Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem

E. Capelas de Oliveira; J. Vanterler da C. Sousa; K. D. Kucche

REVIEW 3 major objections 6 minor 34 references

Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem

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

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Pith's one-line read The paper proves that mild solutions of a fractional nonlinear abstract Cauchy problem are Ulam-Hyers stable under a contraction condition, with explicit stability constants on $[0,T]$ and $[0,\infty)$.

desk verdict Standard Ulam-Hyers stability theorems for a Hilfer abstract Cauchy problem, but the proofs rest on an ill-typed Lipschitz condition and dropped weight factors, so as written the contraction estimates don't hold. read the letter →

arxiv 1908.05296 v1 pith:6MYCEU4F submitted 2019-08-14 math.CA

J. Vanterler da C. Sousa , K. D. Kucche , E. Capelas de Oliveira This is my paper · ORCID

classification math.CA MSC 26A3334G2534A12

keywords Ulam-HyersstabilityUlam-Hyers-RassiasHilferfractionalderivativemildsolutionsabstractCauchyproblemBanachfixedpointtheoremevolutionequationsweightedcontinuousfunctionspace

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 tries to show that the mild-solution integral equation of a fractional nonlinear abstract Cauchy problem is Ulam-Hyers stable: whenever a function solves the equation with an error of at most $\varepsilon$ in the weighted norm $t^{1-\gamma}\|\cdot\|$, there is an exact mild solution within $c\varepsilon$ of it, for an explicit constant $c$. The same program is carried out for Ulam-Hyers-Rassias stability, where the error tolerance is a function $G(t)$ rather than a constant, on both a finite interval $[0,T]$ and the half-line $[0,\infty)$. The engine is the Banach fixed point theorem applied to the operator that defines mild solutions, and the stated conditions are contraction constants built from the resolvent operator, the weight $t^{1-\gamma}$, and the integrable growth of the nonlinearity. If the theorems are correct, they give quantitative error-control statements for a broad class of fractional evolution equations, with Caputo and Riemann-Liouville versions appearing as parameter limits.

What carries the argument

The load-bearing object is the fixed-point operator $\Lambda(\xi)(t)=S_{\alpha,\beta}(t)\xi_0+\int_0^t T_\alpha(t-s)u(s)H(s,\xi(s))\,ds$ acting on the weighted continuous function space $C_{1-\gamma}(I,\Omega)$, whose norm is $\|\xi\|_{C_{1-\gamma}}=\sup_t\|t^{1-\gamma}\xi(t)\|$. The proof shows $\Lambda$ is a contraction either in the metric induced by this norm, for the constant-tolerance theorems, or in the $\varphi$-weighted metric $d_{\varphi,1-\gamma}$, for the Rassias-type theorems, and then invokes the Banach fixed point theorem. The stability estimate is obtained by applying the triangle inequality to the distance between an approximate solution and the fixed point, which reduces the approximate error to $c\varepsilon$ with $c=1/(1-\lambda)$. The kernel $T_\alpha(t)=t^{\alpha-1}G_\alpha(t)$ with $G_\alpha$ defined through the Mainardi-Wright function, together with the resolvent family $S_{\alpha,\beta}$, carries all the fractional-semigroup information.

What would settle it

Take the scalar model $A=0$, $u(t)=1$, $H(t,x)=ax$ with a fixed constant $a$. Compute the exact solution and compare the theorem's predicted Ulam-Hyers bound with the true weighted distance between an approximate solution, such as the exact solution plus a small bump of size $\varepsilon$, and the exact solution. A single instance with $\tilde\lambda<1$ whose weighted distance exceeds $\varepsilon/(1-\tilde\lambda)$ would refute the claim.

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Extended reading notes

Core claim

The central claim is that the integral equation $\xi(t)=S_{\alpha,\beta}(t)\xi_0+\int_0^t T_\alpha(t-s)u(s)H(s,\xi(s))\,ds$ is stable in the Ulam-Hyers sense on $[0,T]$ whenever $\tilde\lambda=\delta T^{1-\gamma}\int_0^T e^{w(T-s)}|u(s)|\ell(s)\,ds<1$, and on $[0,\infty)$ whenever $\tilde\lambda_{\alpha,1-\gamma}=\sup_{t\ge0} t^{1-\gamma}\int_0^t \ell(s)|u(s)|\|T_\alpha(t-s)\|\,ds<1$. Here $S_{\alpha,\beta}$ is the $(\alpha,\beta)$-resolvent family generated by $A$ and $T_\alpha$ is built from the Mainardi-Wright function. The paper also claims the Ulam-Hyers-Rassias analogues with function-valued tolerances $G(t)$ or $\varphi(t)$, under the boundedness conditions (3.5) and (4.9) and the integral inequality $\int_0^t \varphi(s)\,ds\le K\varphi(t)$. In all four theorems the stability constant is explicit and has the form $1/(1-\text{contraction constant})$.

Load-bearing premise

The load-bearing premise is that the nonlinearity is Lipschitz; as written, the condition uses the function-space norm $\|x-y\|_{C_{1-\gamma}}$ for pointwise values, which requires an extra weight-handling step that the proofs do not supply before the contraction estimates follow.

Editorial extensions

If this is right

On the finite interval, the paper implies that any function whose residual $t^{1-\gamma}(\xi-\Lambda\xi)$ is bounded by $\varepsilon$ lies within $\varepsilon/(1-\tilde\lambda)$ of a genuine mild solution, so the stability error is controlled by the data error with an explicit amplification factor.
On the half-line, the uniform contraction condition $\tilde\lambda_{\alpha,1-\gamma}<1$ gives the same control uniformly in time, so stability does not deteriorate as $t\to\infty$.
The parameter limits $\beta\to1$ and $\beta\to0$ recover the same stability theorems for the Caputo and Riemann-Liouville fractional derivatives, and $\alpha=1$ recovers the integer-order abstract Cauchy problem.
The Rassias formulations imply that if the allowed error grows like a function $G(t)$ satisfying the comparison inequality, the exact-solution error grows at the same rate, with constant $C_G$.

Reading between the lines

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

A direct numerical test is a natural next step: for $A=0$, $u\equiv1$, and linear $H(t,x)=ax$, the exact mild solution is available, so the theorem's constant $\varepsilon/(1-\tilde\lambda)$ can be compared with the actual weighted sup-distance for constructed approximate solutions.
The same fixed-point scheme should transfer to $\psi$-Hilfer derivatives and to impulsive or neutral variants of the Cauchy problem, since the argument only needs a resolvent family and the weighted-space contraction estimate.
The stated Lipschitz condition uses the function-space norm $\|x-y\|_{C_{1-\gamma}}$ where $x,y$ are pointwise values; a sympathetic repair would impose a pointwise Lipschitz condition with a weight $s^{\gamma-1}$ and carry the extra factor through the contraction constant, which would keep the theorems intact while making the estimates valid.

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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 claims Ulam-Hyers and Ulam-Hyers-Rassias stability for a fairly broad class of fractional abstract Cauchy problems with Hilfer derivative and (α,β)-resolvent, on both finite and infinite intervals. The combination is new, and the paper honestly closes with open questions about the ψ-Hilfer case and says the Laplace transform for that setting is missing. The fixed-point strategy is standard, and the authors correctly point to limit cases (β→0, β→1, α=1) as particular instances. For a routine extension of known arguments, the topic is genuinely useful to people working in fractional stability theory.

The soft spots are not minor. Condition (2.5) is ill-typed: it quantifies over pointwise values x,y∈Ω but uses the C_{1−γ} norm, which is a norm on functions. As written the assumption has no consistent meaning, especially on [0,∞) where constant embedding would make it infinite. All four theorems rely on this condition, and even under the most charitable reading (an Ω-norm Lipschitz condition), the proofs drop the factor s^{γ−1} when passing from pointwise differences to the C_{1−γ} norm. The constants λ̃, δρKT^{1−γ}, λ̃_{α,1−γ} do not follow without that factor. Theorem 4.2 also has a load-bearing inconsistency: it is stated on [0,∞) but the proof and the condition ρKT^{1−γ}<1 use a finite T, and the final constant differs between statement and proof.

The reader's report and the stress-test note are accurate. I checked the displayed estimates in the proofs of Theorems 3.1, 3.2, 4.1, and 4.2; each one uses ‖H(s,φ(s))−H(s,ξ(s))‖ ≤ ℓ(s)‖φ−ξ‖_{C_{1−γ}} without the required weight. This makes the contraction step unproved in every theorem. The issues are likely fixable—one could either strengthen the Lipschitz condition to the pointwise Ω-norm and carry the weight through, or work with a different norm where the weight is accommodated—but as written the central estimates are not justified.

This paper is for readers who want a ready-made reference for Ulam stability of Hilfer-type abstract Cauchy problems. It deserves a serious referee in the sense that the patch is plausible and the topic is mainstream, but the referee should expect major revision. My own verdict is skeptical as written; I would not cite it until the hypotheses and proofs are made coherent. I'd bring it to a reading group only as an example of how weighted-norm fixed-point arguments can go wrong.

Referee Report

3 major / 6 minor

Summary. The paper studies Ulam-Hyers and Ulam-Hyers-Rassias stability for mild solutions of the fractional nonlinear abstract Cauchy problem with a Hilfer derivative, on finite intervals [0,T] and on [0,∞). The solution operator is defined via an (α,β)-resolvent family, and stability is proved by showing that the associated operator Λ is a contraction in a weighted space of continuous functions and then applying the Banach fixed point theorem. Theorems 3.1 and 3.2 treat the finite interval, Theorems 4.1 and 4.2 the half-line, and the authors note limiting cases β→0, β→1 and α=1 corresponding to Riemann-Liouville, Caputo and integer-order settings.

Significance. If the proofs were correct, the paper would provide a fairly general Ulam-Hyers stability result for Hilfer-type fractional evolution equations, with explicit stability constants and with several classical fractional-derivative settings as limiting cases. The proof strategy is standard and the organization around the resolvent operator is reasonable. The paper does not provide machine-checked proofs or reproducible code, but the explicit constants are a useful feature. However, the central contraction estimates all rely on a Lipschitz condition that is ill-typed as written, and the subsequent estimates omit weight factors that are essential for the C_{1-γ} norm; Theorem 4.2 additionally contains an undefined parameter and an internally inconsistent proof. These issues affect every main theorem, so the results as stated are not established.

major comments (3)

[Section 2, condition (2.5)] Condition (2.5) is ill-typed: it states ||H(t,x)-H(t,y)|| ≤ ℓ(t)||x-y||_{C_{1-γ}} for x,y∈Ω, but ||·||_{C_{1-γ}} is a norm on functions on I, not on elements of Ω. If Ω is embedded as constant functions, then on I=[0,∞) the norm of a nonzero constant difference is infinite when γ<1, making the condition vacuous, and on [0,T] it introduces an extra factor T^{1-γ} that is not carried through the proofs. In the proofs of Theorems 3.1, 3.2, 4.1 and 4.2, the displayed estimate ||H(s,φ(s))-H(s,ξ(s))|| ≤ ℓ(s)||φ-ξ||_{C_{1-γ}} is used; this does not follow from (2.5) as written. If the intended hypothesis is the usual Ω-norm Lipschitz condition, the valid pointwise estimate is ||H(s,φ(s))-H(s,ξ(s))|| ≤ ℓ(s)s^{γ-1}||φ-ξ||_{C_{1-γ}}, and the missing factor s^{γ-1} appears inside the integrals in every contraction estimate. This changes the constants in (3.2), Theorem 3.2, (4.2) and Theorem 4.2, and for γ<1 the integral may even diverge at s=0 unless additional hypotheses on ℓ and u are imposed. The contraction step, on which all stability conclusions rest, is therefore not justified as written.
[Theorem 3.2 proof] Independently of condition (2.5), the proof of Theorem 3.2 contains an unjustified replacement of the global norm ||h-g||_{C_{1-γ}} by C(h,g)φ(s). From ||t^{1-γ}(h(t)-g(t))|| ≤ C(h,g)φ(t) one may conclude pointwise that ||h(s)-g(s)|| ≤ s^{γ-1}C(h,g)φ(s), but not that the global supremum is bounded by the pointwise value C(h,g)φ(s). The displayed chain of inequalities leading to d_{φ,1-γ}(Λh,Λg) ≤ δρKT^{1-γ}d_{φ,1-γ}(h,g) therefore skips a necessary weight factor s^{γ-1} as well as a supremum argument, so the contraction in the Bielecki-type metric is not established.
[Theorem 4.2] Theorem 4.2 is stated on the half-line [0,∞), but its proof and hypothesis (4.10) use a parameter T that is never defined in the theorem; the proof even displays t∈[0,T] at one point. Moreover, after the estimate ≤ ρC(h,g)Kφ(t), the next display introduces a factor T^{1-γ} without any justification, and the concluding bound (4.12) omits that factor. The theorem as stated is therefore not proved: the contraction constant, the fixed-point space, and the final stability bound are inconsistent.

minor comments (6)

[Section 2] After condition (2.5), the functions ℓ and u are declared on [0,T], although later the interval I may be [0,∞); the domains should be I consistently.
[Theorem 3.1 statement] The theorem says the resolvent operator acts on a Banach space (Ω, ||·||_{C_{1-γ}}), which conflates the state space Ω with the function space C_{1-γ}(I,Ω); S_{α,β}(t) maps Ω to Ω, while the contraction argument is in C_{1-γ}(I,Ω).
[Theorem 3.1 proof] The sentence 'since λ~<1, Λ is a contradiction' should read 'Λ is a contraction'.
[Theorem 3.2 proof] The completeness of (C_{1-γ}(I,Ω), d_{φ,1-γ}) is asserted but not proved; while standard, it should be justified or cited.
[Sections 3 and 4] Several displays refer to 'Eq.(6)' (e.g., Theorem 3.4 and the discussion in Remark 3.3) although the equation numbers are not assigned to (2.9); renumber the references to the actual displayed equations.
[Theorem 4.2 statement] The constants in the theorem and proof are written inconsistently: hypothesis (4.10) uses ρKT^{1-γ}<1, the proof derives δ_{φ,1-γ} ≤ 1/(1-T^{1-γ}ρK), and conclusion (4.12) gives 1/(1-Kρ) φ(t) without T^{1-γ}; these should be harmonized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability theorems follow from the stated Lipschitz and contraction hypotheses by the Banach fixed point theorem, with self-citations confined to background definitions and paired with independent references.

full rationale

The derivation chain is self-contained. Theorem 3.1 assumes the contraction condition \tilde{\lambda} = \delta T^{1-\gamma}\int_0^T e^{w(T-s)}|u(s)|\ell(s)\,ds < 1 and, using the Lipschitz hypothesis (2.5) and the resolvent bounds, proves d_{1-\gamma}(\Lambda\varphi,\Lambda\xi) \le \tilde{\lambda}\,d_{1-\gamma}(\varphi,\xi), then obtains d_{1-\gamma}(\theta,v) \le \varepsilon/(1-\tilde{\lambda}) by the triangle inequality. Theorems 3.2, 4.1 and 4.2 follow the same pattern with the corresponding weighted metrics d_{\varphi,1-\gamma} and with the constants K, \rho and C_\varphi determined by the hypotheses. No parameter is fitted to data and no output quantity is an input by construction: the Ulam-Hyers constants are explicit functions of the assumed Lipschitz function, the resolvent bounds and the chosen \varphi, and the fixed point v is produced by Banach's fixed point theorem. The self-citations [1,2,5,6,8] supply background definitions, the weighted function space and auxiliary representation results; Lemma 2.1 is cited jointly with the independent reference [7], and the stability definitions are adapted from the external methodology [10]. The ill-typed Lipschitz condition (2.5) and the missing C_{1-\gamma} weight factors are genuine correctness concerns, but they are not circularity: the argument does not assume the stability conclusion it proves.

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

There are no fitted parameters; the constants δ, w, ρ, K and the functions φ, ℓ, u are part of the hypotheses. The main mathematical input is the Lipschitz condition and the growth bounds on the resolvent. No new entities are postulated.

assumptions (6)

standard math Banach fixed point theorem.
Used in every theorem to obtain existence and uniqueness of the fixed point v.
domain assumption A generates a C0-semigroup (S(t))_{t≥0}; the (α,β)-resolvent S_{α,β} and T_α satisfy the growth bounds ||S_{α,β}(t)|| ≤ δe^{wt} and ||T_α(t)|| ≤ δe^{wt}.
Stated in Section 2 and used in all contraction estimates, e.g., (3.3).
domain assumption The function H satisfies the Lipschitz-type condition (2.5) with respect to the norm ||·||_{C_{1-γ}}.
This condition is the key input for the contraction estimates, but its meaning is unclear because H maps into Ω, not into C_{1-γ}.
domain assumption The functions ℓ, u, and ℓu are locally integrable on I; in the infinite interval case, the relevant integrals converge.
Needed for the fixed point operator to be well defined and for contraction constants to be finite.
domain assumption In Theorems 3.2 and 4.2, there exists a continuous φ satisfying ∫_0^t φ(s)ds ≤ Kφ(t), and G is equivalent to φ in the sense αφ ≤ G ≤ βφ.
This is a standard Bielecki-type condition, but it restricts the growth of the error function G and is not discussed in the paper.
domain assumption Lemma 2.1: Eq (1.1) is equivalent to the integral equation (2.6) and the mild solution is given by (2.7).
Taken from prior work (refs [6,7]), this equivalence is not proved in this paper.

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Pith. "Pith review of Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem." pith.science (2026). https://pith.science/paper/6MYCEU4F

@misc{pith2026190805296,
  author       = {Pith},
  title        = {Pith review of: Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MYCEU4F}},
  note         = {Machine review of arXiv:1908.05296}
}

abstract

Since the main work on Ulam-Hyers dependable stabilities of differential equations to date, numerous significant and applicable papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we will investigate the Ulam--Hyers and Ulam--Hyers--Rassias stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem on the intervals $[0,T]$ and $[0,\infty)$, by means of Banach fixed point theorem.

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Works this paper leans on

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[1] [1]

Sousa, J., Kucche, Kishor D., Capelas de Oliveira, E ., Stability of ψ-Hilfer impulsive fractional diﬀerential equations, Appl

Vanterler da C. Sousa, J., Kucche, Kishor D., Capelas de Oliveira, E ., Stability of ψ-Hilfer impulsive fractional diﬀerential equations, Appl. Math. Lett., 88 (2 019) 73–80

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work page 2018

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