{"id":"a7e3b599-d374-4826-ac1a-0720e230df97","arxiv_id":"1908.05296","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove Ulam-Hyers and Ulam-Hyers-Rassias stability for mild solutions of a Hilfer fractional abstract Cauchy problem on [0,T] and [0,∞) using the Banach fixed point theorem.","lead":"This paper proves stability estimates for solutions of a fractional differential equation in an abstract Banach space, using a contraction mapping argument. The results say that approximate solutions cannot drift far from exact solutions, on both finite and infinite time intervals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lipschitz condition (2.5) is ill-typed and the contraction estimates drop the C_{1-γ} weight; as written Theorems 3.1–4.2 are not proved.","rationale":"The reader's weakest assumption exactly identifies the ill-typed Lipschitz condition (2.5): a function-space norm applied to pointwise Banach-space elements, plus the subsequent omission of the weight factor in every contraction estimate. My independent check of the proof chain in Theorem 3.1 confirms that the inequality ‖(Λφ)(t)−(Λξ)(t)‖ ≤ ∫‖T_α(t−s)‖|u(s)|ℓ(s)ds ‖φ−ξ‖_{C_{1−γ}} does not follow from (2.5) under any consistent interpretation: the condition either gives a weighted pointwise bound with T^{1−γ}s^{γ−1}, or the norm must be converted via s^{γ−1}. With γ<1 the missing factor is singular at s=0, so the contraction constants can be wrong, as the scalar example shows. This is not merely a typo because it changes the sufficient conditions for stability: additional assumptions on ℓ|u| near 0 or a different contraction constant are needed. The paper is likely fixable by replacing (2.5) with a genuine pointwise Ω-Lipschitz condition and redoing the weight estimates, so the work is not worthless; but as written the central proofs are invalid. Thus the reader's conditional verdict is appropriate, and no change to that verdict is needed. Theorem 4.2's additional inconsistency (a statement on [0,∞) proved with a finite T and a condition (4.10) using an undefined T) further supports the need for revision, but the Lipschitz/weight issue is the single most load-bearing concern.","tokens_in":13205,"tokens_out":8545,"duration_ms":80322,"concrete_test":"Re-derive the key contraction inequality of Theorem 3.1 with condition (2.5) replaced by the standard pointwise Lipschitz hypothesis ‖H(t,x)−H(t,y)‖_Ω ≤ ℓ(t)‖x−y‖_Ω, and with the fixed-point metric d_{1−γ}. The valid estimate contains ‖φ−ξ‖_C ∫_0^t s^{γ−1}e^{w(t−s)}|u(s)|ℓ(s)ds, not the factorless integral in the proof. For the concrete scalar case Ω=ℝ, γ=1/2, T=1, u≡1, ℓ≡1, w=0, δ=1, A=0 with normalized kernel ‖T_α‖≤1, take h(s)=s^{−1/2}a and g(s)=s^{−1/2}b in C_{1−γ}. Then d_{1−γ}(h,g)=|a−b|, while the pointwise difference is |h(s)−g(s)|=s^{−1/2}|a−b|, not |a−b|. Direct computation gives d_{1−γ}(Λh,Λg)=sup_{t≤1} t^{1/2}∫_0^t s^{−1/2}ds |a−b|=2|a−b|, whereas (3.2) gives λ~=δT^{1−γ}∫_0^T|u|ℓ ds=1. The discrepancy refutes the contraction estimate as stated and shows the missing s^{γ−1} weight is load-bearing.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Condition (2.5) quantifies over pointwise values x,y ∈ Ω but measures them in the function-space norm ‖·‖_{C_{1-γ}}. For a point x∈Ω the expression ‖x−y‖_{C_{1-γ}} is undefined unless Ω is embedded as constant functions; on [0,T] that embedding gives T^{1−γ}‖x−y‖_Ω and on [0,∞) it is infinite for γ<1, so the assumption has no consistent meaning on the infinite interval. All four theorems then use (2.5) to write ‖H(s,φ(s))−H(s,ξ(s))‖ ≤ ℓ(s)‖φ−ξ‖_{C_{1-γ}}. If (2.5) is read with the constant embedding, the valid pointwise estimate at time s carries the extra factor T^{1−γ}s^{γ−1}; if it is read as the intended Ω-Lipschitz condition, the proof must first pass from pointwise values to the weighted norm via ‖φ(s)−ξ(s)‖_Ω ≤ s^{γ−1}‖φ−ξ‖_C. Either way the displayed estimates in Theorems 3.1, 3.2, 4.1 and 4.2 omit the factor s^{γ−1} inside the integral, and the constants λ~, δρKT^{1−γ}, λ~_{α,1−γ} do not follow. For γ<1 the omitted factor is singular at s=0, so the contraction constant may even be infinite unless ℓ|u| compensates; the stated hypotheses do not require this. Consequently the contraction step, on which every stability conclusion rests, is not established as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13536,"tokens_out":6469,"duration_ms":60836,"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":[{"comment":"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.","section":"Section 2, condition (2.5)"},{"comment":"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.","section":"Theorem 3.2 proof"},{"comment":"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.","section":"Theorem 4.2"}],"minor_comments":[{"comment":"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.","section":"Section 2"},{"comment":"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).","section":"Theorem 3.1 statement"},{"comment":"The sentence 'since λ~<1, Λ is a contradiction' should read 'Λ is a contraction'.","section":"Theorem 3.1 proof"},{"comment":"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.","section":"Theorem 3.2 proof"},{"comment":"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.","section":"Sections 3 and 4"},{"comment":"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.","section":"Theorem 4.2 statement"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is plausible, and the flaws identified above appear fixable by reformulating condition (2.5) as a pointwise Ω-Lipschitz hypothesis, carrying the weight s^{γ-1} through the estimates, and repairing the statement and proof of Theorem 4.2. The high density of self-citations for background material is worth watching, but it is not by itself disqualifying. I would consider a revised version in which the contraction estimates are re-derived with all weight factors and the half-line theorem is stated consistently."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":14085,"tokens_out":3909,"would_cite":false,"duration_ms":39562,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26A33","34G25","34A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $[0,\\infty)$.","keywords":["Ulam-Hyers stability","Ulam-Hyers-Rassias stability","Hilfer fractional derivative","mild solutions","abstract Cauchy problem","Banach fixed point theorem","fractional evolution equations","weighted continuous function space"],"falsifier":"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.","tokens_in":13001,"feed_emoji":"📏","tokens_out":10346,"duration_ms":94540,"temperature":0.7,"pith_summary":"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.","feed_headline":"Approximate solutions stay ε-close to exact ones","feed_subtitle":"Paper proves Ulam-Hyers stability for Hilfer-fractional abstract Cauchy problems on finite and infinite intervals.","key_machinery":"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.","core_discovery":"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 $1/(1-\\text{contraction constant})$.","pith_inferences":["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."],"forward_implications":["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 $C_G$."],"supporting_citations":[{"why":"Supplies the fixed-point approach and the definitions of Ulam-Hyers-Rassias stability for integral equations that the paper adapts.","marker":"[10]"},{"why":"Provides the Mittag-Leffler-Ulam stability notions for fractional evolution equations that this paper extends to the present setting.","marker":"[9]"},{"why":"Defines the $(\\alpha,\\beta)$-resolvent operator function that carries the semigroup part of the mild solution.","marker":"[4]"},{"why":"Supplies the mild-solution representation for Hilfer fractional evolution equations via the Mainardi-Wright function.","marker":"[6]"},{"why":"Supports the equivalence between the fractional abstract Cauchy problem and the integral equation used to define mild solutions.","marker":"[7]"},{"why":"Provides the weighted-space formulation and Ulam-Hyers stability framework for fractional Volterra integro-differential equations.","marker":"[8]"},{"why":"Supplies the weighted continuous function space $C_{1-\\gamma}(I,\\Omega)$ and its norm used throughout the contraction estimates.","marker":"[1]"}],"fun_headline_variants":["Fractional Cauchy problem: mild solutions are Ulam-Hyers stable","ε-close approximations hold for fractional Cauchy mild solutions","Stability of fractional abstract Cauchy solutions on long intervals","Ulam-Hyers stability proved for fractional nonlinear Cauchy","Mild solutions of fractional Cauchy stay ε-close to exact"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fractional Cauchy problem: mild solutions are Ulam-Hyers stable","ε-close approximations hold for fractional Cauchy mild solutions","Stability of fractional abstract Cauchy solutions on long intervals","Ulam-Hyers stability proved for fractional nonlinear Cauchy","Mild solutions of fractional Cauchy stay ε-close to exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1332,"prompt_tokens":931,"completion_tokens":401,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":320}},"tokens_in":547,"tokens_out":401,"duration_ms":4281,"temperature":1.0,"reasoning_tokens":320,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:18:15.218452+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fixed-point approach and the definitions of Ulam-Hyers-Rassias stability for integral equations that the paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Mittag-Leffler-Ulam stability notions for fractional evolution equations that this paper extends to the present setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the $(\\alpha,\\beta)$-resolvent operator function that carries the semigroup part of the mild solution."},{"cited_title":"Existence of mild solutions to Hilfer fractional evolution equations in Banach space","cited_arxiv_id":"1812.02213","evidence_quote":"Supplies the mild-solution representation for Hilfer fractional evolution equations via the Mainardi-Wright function."},{"cited_title":"J., Existence of mild solution for evolution equatio n with Hilfer frac- tional derivative, Appl","cited_arxiv_id":null,"evidence_quote":"Supports the equivalence between the fractional abstract Cauchy problem and the integral equation used to define mild solutions."},{"cited_title":"Sousa, J., Capelas de Oliveira, E., Ulam–Hyers stab ility of a nonlinear fractional Volterra integro-differential equation, Appl","cited_arxiv_id":null,"evidence_quote":"Provides the weighted-space formulation and Ulam-Hyers stability framework for fractional Volterra integro-differential equations."},{"cited_title":"Sousa, J., Kucche, Kishor D., Capelas de Oliveira, E ., Stability of ψ-Hilfer impulsive fractional differential equations, Appl","cited_arxiv_id":null,"evidence_quote":"Supplies the weighted continuous function space $C_{1-\\gamma}(I,\\Omega)$ and its norm used throughout the contraction estimates."}],"review_version":1}