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Papers by donatella bongiorno

$Research paper thumbnail of Exact solutions of some fractal differential equations$

Applied mathematics and computation, Jul 1, 2024

Journal of The Australian Mathematical Society, Oct 15, 2013

Fractal Analysis - Applications and Updates [Working Title]

The ordinary calculus is usually inapplicable to fractal sets, therefore we introduce the various... more The ordinary calculus is usually inapplicable to fractal sets, therefore we introduce the various approaches made so far to describe the theory of derivation and integration on a fractal set. In particular we study the Riemann type integrals (s-Riemann integral, s-HK integral, s-first return integral) defined on a closed fractal subset of the real line with finite positive s-dimensional Hausdorff measure (s-set) with particular attention to the Fundamental Theorem of Calculus. Moreover we pay attention to the relation between the s-HK integral, the s-first return integral and the Lebesgue integral respectively. Finally we give a descriptive characterization of the primitives of a s-HK integrable function.

$Research paper thumbnail of Uniform convergence for sequences of best L^{p} approximation$

arXiv (Cornell University), Nov 30, 2021

Dottorato di ricerca in matematica. 11. ciclo. Coordinatore Gaetana RestucciaConsiglio Nazionale ... more Dottorato di ricerca in matematica. 11. ciclo. Coordinatore Gaetana RestucciaConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 , Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal

$Research paper thumbnail of The Henstock-Kurzweil-Stieltjes type integral for real functions on a fractal subset of the real line$

The aim of this paper is to introduce an Henstock-Kurzweil type integration process for real func... more The aim of this paper is to introduce an Henstock-Kurzweil type integration process for real functions on a fractal subset E of the real line

We extend the first-return integration process, introduced in [5] by U.B. Darji and M.J. Evans, a... more We extend the first-return integration process, introduced in [5] by U.B. Darji and M.J. Evans, and prove that each Lebesgue-improper integrable function f : [a, b] --> R is first-return integrable in this generalized sense to (Li)int_a^b f(t) dt

$Research paper thumbnail of Derivatives not first return integrable on a fractal set$

Ricerche di Matematica, 2018

Mathematica Slovaca, 2017

$Research paper thumbnail of A regularity condition in Sobolev spaces <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>W</mi><mrow><mi mathvariant="normal">l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mn>1</mn><mo separator="true">,</mo><mi>p</mi></mrow></msubsup><mo stretchy="false">(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W^{1,p}_{\mathrm loc}({\mathbb R}^n)</annotation></semantics></math>Wloc1,p(Rn) with <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 ≤ p \lt n</annotation></semantics></math>1≤p<n$

Illinois Journal of Mathematics, 2002

Extending Malý's geometric definition of absolutely continuous functions of n variables (in a sen... more Extending Malý's geometric definition of absolutely continuous functions of n variables (in a sense equivalent to that of Rado-Reichelderfer), we define classes of p-absolutely continuous functions (1 ≤ p < n) and show that this weaker notion of absolute continuity still implies differentiability almost everywhere, although it does not imply continuity or Lusin's condition (N).

Topology and its Applications, 2009

Journal of Mathematical Analysis and Applications, 2017

Real Analysis Exchange, 2015

Commentationes Mathematicae Universitatis Carolinae, 1998

Fractals, 2015

The aim of this paper is to formulate the best version of the Fundamental theorem of Calculus for... more The aim of this paper is to formulate the best version of the Fundamental theorem of Calculus for real functions on a fractal subset of the real line. In order to do that an integral of Henstock–Kurzweil type is introduced.

Vector Measures, Integration and Related Topics, 2009

We prove that for each Bochner integrable function f there exists a trajectory yielding the Bochn... more We prove that for each Bochner integrable function f there exists a trajectory yielding the Bochner integral of f , and that on infinite-dimensional Banach spaces there exist Pettis integrable functions f such that no trajectory yields the Pettis integral of f .

Nonlinear Analysis: Theory, Methods & Applications, 2009

We study some slight modifications of the class α-ACn(Ω,Rm) introduced in [D. Bongiorno, Absolute... more We study some slight modifications of the class α-ACn(Ω,Rm) introduced in [D. Bongiorno, Absolutely continuous functions in Rn, J. Math. Anal. and Appl. 303 (2005) 119–134]. In particular we prove that the classes α-ACλn(Ω,Rm), 0λ1, introduced in [C. Di Bari, C. Vetro, A remark on absolutely continuous functions in Rn, Rend. Circ. Matem. Palermo 55 (2006) 296–304] are independent by

Journal of the Australian Mathematical Society, 2013

An extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces witho... more An extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces without the Radon–Nikodým property.