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Title: Linear integral equations in the space of regulated functions (English)
Author: Tvrdý, Milan
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 123
Issue: 2
Year: 1998
Pages: 177-212
Summary lang: English
Category: math
Summary: n this paper we investigate systems of linear integral equations in the space ${{\Bbb G}^n_L}$ of $n$-vector valued functions which are regulated on the closed interval ${[0,1]}$ (i.e. such that can have only discontinuities of the first kind in ${[0,1]}$) and left-continuous in the corresponding open interval $(0,1).$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - \int_0^1B(t,s)[\text{d} x(s)] = f(t), where $f\in{{\Bbb G}^n_L}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${{\Bbb G}^n_L}$, the entries of $B(t,\ldotp)$ have a bounded variation on ${[0,1]}$ for any $t\in{[0,1]}$ and the mapping $t\in{[0,1]} \to B(t,\ldotp)$ is regulated on ${[0,1]}$ and left-continuous on $(0,1)$ as the mapping with values in the space of $n\times n$-matrix valued functions of bounded variation on ${[0,1]}.$ The integral stands for the Perron-Stieltjes one treated as the special case of the Kurzweil-Henstock integral. \endgraf In particular, we prove basic existence and uniqueness results for the given equation and obtain the explicit form of its adjoint equation. A special attention is paid to the Volterra (causal) type case. It is shown that in that case the given equation possesses a unique solution for any right-hand side from ${{\Bbb G}^n_L}$, and its representation by means of resolvent operators is given. \endgraf The results presented cover e.g. the results known for systems of linear generalized differential equations x(t) - x(0) - \int_0^t [\text{d} A(s)] x(s) = f(t) - f(0) as well as systems of Stieltjes integral equations x(t) - \int_0^1 [\text{d}_s K(t,s)] x(s) = g(t) \quad\text{or}\quad x(t) - \int_0^t [\text{d}_s K(t,s)] x(s) = g(t). (English)
Keyword: regulated function
Keyword: Fredholm-Stieltjes integral equation
Keyword: Volterra-Stieltjes integral equation
Keyword: compact operator
Keyword: Perron-Stieltjes integral
Keyword: Kurzweil-Henstock integral
Keyword: existence
Keyword: uniqueness
Keyword: resolvent operators
Keyword: Kurzweil integral
MSC: 26A39
MSC: 26A42
MSC: 45A05
MSC: 45B05
MSC: 45D05
MSC: 45F05
MSC: 47G10
idZBL: Zbl 0941.45001
idMR: MR1673977
DOI: 10.21136/MB.1998.126306
Date available: 2009-09-24T21:30:48Z
Last updated: 2020-07-29
Stable URL:
Reference: [Au] Aumann G.: Reelle Funktionen.Springer-Verlag, Berlin, 1909. MR 0061652
Reference: [Ba] Barbanti L.: Linear Volterra-Stieltjes integral equations and control.Equadiff 82. Proceedings, Würzburg 1982. Lecture Notes in Mathematics 1017, Springer-Verlag, Berlin, 1983, pp. 07-72. MR 0726569
Reference: [Fi] Fichmann L.: Volterra-Stieltjes integral equations and equations of the neutral type.Thesis, University of Sao Paulo, 1984. (In Portuguese.)
Reference: [Fra] Fraňková D.: Regulated functions.Math. Bohem. 116 (1991), 20-59. MR 1100424
Reference: [Hi] Hildebrandt T. H.: Introduction to the Theory of Integration.Academic Press, New York, 1903. MR 0154957
Reference: [Hö1] Hönig Gh. S.: Volterra-Stieltjes Integral Equations.Mathematics Studies 16, North Holland, Amsterdam, 1975.
Reference: [Hö2] Hönig, Ch. S.: Volterra-Stieltjes integral equations.Functional Differential Equations and Bifurcation, Proceedings of the Sao Carlos Conference 1979. Lecture Notes in Mathematics 799, Springer-Verlag, Berlin, 1980, pp. 173-216. MR 0585488
Reference: [Ku1] Kurzweil J.: Generalized ordinary differential equations and continuous dependence on a parameter.Czechoslovak Math. J. 7 (82) (1957), 418-449. Zbl 0090.30002, MR 0111875
Reference: [Ku2] Kurzweil J.: Nichtabsolut konvergente Integrale.Teubner, Leipzig, 1980. Zbl 0441.28001, MR 0597703
Reference: [Sa] Saks S: Theory of the Integral.Monografie Matematyczne. Warszawa, 1937. Zbl 0017.30004
Reference: [Sche] Schechter M.: Principles of Functional Analysis.Academic Press, New York, 1973. MR 0467221
Reference: [Sch1] Schwabik Š.: Generalized Differential Equations (Fundamental Results).Rozpravy ČSAV, Řada MPV, 95 (6). Academia, Praha. 1985. Zbl 0594.34002, MR 0823224
Reference: [Sch2] Schwabik Š.: On the relation between Young's and Kurzweil's concept of Stieltjes integral.Časopis Pěst. Mat. 98 (1973), 237-251. Zbl 0266.26006, MR 0322113
Reference: [SchЗ] Schwabik Š.: Generalized Ordinary Differential Equations.World Scientific, Singapore, 1992. Zbl 0781.34003, MR 1200241
Reference: [Sch4] Schwabik Š.: Linear operators in the space of regulated functions.Math. Bohem. 117 (1992), 79-92. Zbl 0790.47023, MR 1154057
Reference: [STV] Schwabik Š., Tvrdý M., Vejvoda. O.: Differential and Integral Equations: Boundary Value Problems and Adjoints.Academia and D. Reidel, Praha-Dordrecht, 1979. Zbl 0417.45001, MR 0542283
Reference: [Tv1] Tvrdý M.: Boundary value problems for generalized linear integro-differential equations with left-continuous solutions.Časopis Pěst. Mat. 99 (1974), 147-157. MR 0405041
Reference: [Tv2] Tvrdý M.: Regulated functions and the Perron-Stieltjes integral.Časopis Pӗst. Mat. 114 (1989), 187-209. MR 1063765
Reference: [Wa] Ward A. J.: The Perron-Stieltjes integral.Math. Z. 41 (1936), 578-604. Zbl 0014.39702, MR 1545641, 10.1007/BF01180442


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