# Article

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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
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).
References:
[Au] Aumann G.: Reelle Funktionen. Springer-Verlag, Berlin, 1909. MR 0061652
[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
[Fi] Fichmann L.: Volterra-Stieltjes integral equations and equations of the neutral type. Thesis, University of Sao Paulo, 1984. (In Portuguese.)
[Fra] Fraňková D.: Regulated functions. Math. Bohem. 116 (1991), 20-59. MR 1100424
[Hi] Hildebrandt T. H.: Introduction to the Theory of Integration. Academic Press, New York, 1903. MR 0154957
[Hö1] Hönig Gh. S.: Volterra-Stieltjes Integral Equations. Mathematics Studies 16, North Holland, Amsterdam, 1975.
[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
[Ku1] Kurzweil J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (82) (1957), 418-449. MR 0111875 | Zbl 0090.30002
[Ku2] Kurzweil J.: Nichtabsolut konvergente Integrale. Teubner, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[Sa] Saks S: Theory of the Integral. Monografie Matematyczne. Warszawa, 1937. Zbl 0017.30004
[Sche] Schechter M.: Principles of Functional Analysis. Academic Press, New York, 1973. MR 0467221
[Sch1] Schwabik Š.: Generalized Differential Equations (Fundamental Results). Rozpravy ČSAV, Řada MPV, 95 (6). Academia, Praha. 1985. MR 0823224 | Zbl 0594.34002
[Sch2] Schwabik Š.: On the relation between Young's and Kurzweil's concept of Stieltjes integral. Časopis Pěst. Mat. 98 (1973), 237-251. MR 0322113 | Zbl 0266.26006
[SchЗ] Schwabik Š.: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[Sch4] Schwabik Š.: Linear operators in the space of regulated functions. Math. Bohem. 117 (1992), 79-92. MR 1154057 | Zbl 0790.47023
[STV] Schwabik Š., Tvrdý M., Vejvoda. O.: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia and D. Reidel, Praha-Dordrecht, 1979. MR 0542283 | Zbl 0417.45001
[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
[Tv2] Tvrdý M.: Regulated functions and the Perron-Stieltjes integral. Časopis Pӗst. Mat. 114 (1989), 187-209. MR 1063765
[Wa] Ward A. J.: The Perron-Stieltjes integral. Math. Z. 41 (1936), 578-604. DOI 10.1007/BF01180442 | MR 1545641 | Zbl 0014.39702

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