| 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:
|
http://hdl.handle.net/10338.dmlcz/126306 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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[Fra] Fraňková D.: Regulated functions.Math. Bohem. 116 (1991), 20-59. MR 1100424 |
| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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[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:
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| Reference:
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| Reference:
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| . |
