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Perron-Stieltjes integral; Kurzweil integral; distributional coefficients; regulated functions; generalized linear differential equation; existence; uniqueness; variation-of-constants formula; distribution
In the paper existence and uniqueness results for the linear differential system on the interval [0,1] $A_1(A_0x)'-A'_2x=f'$ with distributional coefficients and solutions from the space of regulated functions are obtained.
[An-Li] Antosik P., Ligęza J.: Products of measures and functions of bounded variation. Proceedings of the conf. on generalized functions and operational calculus, Varna 1975 (1979), 20-26. MR 0547326
[At] Atkinson F. V.: Discrete and Continuous Boundary Value Problems. Academic Press, New York-London, 1964. MR 0176141
[Au] Aumann G.: Reelle Funktionen. Springer-Verlag, Berlin-Heidelberg-New York, 1969. MR 0061652 | Zbl 0181.05801
[Fra] Fraňková D.: Regulated functions. Mathematica Bohemica 116(1991), 20-59. MR 1100424
[Ha] Halperin I.: Introduction to the Theory of Distributions. University of Toronto Press, Toronto, 1952. MR 0045933 | Zbl 0046.12603
[Hi] Hildebrandt T.H.: Introduction to the Theory of Integration. Academic Press, New York-London, 1963. MR 0154957 | Zbl 0112.28302
[Hö] Hönig Ch.S.: Volterra-Stieltjes Integral Equations. Mathematics Studies 16, North-Holland, Amsterdam, 1975. MR 0499969
[Ku1] Kurzweil J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (82) (1957), 418-449. MR 0111875 | Zbl 0090.30002
[Ku2] Kurzweil J.: Linear Differential Equations with Distributions as Coefficients. Bull. Acad. Polon. Sci. Ser. math. astr. et phys. 7 (1959), 557-560. MR 0111887 | Zbl 0117.34401
[Ku3] Kurzweil J.: Nichtabsolut konvergente Integrale. BSB B.G.Teubner Verlagsgesselschaft, Leipzig, 1980. MR 0597703 | Zbl 0441.28001
[Li1] Ligęza J.: On generalized solutions of linear differential equations of order n. Prace Mat., Uniw. Sląski v Katowicach 3 (1973), 101-108. MR 0326038
[Li2] Ligęza J.: On distributional solutions of some systems of linear differential equations. Časopis pěst. mat. 102 (1977), 37-41. MR 0460757
[Li3] Ligęza J.: Weak Solutions of Ordinary Differential Equations. Uniwersytet Sląski, Katowice, 1986. MR 0868863
[Li4] Ligęza J.: Existence and uniqueness of distributional solutions of ordinary differential equations in the class of regulated functions. EQUADIFF 7 - Enlarged Abstracts (7th Czechoslovak Conference on Differential Equations and Their Applications, Praha) (1989), 48-50.
[Pa-De] Pandit S.G., Deo S.G.: Differential Equations Involving Impulses. Lecture Notes in Mathematics, 954, Springer-Verlag, Berlin-Heidelberg-New York, 1982.
[Pe1] Persson J.: Linear distribution differential equations. Comment.Math. Univ.St.Pauli 33 (1984), 119-126. MR 0741385 | Zbl 0552.34004
[Pe2] Persson J.: Fundamental theorems for linear measure differential equations. Mathematica Scandinavica 62 (1988), 19-43. MR 0961581 | Zbl 0642.34004
[Pf1] Pfaff R.: Gewöhnliche lineare Differentialgleichungen n-ter Ordnung mit Distributionskoeffizienten. Proc. Roy. Soc. Edinburgh 85A (1980), 291-298. MR 0574022 | Zbl 0424.34011
[Pf2] Pfaff R.: Generalized systems of linear differential equations. Proc. Roy. Soc. Edinburgh 89A (1981), 1-14. MR 0628123 | Zbl 0475.34005
[Ru] Rudin W.: Functional Analysis. McGraw-Hill, New York, 1973. MR 0365062 | Zbl 0253.46001
[Sa] Saks S.: Theory of the Integral. Monografie Matematyczne, Warszawa-Lwów, 1937. Zbl 0017.30004
[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.: Generalized Differential Equations (Special Results). Rozpravy ČSAV, řada MPV, 99 (3), Academia, Praha, 1989. MR 1002180 | Zbl 0821.34002
[Sch3] Schwabik Š.: Linear operators in the space of regulated functions. Mathematica Bohemica 117 (1992), 79-92. MR 1154057 | Zbl 0790.47023
[S-T-V] Schwabik Š., Tvrdý M., Vejvoda O.: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia and D. Reidel, Praha and Dordrecht, 1979. MR 0542283
[Tv1] Tvrdý M.: Regulated functions and the Perron-Stieltjes integral. Časopis pěst. mat. 114 (1989), 187-209. MR 1063765
[Tv2] Tvrdý M.: Generalized differential equations in the space of regulated functions (Boundary value problems and controllability). Mathematica Bohemica 116(1991), 225-244. MR 1126445 | Zbl 0744.34021
[Wa] Ward A J.: The Perron-Stieltjes integral. Math.Zeitschr. 41 (1936), 578-604. DOI 10.1007/BF01180442 | MR 1545641 | Zbl 0014.39702
[Za-Se] Zavalishchin S.G., Sesekin A.N.: Impulse Processes, Models and Applications. (in Russian), Nauka, Moscow, 1991. MR 1126685
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