Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral
Federson, Márcia ; Bianconi, Ricardo
Archivum Mathematicum, Tome 037 (2001), p. 307-328 / Harvested from Czech Digital Mathematics Library
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In 1990, Hönig proved that the linear Volterra integral equation \[ x\left( t\right) -\,(K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right)\,ds=f\left( t\right)\,,\qquad t\in \left[ a,b\right]\,, \] where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[ a,b\right] $ of the real line $\mathbb R$, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation \[ x\left( t\right) - (K)\int \nolimits _{\left[ a,t\right] }\alpha \left( t,s\right) x\left( s\right) dg\left( s\right) =f\left( t\right) ,\qquad t\in \left[ a,b\right]\,, \] in a real-valued context.

Publié le : 2001-01-01
Classification:  26A39,  45A05 
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     author = {M\'arcia Federson and Ricardo Bianconi},
     title = {Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral},
     journal = {Archivum Mathematicum},
     volume = {037},
     year = {2001},
     pages = {307-328},
     zbl = {1090.45001},
     mrnumber = {1879454},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107809}
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Federson, Márcia; Bianconi, Ricardo. Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral. Archivum Mathematicum, Tome 037 (2001) pp. 307-328. http://gdmltest.u-ga.fr/item/107809/

Federson M. Sobre a existência de soluções para Equações Integrais Lineares com respeito a Integrais de Gauge, Doctor Thesis, Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil, 1998, in portuguese. (1998)

Federson M. The Fundamental Theorem of Calculus for multidimensional Banach space-valued Henstock vector integrals, Real Anal. Exchange 25 (1), (1999-2000), 469–480. (1999) | MR 1758903

Federson M. Substitution formulas for the vector integrals of Kurzweil and of Henstock, Math. Bohem., (2001), in press.

Federson M.; Bianconi R. Linear integral equations of Volterra concerning the integral of Henstock, Real Anal. Exchange 25 (1), (1999-2000), 389–417. (1999) | MR 1758896

Gilioli A. Natural ultrabornological non-complete, normed function spaces, Arch. Math. 61 (1993), 465–477. (1993) | MR 1241052 | Zbl 0783.46003

Henstock R. A Riemann-type integral of Lebesgue power, Canad. J. Math. 20 (1968), 79–87. (1968) | MR 0219675 | Zbl 0171.01804

Henstock R. The General Theory of Integration, Oxford Math. Monogr., Clarendon Press, Oxford, 1991. (1991) | MR 1134656 | Zbl 0745.26006

Hönig C. S. The Abstract Riemann-Stieltjes Integral and its Applications to Linear Differential Equations with Generalized Boundary Conditions, Notas do Instituto de Matemática e Estatística da Universidade de São Paulo, Série Matemática, 1, 1973. (1973) | MR 0460561 | Zbl 0372.34038

Hönig C. S. Volterra-Stieltjes integral equations, Math. Studies 16, North-Holland Publ. Comp., Amsterdam, 1975. (1975) | MR 0499969

Hönig C. S. Equations intégrales généralisées et applications, Publ. Math. Orsay, 83-01, exposé 5, 1–50. | Zbl 0507.45017

Hönig C. S. On linear Kurzweil-Henstock integral equations, Seminário Brasileiro de Análise 32 (1990), 283–298. (1990)

Hönig C. S. On a remarkable differential characterization of the functions that are Kurzweil-Henstock integrals, Seminário Brasileiro de Análise 33 (1991), 331–341. (1991)

Hönig C. S. A Riemannian characterization of the Bochner-Lebesgue integral, Seminário Brasileiro de Análise 35 (1992), 351–358. (1992)

Kurzweil J. Nichtabsolut Konvergente Integrale, Leipzig, 1980. (1980) | MR 0597703 | Zbl 0441.28001

Lee P. Y. Lanzhou Lectures on Henstock Integration, World Sci. Singapore, 1989. (1989) | MR 1050957 | Zbl 0699.26004

Lin; Ying-Jian On the equivalence of the McShane and Lebesgue integrals, Real Anal. Exchange 21 (2), (1995-96), 767–770. (1995) | MR 1407292

Maclane S.; Birkhoff G. Algebra, The MacMillan Co., London, 1970. (1970) | MR 0214415

Mawhin J. Introduction a l’Analyse, Cabay, Louvain-la-Neuve, 1983. (1983)

Mcleod R. M. The Generalized Riemann Integral, Carus Math. Monogr. 20, The Math. Ass. of America, 1980. (1980) | MR 0588510 | Zbl 0486.26005

Mcshane E. J. A unified theory of integration, Amer. Math. Monthly 80 (1973), 349–359. (1973) | MR 0318434 | Zbl 0266.26008

Pfeffer W. F. The Riemann Approach to Integration, Cambridge, 1993. (1993) | MR 1268404 | Zbl 0804.26005

Schwabik S. The Perron integral in ordinary differential equations, Differential Integral Equations 6 (4) (1993), 863–882. (1993) | MR 1222306 | Zbl 0784.34006

Schwabik S. Abstract Perron-Stieltjes integral, Math. Bohem. 121 (4), (1996), 425–447. (1996) | MR 1428144 | Zbl 0879.28021

Tvrdý M. Linear integral equations in the space of regulated functions, Math. Bohem. 123 (2), (1998), 177–212. (1998) | MR 1673977 | Zbl 0941.45001