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Generalized solutions of boundary value problems for ordinary linear differential equations of second order in the Colombeau algebra

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 Abstract: It is shown that from the fact that a homogeneous problem has a unique trivial solution it follows that a non-homogeneous problem has a solution in the Colombeau algebra.
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  • Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland
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Bibliografia
[1] P. Antosik, J. Mikusiński and R. Sikorski, Theory of Distributions, The Sequential Approach, Elsevier-PWN, Amsterdam-Warszawa, 1973.
[2] J. F. Colombeau, New Generelized Functions and Multiplication of Distributions, North-Holland, Amsterdam, 1984.
[3] J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland, Amsterdam, 1985.
[4] J. F. Colombeau, Multiplication of distributions, Bull. Amer. Math. Soc. 23 (1990), 251-268.
[5] S. G. Deo and S. G. Pandit, Differential Systems Involving Impulses, Lecture Notes in Math. 954, Springer, 1982.
[6] T. Dłotko, Application of the notion of rotation of a vector field in the theory of differential equations and their generalizations, Prace Naukowe Uniw. Śląsk. w Katowicach 32 (1971) (in Polish).
[7] T. H. Hildebrandt, On systems of linear differential Stieltjes integral equations, Illinois J. Math. 3 (1959), 352-373.
[8] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 17 (1957), 418-449.
[9] A. Lasota and F. H. Szafraniec, Applications of differential equations with distributional coefficients to optimal control theory, Prace Mat. 12, Kraków UJ, 31-37.
[10] J. Ligęza, Generalized solutions of ordinary linear differential equations in the Colombeau algebra, Math. Bohemica 2 (118) (1993), 123-146.
[11] J. Ligęza, Weak solutions of ordinary differential equations, Prace Nauk. Uniw. Śląsk. w Katowicach 842 (1986).
[12] J. Ligęza, On generalized solutions of boundary value problems for non-linear differential equations of second order, Prace Mat. Uniw. Śląsk. w Katowicach 9 (1977), 20-29.
[13] J. Persson, The Cauchy system for linear distribution differential equations, Funcial. Ekvac. 30 (1987), 162-168.
[14] Š. Schwabik, M. Tvrdý and O. Vejvoda, Differential and integral equations, Boundary Value Problems and Adjoints, Academia, Praha 1979.
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