# Linear integral equations in the space of regulated functions

Mathematica Bohemica (1998)

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

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topTvrdý, 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 -

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