodory dynamics. Fract. Calc. Appl. Anal. 22, No 3 (2019), 722–749; DOI: 10.1515/fca-2019-0040

work page doi:10.1515/fca-2019-0040 2019
[5]

Chikriy and I.I

A.A. Chikriy and I.I. Matichin, Presentation of solutio ns of linear sys- tems with fractional derivatives in the sense of Riemann–Li ouville, Ca- puto and Miller–Ross. J. Autom. Inf. Sci. 40, No 6 (2008), 1–11; DOI: 10.1615/JAutomatInfScien.v40.i6.10

work page doi:10.1615/jautomatinfscien.v40.i6.10 2008
[6]

Cong and H.T

N.D. Cong and H.T. Tuan, Generation of nonlocal fraction al dynamical systems by fractional diﬀerential equations. J. Integral Equations Appl. 29, No 4 (2017), 585–608; DOI: 10.1216/JIE-2017-29-4-585. ON REPRESENTATION FORMULAS FOR SOLUTIONS . . . 19

work page doi:10.1216/jie-2017-29-4-585 2017
[7]

Diethelm, The Analysis of Fractional Diﬀerential Equations

K. Diethelm, The Analysis of Fractional Diﬀerential Equations. An Application-Oriented Exposition Using Diﬀerential Opera tors of Ca- puto Type . Volume 2004 of Lecture Notes in Mathematics, Springer- Verlag, Berlin (2010)

work page 2010

Show all 24 references

[8]

Duan, A generalization of the Mittag-Leﬄer function a nd solution of system of fractional diﬀerential equations

J. Duan, A generalization of the Mittag-Leﬄer function a nd solution of system of fractional diﬀerential equations. Adv. Diﬀer. Eq. Art. no. 239 (2018); DOI: 10.1186/s13662-018-1693-9

2018 doi
[9]

Gomoyunov, Fractional derivatives of convex Lyapu nov functions and control problems in fractional order systems

M.I. Gomoyunov, Fractional derivatives of convex Lyapu nov functions and control problems in fractional order systems. Frac. Calc. Appl. Anal. 21, No 5 (2018), 1238–1261; DOI: 10.1515/fca-2018-0066

2018 doi
[10]

Gomoyunov, Approximation of fractional order con ﬂict-controlled systems

M.I. Gomoyunov, Approximation of fractional order con ﬂict-controlled systems. Progr. Fract. Diﬀer. Appl. 5, No 2 (2019), 143–155; DOI: 10.18576/pfda/050205

2019 doi
[11]

Gomoyunov, Solution to a zero-sum diﬀerential game with frac- tional dynamics via approximations

M.I. Gomoyunov, Solution to a zero-sum diﬀerential game with frac- tional dynamics via approximations. Dyn. Games Appl. (2019), 1–27; DOI: 10.1007/s13235-019-00320-4

2019 doi
[12]

Gomoyunov and N.Yu

M.I. Gomoyunov and N.Yu. Lukoyanov, Guarantee optimiz a- tion in functional-diﬀerential systems with a control after ef- fect. J. Appl. Math. Mech. 76, No 4 (2012), 369–377; DOI: 10.1016/j.jappmathmech.2012.09.002

2012 doi
[13]

Gomoyunov and N.Yu

M.I. Gomoyunov and N.Yu. Lukoyanov, On the numerical so lution of diﬀerential games for neutral-type linear systems. Proc. Steklov Inst. Math. 301, Suppl 1 (2018), 44–56; DOI: 10.1134/S0081543818050048

2018 doi
[14]

Idczak and R

D. Idczak and R. Kamocki, On the existence and uniquenes s and for- mula for the solution or R–L fractional Cauchy problem in Rn. Fract. Calc. Appl. Anal. 14, No 4 (2011), 538–553; DOI: 10.2478/s13540-011- 0033-5

2011 doi
[15]

Kaczorek and D

T. Kaczorek and D. Idczak, Cauchy formula for the time-v arying linear systems with Caputo derivative. Fract. Calc. Appl. Anal. 20, No 2 (2017), 494–505; DOI: 10.1515/fca-2017-0025

2017 doi
[16]

Kilbas, H.M

A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applica- tions of Fractional Diﬀerential Equations . Volume 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (20 06)
[17]

Krasovskii and N.N

A.N. Krasovskii and N.N. Krasovskii, Control under Lack of Informa- tion. Birkh¨ auser, Boston (1995)

1995
[18]

Krasovskii and N.Yu

N.N. Krasovskii and N.Yu. Lukoyanov, Problem of conﬂic t control with hereditary information. J. Appl. Math. Mech. 60, No 6 (1996), 869–882; DOI: 10.1016/S0021-8928(96)00109-8

1996 doi
[19]

Gomoyunov, Diﬀerential games on minmax of the positional quality index

N.Yu Lukoyanov and M.I. Gomoyunov, Diﬀerential games on minmax of the positional quality index. Dyn. Games Appl. 9, No 3 (2018), 780– 799; DOI: 10.1007/s13235-018-0281-7. 20 M. I. Gomoyunov

2018 doi
[20]

Lukoyanov and T.N

N.Yu. Lukoyanov and T.N. Reshetova, Problems of conﬂic t control of high dimensionality functional systems. J. Appl. Math. Mech. 62, No 4 (1998), 545–554; DOI: 10.1016/S0021-8928(98)00071-9

1998 doi
[21]

Pskhu, Initial-value problem for a linear ordinar y diﬀerential equation of noninteger order

A.V. Pskhu, Initial-value problem for a linear ordinar y diﬀerential equation of noninteger order. Sb. Math. 202, No 4 (2011), 571–582; DOI: 10.4213/sm7645

2011 doi
[23]

Zahariev and H

A. Zahariev and H. Kiskinov, Existence of fundamental m atrix for neutral linear fractional system with distributed delays. Int. J. Pure Appl. Math. 119, No 1 (2018), 31–51; DOI: 10.12732/ijpam.v119i1.3

2018 doi
[24]

Zhang and D

H. Zhang and D. Wu, Variation of constant formulae for ti me invari- ant and time varying Caputo fractional delay diﬀerential sys tems. J. Math. Res. Appl. 34, No 5 (2014), 549–560; DOI: 10.3770/j.issn:2095- 2651.2014.05.006. Krasovskii Institute of Mathematics and Mechanics U...

2014

Pith tools

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

[1] [22]

Samko, A.A

S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications . Gordon and Breach Science Pub- lishers, Yverdon (1993)

work page 1993

[2] [1]

Atanackovic, D

T. Atanackovic, D. Dolicanin, S. Pilipovic, and B. Stank ovic, Cauchy problems for some classes of linear fractional diﬀerential e qua- tions. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1039–1059; DOI: 10.2478/s13540-014-0213-1

work page doi:10.2478/s13540-014-0213-1 2014

[3] [2]

Bonilla, M

B. Bonilla, M. Rivero, and J.J. Trujillo, On systems of li near fractional diﬀerential equations with constant coeﬃcients. Appl. Math. Comput. 187, No 1 (2007), 68–78; DOI: 10.1016/j.amc.2006.08.104

work page doi:10.1016/j.amc.2006.08.104 2007

[4] [3]

Bourdin, Cauchy–Lipschitz theory for fractional mul ti-order dy- namics: State-transition matrices, Duhamel formulas and d ual- ity theorems

L. Bourdin, Cauchy–Lipschitz theory for fractional mul ti-order dy- namics: State-transition matrices, Duhamel formulas and d ual- ity theorems. Diﬀer. Integral Equ. 31, No 7-8 (2018), 559–594; https://projecteuclid.org/euclid.die/1526004031

work page arXiv 2018

[5] [4]

L. Bourdin, Weighted H¨ older continuity of Riemann–Lio uville frac- tional integrals — Application to regularity of solutions t o fractional Cauchy problems with Carath´ eodory dynamics. Fract. Calc. Appl. Anal. 22, No 3 (2019), 722–749; DOI: 10.1515/fca-2019-0040

work page doi:10.1515/fca-2019-0040 2019

[6] [5]

Chikriy and I.I

A.A. Chikriy and I.I. Matichin, Presentation of solutio ns of linear sys- tems with fractional derivatives in the sense of Riemann–Li ouville, Ca- puto and Miller–Ross. J. Autom. Inf. Sci. 40, No 6 (2008), 1–11; DOI: 10.1615/JAutomatInfScien.v40.i6.10

work page doi:10.1615/jautomatinfscien.v40.i6.10 2008

[7] [6]

Cong and H.T

N.D. Cong and H.T. Tuan, Generation of nonlocal fraction al dynamical systems by fractional diﬀerential equations. J. Integral Equations Appl. 29, No 4 (2017), 585–608; DOI: 10.1216/JIE-2017-29-4-585. ON REPRESENTATION FORMULAS FOR SOLUTIONS . . . 19

work page doi:10.1216/jie-2017-29-4-585 2017

[8] [7]

Diethelm, The Analysis of Fractional Diﬀerential Equations

K. Diethelm, The Analysis of Fractional Diﬀerential Equations. An Application-Oriented Exposition Using Diﬀerential Opera tors of Ca- puto Type . Volume 2004 of Lecture Notes in Mathematics, Springer- Verlag, Berlin (2010)

work page 2010