Linear integral equations in the space of regulated functions

Milan Tvrdý

Mathematica Bohemica (1998)

  • Volume: 123, Issue: 2, page 177-212
  • ISSN: 0862-7959

Abstract

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n this paper we investigate systems of linear integral equations in the space 𝔾 L n 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) - 01B(t,s)[d x(s)] = f(t), where f ∈ 𝔾 L n , the columns of the n × n -matrix valued function A belong to 𝔾 L n , the entries of B ( t , . ) have a bounded variation on [ 0 , 1 ] for any t ∈ [ 0 , 1 ] and the mapping t ∈ [ 0 , 1 ] → B ( t , . ) is regulated on [ 0 , 1 ] and left-continuous on ( 0 , 1 ) as the mapping with values in the space of n × 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. 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 𝔾 L n , and its representation by means of resolvent operators is given. The results presented cover e.g. the results known for systems of linear generalized differential equations x(t) - x(0) - 0t [d A(s)] x(s) = f(t) - f(0) as well as systems of Stieltjes integral equations x(t) - 01 [ds K(t,s)] x(s) = g(t)    or   x(t) - 0t [ds K(t,s)] x(s) = g(t).

How to cite

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Tvrdý, Milan. "Linear integral equations in the space of regulated functions." Mathematica Bohemica 123.2 (1998): 177-212. <http://eudml.org/doc/248299>.

@article{Tvrdý1998,
abstract = {n this paper we investigate systems of linear integral equations in the space $\{\{\mathbb \{G\}\}^n_L\}$ 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) - 01B(t,s)[d x(s)] = f(t), where $f\in \{\{\mathbb \{G\}\}^n_L\}$, the columns of the $n\times n$-matrix valued function $A$ belong to $\{\{\mathbb \{G\}\}^n_L\}$, the entries of $B(t,.)$ have a bounded variation on $\{[0,1]\}$ for any $t\in \{[0,1]\}$ and the mapping $t\in \{[0,1]\} \rightarrow B(t,.)$ 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. 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 $\{\{\mathbb \{G\}\}^n_L\}$, and its representation by means of resolvent operators is given. The results presented cover e.g. the results known for systems of linear generalized differential equations x(t) - x(0) - 0t [d A(s)] x(s) = f(t) - f(0) as well as systems of Stieltjes integral equations x(t) - 01 [ds K(t,s)] x(s) = g(t)    or   x(t) - 0t [ds K(t,s)] x(s) = g(t).},
author = {Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators; Kurzweil integral; regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators},
language = {eng},
number = {2},
pages = {177-212},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear integral equations in the space of regulated functions},
url = {http://eudml.org/doc/248299},
volume = {123},
year = {1998},
}

TY - JOUR
AU - Tvrdý, Milan
TI - Linear integral equations in the space of regulated functions
JO - Mathematica Bohemica
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 123
IS - 2
SP - 177
EP - 212
AB - n this paper we investigate systems of linear integral equations in the space ${{\mathbb {G}}^n_L}$ 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) - 01B(t,s)[d x(s)] = f(t), where $f\in {{\mathbb {G}}^n_L}$, the columns of the $n\times n$-matrix valued function $A$ belong to ${{\mathbb {G}}^n_L}$, the entries of $B(t,.)$ have a bounded variation on ${[0,1]}$ for any $t\in {[0,1]}$ and the mapping $t\in {[0,1]} \rightarrow B(t,.)$ 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. 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 ${{\mathbb {G}}^n_L}$, and its representation by means of resolvent operators is given. The results presented cover e.g. the results known for systems of linear generalized differential equations x(t) - x(0) - 0t [d A(s)] x(s) = f(t) - f(0) as well as systems of Stieltjes integral equations x(t) - 01 [ds K(t,s)] x(s) = g(t)    or   x(t) - 0t [ds K(t,s)] x(s) = g(t).
LA - eng
KW - regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators; Kurzweil integral; regulated function; Fredholm-Stieltjes integral equation; Volterra-Stieltjes integral equation; compact operator; Perron-Stieltjes integral; Kurzweil-Henstock integral; existence; uniqueness; resolvent operators
UR - http://eudml.org/doc/248299
ER -

References

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  1. Aumann G., Reelle Funktionen, Springer-Verlag, Berlin, 1909. (1909) MR0061652
  2. 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. (1982) MR0726569
  3. Fichmann L., Volterra-Stieltjes integral equations and equations of the neutral type, Thesis, University of Sao Paulo, 1984. (In Portuguese.) (1984) 
  4. Fraňková D., Regulated functions, Math. Bohem. 116 (1991), 20-59. (1991) MR1100424
  5. Hildebrandt T. H., Introduction to the Theory of Integration, Academic Press, New York, 1903. (1903) MR0154957
  6. Hönig Gh. S., Volterra-Stieltjes Integral Equations, Mathematics Studies 16, North Holland, Amsterdam, 1975. (1975) 
  7. 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. (1979) MR0585488
  8. Kurzweil J., Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (82) (1957), 418-449. (1957) Zbl0090.30002MR0111875
  9. Kurzweil J., Nichtabsolut konvergente Integrale, Teubner, Leipzig, 1980. (1980) Zbl0441.28001MR0597703
  10. Saks S, Theory of the Integral, Monografie Matematyczne. Warszawa, 1937. (1937) Zbl0017.30004
  11. Schechter M., Principles of Functional Analysis, Academic Press, New York, 1973. (1973) MR0467221
  12. Schwabik Š., Generalized Differential Equations (Fundamental Results), Rozpravy ČSAV, Řada MPV, 95 (6). Academia, Praha. 1985. (1985) Zbl0594.34002MR0823224
  13. Schwabik Š., On the relation between Young's and Kurzweil's concept of Stieltjes integral, Časopis Pěst. Mat. 98 (1973), 237-251. (1973) Zbl0266.26006MR0322113
  14. Schwabik Š., Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. (1992) Zbl0781.34003MR1200241
  15. Schwabik Š., Linear operators in the space of regulated functions, Math. Bohem. 117 (1992), 79-92. (1992) Zbl0790.47023MR1154057
  16. Schwabik Š., Tvrdý M., Vejvoda. O., Differential and Integral Equations: Boundary Value Problems and Adjoints, Academia and D. Reidel, Praha-Dordrecht, 1979. (1979) Zbl0417.45001MR0542283
  17. Tvrdý M., Boundary value problems for generalized linear integro-differential equations with left-continuous solutions, Časopis Pěst. Mat. 99 (1974), 147-157. (1974) MR0405041
  18. Tvrdý M., Regulated functions and the Perron-Stieltjes integral, Časopis Pӗst. Mat. 114 (1989), 187-209. (1989) MR1063765
  19. Ward A. J., 10.1007/BF01180442, Math. Z. 41 (1936), 578-604. (1936) Zbl0014.39702MR1545641DOI10.1007/BF01180442

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