| Title:
|
Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral (English) |
| Author:
|
Federson, M. |
| Author:
|
Bianconi, R. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
37 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
307-328 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
linear integral equations |
| Keyword:
|
Kurzweil-Henstock integrals |
| MSC:
|
26A39 |
| MSC:
|
45A05 |
| idZBL:
|
Zbl 1090.45001 |
| idMR:
|
MR1879454 |
| . |
| Date available:
|
2008-06-06T22:29:26Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107809 |
| . |
| Reference:
|
[1] 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. |
| Reference:
|
[2] Federson M.: The Fundamental Theorem of Calculus for multidimensional Banach space-valued Henstock vector integrals.Real Anal. Exchange 25 (1), (1999-2000), 469–480. MR 1758903 |
| Reference:
|
[3] Federson M.: Substitution formulas for the vector integrals of Kurzweil and of Henstock.Math. Bohem., (2001), in press. |
| Reference:
|
[4] Federson M., Bianconi R.: Linear integral equations of Volterra concerning the integral of Henstock.Real Anal. Exchange 25 (1), (1999-2000), 389–417. MR 1758896 |
| Reference:
|
[5] Gilioli A.: Natural ultrabornological non-complete, normed function spaces.Arch. Math. 61 (1993), 465–477. Zbl 0783.46003, MR 1241052 |
| Reference:
|
[6] Henstock R.: A Riemann-type integral of Lebesgue power.Canad. J. Math. 20 (1968), 79–87. Zbl 0171.01804, MR 0219675 |
| Reference:
|
[7] Henstock R.: The General Theory of Integration.Oxford Math. Monogr., Clarendon Press, Oxford, 1991. Zbl 0745.26006, MR 1134656 |
| Reference:
|
[8] 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. Zbl 0372.34038, MR 0460561 |
| Reference:
|
[9] Hönig C. S.: Volterra-Stieltjes integral equations.Math. Studies 16, North-Holland Publ. Comp., Amsterdam, 1975. MR 0499969 |
| Reference:
|
[10] Hönig C. S.: Equations intégrales généralisées et applications.Publ. Math. Orsay, 83-01, exposé 5, 1–50. Zbl 0507.45017 |
| Reference:
|
[11] Hönig C. S.: On linear Kurzweil-Henstock integral equations.Seminário Brasileiro de Análise 32 (1990), 283–298. |
| Reference:
|
[12] Hönig C. S.: On a remarkable differential characterization of the functions that are Kurzweil-Henstock integrals.Seminário Brasileiro de Análise 33 (1991), 331–341. |
| Reference:
|
[13] Hönig C. S.: A Riemannian characterization of the Bochner-Lebesgue integral.Seminário Brasileiro de Análise 35 (1992), 351–358. |
| Reference:
|
[14] Kurzweil J.: Nichtabsolut Konvergente Integrale.Leipzig, 1980. Zbl 0441.28001, MR 0597703 |
| Reference:
|
[15] Lee P. Y.: Lanzhou Lectures on Henstock Integration.World Sci. Singapore, 1989. Zbl 0699.26004, MR 1050957 |
| Reference:
|
[16] Lin, Ying-Jian: On the equivalence of the McShane and Lebesgue integrals.Real Anal. Exchange 21 (2), (1995-96), 767–770. MR 1407292 |
| Reference:
|
[17] MacLane S., Birkhoff G.: Algebra.The MacMillan Co., London, 1970. MR 0214415 |
| Reference:
|
[18] Mawhin J.: Introduction a l’Analyse.Cabay, Louvain-la-Neuve, 1983. |
| Reference:
|
[19] McLeod R. M.: The Generalized Riemann Integral.Carus Math. Monogr. 20, The Math. Ass. of America, 1980. Zbl 0486.26005, MR 0588510 |
| Reference:
|
[20] McShane E. J.: A unified theory of integration.Amer. Math. Monthly 80 (1973), 349–359. Zbl 0266.26008, MR 0318434 |
| Reference:
|
[21] Pfeffer W. F.: The Riemann Approach to Integration.Cambridge, 1993. Zbl 0804.26005, MR 1268404 |
| Reference:
|
[22] Schwabik S.: The Perron integral in ordinary differential equations.Differential Integral Equations 6 (4) (1993), 863–882. Zbl 0784.34006, MR 1222306 |
| Reference:
|
[23] Schwabik S.: Abstract Perron-Stieltjes integral.Math. Bohem. 121 (4), (1996), 425–447. Zbl 0879.28021, MR 1428144 |
| Reference:
|
[24] Tvrdý M.: Linear integral equations in the space of regulated functions.Math. Bohem. 123 (2), (1998), 177–212. Zbl 0941.45001, MR 1673977 |
| . |
