| Title:
|
Linear distributional differential equations of the second order (English) |
| Author:
|
Tvrdý, Milan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
119 |
| Issue:
|
4 |
| Year:
|
1994 |
| Pages:
|
415-436 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with the linear differential equation (0.1) $(pu')'+q'u=f''$ with distributional coefficients and solutions from the space of regulated functions. Our aim is to get the basic existence and uniqueness results for the equation (0.1) and to generalize the known results due to F. V. Atkinson [At], J. Ligeza [Li1]-[Li3], R. Pfaff ([Pf1], [Pf2]), A. B. Mingarelli [Mi] as well as the results from the paper [Pe-Tv] concerning the equation (0.1). (English) |
| Keyword:
|
regulated functions |
| Keyword:
|
Perron-Stieltjes integral |
| Keyword:
|
Kurzweil integral |
| Keyword:
|
generalized differential equation |
| Keyword:
|
linear second order equation |
| Keyword:
|
distributional coefficients |
| Keyword:
|
existence |
| Keyword:
|
boundary value problem |
| Keyword:
|
numerical approximation |
| Keyword:
|
uniqueness |
| Keyword:
|
distribution |
| MSC:
|
34A12 |
| MSC:
|
34A37 |
| MSC:
|
34B05 |
| MSC:
|
46F10 |
| MSC:
|
46F99 |
| MSC:
|
65J99 |
| idZBL:
|
Zbl 0819.34007 |
| idMR:
|
MR1316594 |
| DOI:
|
10.21136/MB.1994.126120 |
| . |
| Date available:
|
2009-09-24T21:07:50Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126120 |
| . |
| Reference:
|
[At] Atkinson F. V.: Discrete and Continuous Boundary Value Problems.Academic Press, New York-London, 1964. MR 0176141 |
| Reference:
|
[Ha] Halperin I.: Introduction to the Theory of Distributions.University of Toronto Press, Toronto, 1952. Zbl 0046.12603, MR 0045933 |
| Reference:
|
[Ho] Hollander K.: Die numerische Behandlung eines singulären Randwertproblems nach der Differenzenmethode.ZAMM Z. Angew. Math. u. Mech. 65 (1985), T269-T271. MR 0798964 |
| Reference:
|
[Ho] Hönig Gh.S.: Volterra-Stieltjes Integral Equations.Mathematics Studies 16, North-Holland, Amsterdam, 1975. |
| Reference:
|
[Ku] Kurzweil J.: Generalized ordinary differential equations and continuous dependence on a parameter.Czech. Math. J. 7 (82) (1957), 418-449. Zbl 0090.30002, MR 0111875 |
| Reference:
|
[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 |
| Reference:
|
[Li2] Ligęza J.: On distributional solutions of some systems of linear differential equations.Časopis pěst. mat. 102 (1977), 37-41. MR 0460757 |
| Reference:
|
[Li3] Ligęza J.: Weak Solutions of Ordinary Differential Equations.Uniwersytet Sląski, Katowice, 1986. MR 0868863 |
| Reference:
|
[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, pp. 48-50. |
| Reference:
|
[Li5] Ligęza J.: Product of measures and regulated functions.Generalized Functions and Convergence (Memorial Volume for Professor Jan Mikusiński, 13-18 June 1988) (Piotr Antosik and Andrzej Kamiński Katowice, eds.). World Scientific, Praha, Singapore-New Jersey-London-Hong Kong, 1988, pp. 175-179. MR 1085505 |
| Reference:
|
[Mi] Mingarelli A.B.: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions.Lecture Notes in Mathematics, 989, Springer-Verlag, Berlin-Heidelberg-New York, 1983. Zbl 0516.45012, MR 0706255 |
| Reference:
|
[Pa-De] Pandit S.G., Deo S.G.: Differential Equations Involving Impulses.Lecture Notes in Mathematics, 954, Springer-Verlag, Berlin-Heidelberg-New York, 1982. |
| Reference:
|
[Pe1] Persson J.: Linear distribution differential equations.Comment. Math. Univ. St. Pauli 33 (1984), 119-126. Zbl 0552.34004, MR 0741385 |
| Reference:
|
[Pe2] Persson J.: Fundamental theorems for linear measure differential equations.Mathematica Scandinavica 62 (1988), 19-43. Zbl 0642.34004, MR 0961581, 10.7146/math.scand.a-12208 |
| Reference:
|
[Pe-Tv] Pelant M., Tvrdý M.: Linear distributional differential equations in the space of regulated functions.Mathematica Bohemica 118 (1993), to appear. MR 1251883 |
| Reference:
|
[Pf1] Pfaff R.: Gewöhnliche lineare Differentialgleichungen n-ter Ordnung mit Distributionskoeffizienten.Proc. Roy. Soc. Edinburgh 85A (1980), 291-298. Zbl 0424.34011, MR 0574022 |
| Reference:
|
[Pf2] Pfaff R.: Generalized systems of linear differential equations.Proc. Roy. Soc. Edinburgh 89A (1981), 1-14. Zbl 0475.34005, MR 0628123 |
| Reference:
|
[Sch1] Schwabik Š.: Generalized Differential Equations (Fundamental Results).Rozpravy ČSAV, řada MPV, 95 (6), Academia, Praha, 1985. Zbl 0594.34002, MR 0823224 |
| Reference:
|
[Sch2] Schwabik Š.: Generalized Differential Equations (Special Results).Rozpravy ČSAV, řada MPV, 99 (3), Academia, Praha, 1989. Zbl 0821.34002, MR 1002180 |
| Reference:
|
[Sch3] Schwabik Š.: Generalized Ordinary Differential Equations.World Scientific, Singapore, 1992. Zbl 0781.34003, MR 1200241 |
| Reference:
|
[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 |
| Reference:
|
[Tv1] Tvrdý M.: Regulated functions and the Perron-Stieltjes integral.Časopis pěst. mat. 114 (1989), 187-209. MR 1063765 |
| Reference:
|
[Tv2] Tvrdý M.: Generalized differential equations in the space of regulated functions (Boundary value problems and controllability).Mathematica Bohemica 116 (1991), 225-244. Zbl 0744.34021, MR 1126445 |
| Reference:
|
[Za-Se] Zavalishchin S.G., Sesekin A.N.: Impulse Processes, Models and Applications.Nauka, Moscow, 1991. (In Russian.) MR 1126685 |
| . |
