{"id":"0c111d2b-3488-4764-8f71-b621d50c2edf","arxiv_id":"1908.07793","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims unique solutions and Ulam-Hyers-Mittag-Leffler stability for implicit impulsive Psi-Hilfer fractional differential equations with delay, but the stability proof relies on a false monotonicity assertion.","lead":"This paper states existence, uniqueness, and Ulam-Hyers-Mittag-Leffler stability theorems for a class of impulsive fractional differential equations that combine a Psi-Hilfer derivative, an implicit nonlinearity, and a time delay. A smart generalist might read it to see whether yet another fractional calculus stability extension actually proves what it claims.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The comparison operator Q in the stability proof is not a valid majorant of z: the delayed term in (4.10) carries the weight at s rather than at h(s), so z ≤ Q(z) does not follow and the UHML bound collapses.","rationale":"The reader's REJECT verdict is correct, and I identify a distinct load-bearing flaw. The stability proof's comparison argument asserts z ≤ Q(z) directly from (4.10), but the delayed term in (4.10) is weighted by (Ψ(s) − Ψ(0))^{1−ρ}, not by (Ψ(h(s)) − Ψ(0))^{1−ρ}. Consequently the operator Q in (4.11) is not a majorant of the right-hand side needed to dominate z; the inequality direction is wrong, and Lemma 2.3 cannot be applied. This invalidates the UHML stability conclusion independently of the paper's later monotonicity claim. That later claim is also false: for 0 < α < 1, fractional integrals of nonnegative functions can decrease, and the proof's kernel-difference lower bound drops a negative cross term. Even if Q were repaired, the delay-removal step z*(h(t)) ≤ z*(t) would still lack a valid proof. Other defects include the incorrect Γ((n+1)α) denominator in (4.9), the undefined constant m, and the dropped ζ factor in the estimate of E(s); these are secondary. The existence-uniqueness argument is a standard contraction, though it shares the same weight-mismatch issue in bounding the delayed term and should be rechecked. As written, Theorem 4.1(2) is not established, so the manuscript should not be accepted in its present form.","tokens_in":15107,"tokens_out":26155,"duration_ms":251054,"concrete_test":"Recompute the comparison step for a concrete instance: take α = 1/2, β = 0 (so ρ = 1/2), Ψ(t) = t, and h(t) = t/2. Then 1 − ρ = 1/2 and q(s) = (s/(s/2))^{1/2} = √2, so the delayed contribution in (4.10) is √2 z(h(s)) ds, while Q uses z(h(s)) ds. Substituting this into (4.10) and (4.11) gives RHS_{4.10}(t) − (Qz)(t) = K(Ψ(b) − Ψ(0))^{1/2}/((1 − L_f)Γ(1/2)) ∫_0^t (t − s)^{−1/2}(√2 − 1)z(h(s)) ds ≥ 0, with strict positivity whenever z(h(s)) > 0. Therefore z ≤ RHS_{4.10} does not imply z ≤ Qz; the claimed application of Lemma 2.3 fails at this step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1(2) rests on showing that z(t) = (Ψ(t) − Ψ(0))^{1−ρ}|v(t) − u(t)| is bounded by the fixed point z* of the operator Q defined in (4.11). The only link is the sentence “from (4.10) we have z ≤ Q(z)”. That link is false. In (4.10), the delayed part of the integrand is (Ψ(s) − Ψ(0))^{1−ρ}|v(h(s)) − u(h(s))|, whereas Q in (4.11) contains z(h(s)) = (Ψ(h(s)) − Ψ(0))^{1−ρ}|v(h(s)) − u(h(s))|. Because h(s) ≤ s and Ψ is increasing, the former is larger by the factor q(s) = ((Ψ(s) − Ψ(0))/(Ψ(h(s)) − Ψ(0)))^{1−ρ} ≥ 1 when h(s) > 0, and the term vanishes when h(s) ≤ 0. Hence the right-hand side of (4.10) is not Qz(t) but Qz(t) plus a nonnegative integral of (q(s) − 1)z(h(s)). From z ≤ RHS and RHS ≥ Qz one cannot infer z ≤ Qz, so Lemma 2.3 is inapplicable. This gap occurs before, and independently of, the later assertion that z* is increasing; that assertion is also false, since a fractional integral of order α < 1 of a nonnegative function need not be monotone and the displayed Abel-kernel lower bound discards a negative cross term. The comparison failure alone is fatal to the stability conclusion as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15364,"tokens_out":7571,"duration_ms":147229,"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":[{"comment":"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":"Section 4, Eqs. (4.10)–(4.11)"},{"comment":"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.","section":"Section 4, after Eq. (4.14)"},{"comment":"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.","section":"Theorem 4.1(2) and Remark 4.2"}],"minor_comments":[{"comment":"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":"Section 4, Eq. (4.9)"},{"comment":"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":"Section 4, Eq. (4.11)"},{"comment":"The phrase “z* is increasing operator” should read “z* is an increasing function”; the word operator is inappropriate for a real-valued function.","section":"Section 4, after Eq. (4.14)"},{"comment":"The sentence “This section deals with the In this section, we derive...” contains a duplicated fragment and should be rewritten.","section":"Section 4, opening sentence"},{"comment":"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.","section":"Example 5.1, inequality (5.3)"},{"comment":"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.","section":"References"}],"recommendation":"reject","confidential_remarks":"The existence-uniqueness part is standard, while the stability theorem, which is the main novelty, rests on a comparison inequality that is false as written. The gap is not a local typo: it affects the core mechanism of the proof, and the monotonicity of the comparison fixed point is also asserted without a valid justification. For these reasons I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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_{01$), so the lower bound used to prove $z^*$ increasing breaks down; if the actual fixed point is nevertheless increasing, the stability claim would survive.","tokens_in":14815,"feed_emoji":"🧮","tokens_out":14687,"duration_ms":291882,"temperature":0.7,"pith_summary":"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.","feed_headline":"Stability and uniqueness proved for delayed fractional equations","feed_subtitle":"A theorem for impulsive implicit Ψ-Hilfer equations with delay, with Ulam-Hyers stability as a special case.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Ψ-Hilfer delay-equation UHML stability setting that this paper extends to impulses and implicit equations.","marker":"[16]"},{"why":"Introduces the Ulam-Hyers-Mittag-Leffler stability notion and the Picard-operator/Gronwall strategy for delay equations.","marker":"[14]"},{"why":"Gives the extended Gronwall inequality used as Lemma 2.4 to turn the comparison envelope into the explicit Mittag-Leffler bound.","marker":"[23]"},{"why":"Provides the Cauchy-type representation and the identity that lets the fractional integral equation (3.2) imply the original derivative equation (1.1).","marker":"[25]"},{"why":"Defines the Ψ-Hilfer derivative and the weighted space C_{1-ρ;Ψ} that carries the solution space and norms.","marker":"[19]"},{"why":"Supplies Lemma 3.1, the formula for solutions between impulses, used to derive the equivalent integral equation.","marker":"[24]"},{"why":"Gives the abstract Gronwall lemma used to conclude z ≤ Qz implies z ≤ z*.","marker":"[22]"},{"why":"Defines Picard operators, used to identify the unique fixed point of the comparison operator.","marker":"[21]"}],"fun_headline_variants":["Delayed Ψ-Hilfer equations: uniqueness and U-H stability proven","Ulam-Hyers-Mittag-Leffler stability for impulsive Ψ-Hilfer delay","Fixed-point proof of uniqueness for Ψ-Hilfer delay problems","Gronwall-based stability for impulsive fractional delay equations","Ψ-Hilfer delay: existence, uniqueness, and stability results"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Delayed Ψ-Hilfer equations: uniqueness and U-H stability proven","Ulam-Hyers-Mittag-Leffler stability for impulsive Ψ-Hilfer delay","Fixed-point proof of uniqueness for Ψ-Hilfer delay problems","Gronwall-based stability for impulsive fractional delay equations","Ψ-Hilfer delay: existence, uniqueness, and stability results"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2923,"prompt_tokens":897,"completion_tokens":2026,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":1933}},"tokens_in":513,"tokens_out":2026,"duration_ms":15096,"temperature":1.0,"reasoning_tokens":1933,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:56:24.751392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Ψ-Hilfer delay-equation UHML stability setting that this paper extends to impulses and implicit equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Ulam-Hyers-Mittag-Leffler stability notion and the Picard-operator/Gronwall strategy for delay equations."},{"cited_title":"Vanterler da C","cited_arxiv_id":null,"evidence_quote":"Provides the Cauchy-type representation and the identity that lets the fractional integral equation (3.2) imply the original derivative equation (1.1)."},{"cited_title":"Vanterler da C","cited_arxiv_id":null,"evidence_quote":"Defines the Ψ-Hilfer derivative and the weighted space C_{1-ρ;Ψ} that carries the solution space and norms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the abstract Gronwall lemma used to conclude z ≤ Qz implies z ≤ z*."},{"cited_title":"World scientific (2014)","cited_arxiv_id":null,"evidence_quote":"Defines Picard operators, used to identify the unique fixed point of the comparison operator."}],"review_version":1}