Solvability of discrete approximations for linear and nonlinear differential-operator equations in Banach spaces within a projection-algebraic approach

M.Lustyk; M.I.Vovk; A.K.Prykarpatsky

The projection-algebraic approach to discrete approximations, proposed before in work \cite{MPS}, for linear and nonlinear differential operator equations in Banach spaces is analyzed. The convergence analysis of the corresponding finite-dimensional expressions, based on the functional-analytic properties of discrete approximations and methods of operator theory in Banach spaces, is studied. Based on a generalized Leray-Schauder type fixed-point theorem the projection-algebraic scheme of discrete approximations is investigated, its solvability and convergence for a special class of nonlinear operator equations are analyzed. Application of the results to the Lagrangian functional interpolation scheme of the projection-algebraic method of discrete approximations in case of linear differential operator equations is presented.

1. Luśtyk M., Bihun O. Approximation properties of the Lie-algebraic scheme, Mat. Stud., 20 (2003), №1, 85-91.

2. Luśtyk M. The Lie-Algebraic Discrete Approximation Scheme for Evolution Equations with Dirichlet/Neumann Data, Universitatis Iagellonicae Acta Mathematica, 40 (2002), 117-124.

3. Mitropolski Yu.A., Prykarpatsky A.K., Samoilenko V.H. A Lie-algebraic scheme of discrete approximations of dynamical systems of mathematical physics, Ukrainian Math. Journal, 40 (1988), 453-458.

4. Bihun O., Lustyk M. Numerical tests and theoretical estimations for a Lie-algebraic scheme of discrete approximations, Visnyk of the Lviv National University, Applied Mathematics and Computer Science Series, 6 (2003), 23-29.

5. Calogero F. Interpolation, differentiation and solution of eigenvalue problems in more than one dimension, Lett. Nuovo Cimento, 38 (1983), №13, 453-459.

6. Casas F. Solution of linear partial differential equations by Lie algebraic methods, J. of Pure and Appl. Math. 76 (1996), 159-170.

7. Luśtyk M. Lie-algebraic discsrete approximation for nonlinear evolution equations, Journal of Mathematical Sciences, 109 (2002), №1, 1169-1172.

8. Samoylenko V.Hr. Algebraic scheme of discrete approximations for dynamical systems of mathematical physics and the accuracy estimation, Asymptotic methods in mathematical physics problems, Kiev, Institute of Mathematics of NAS, (1988), 144-151 (in Russian).

9. Trenogin V.A. Functional analysis, Moscow, Nauka Publ., 1980 (in Russian)

10. Michael E. Continuous selections. Parts 1-3. The Annals of Mathematics, 2nd Ser., 63 (1956), №2, 361-382; 2nd Ser., 64 (1956), №3, 562-580; 2nd Ser., 65 (1957), №2, 375-390.

11. Wei J., Norman E. On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327-334.

12. Kato T. The theory of linear operators, NY, Springer, 1962.

13. Babenko K. Numerical Analysis, Moscow, Nauka, 1984 (in Russian).

14. Krasnoselskiy M.A., Vainikko G.M., Zabreiko P.P. and others. Approximate solution of operator equations, Moscow., Nauka Publ., 1969 (in Russian).

15. Gaevsky~H., Greger K., Zakharias K. Nonlinear operator equations and operator differential equations. Moscow, Mir, 1978, 336p. (in Russian).

16. Prykarpatsky A.K. A Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem. Preprint ICTP, IC/2007/028, Trieste, Italy, 2007.

17. Prykarpatsky A.K. An infinite-dimernsional Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem and some applications, Ukrainian Mathem. Journal, 60 (2008), №1, 114-120.

18. Serre J.-P. Lie algebras and Lie groups. Benjamin, New York, 1966.

	Solvability of discrete approximations for linear and nonlinear differential-operator equations in Banach spaces within a projection-algebraic approach
Author	M.Lustyk, M.I.Vovk, A.K.Prykarpatsky lustyk@wms.mat.agh.edu.pl, pryk.anat@ua.fm, prykanat@cybergal.com The AGH University of Science and Technology, Krakow, 30059 Poland, Institute of Fundamental Research at the "Lviv Polytechnika", National University, 79013, Lviv Ukraine, Institute of Mathematics of NAS, Kyiv, 00104, and Ivan Franko State Pedagogical University, Lviv region, Drohobych, 82100 Ukraine,
Abstract	The projection-algebraic approach to discrete approximations, proposed before in work \cite{MPS}, for linear and nonlinear differential operator equations in Banach spaces is analyzed. The convergence analysis of the corresponding finite-dimensional expressions, based on the functional-analytic properties of discrete approximations and methods of operator theory in Banach spaces, is studied. Based on a generalized Leray-Schauder type fixed-point theorem the projection-algebraic scheme of discrete approximations is investigated, its solvability and convergence for a special class of nonlinear operator equations are analyzed. Application of the results to the Lagrangian functional interpolation scheme of the projection-algebraic method of discrete approximations in case of linear differential operator equations is presented.
Keywords	discrete approximation, nonlinear differential-operator equation, Banach space
DOI	doi:10.30970/ms.32.1.21-32
Reference	1. Luśtyk M., Bihun O. Approximation properties of the Lie-algebraic scheme, Mat. Stud., 20 (2003), №1, 85-91. 2. Luśtyk M. The Lie-Algebraic Discrete Approximation Scheme for Evolution Equations with Dirichlet/Neumann Data, Universitatis Iagellonicae Acta Mathematica, 40 (2002), 117-124. 3. Mitropolski Yu.A., Prykarpatsky A.K., Samoilenko V.H. A Lie-algebraic scheme of discrete approximations of dynamical systems of mathematical physics, Ukrainian Math. Journal, 40 (1988), 453-458. 4. Bihun O., Lustyk M. Numerical tests and theoretical estimations for a Lie-algebraic scheme of discrete approximations, Visnyk of the Lviv National University, Applied Mathematics and Computer Science Series, 6 (2003), 23-29. 5. Calogero F. Interpolation, differentiation and solution of eigenvalue problems in more than one dimension, Lett. Nuovo Cimento, 38 (1983), №13, 453-459. 6. Casas F. Solution of linear partial differential equations by Lie algebraic methods, J. of Pure and Appl. Math. 76 (1996), 159-170. 7. Luśtyk M. Lie-algebraic discsrete approximation for nonlinear evolution equations, Journal of Mathematical Sciences, 109 (2002), №1, 1169-1172. 8. Samoylenko V.Hr. Algebraic scheme of discrete approximations for dynamical systems of mathematical physics and the accuracy estimation, Asymptotic methods in mathematical physics problems, Kiev, Institute of Mathematics of NAS, (1988), 144-151 (in Russian). 9. Trenogin V.A. Functional analysis, Moscow, Nauka Publ., 1980 (in Russian) 10. Michael E. Continuous selections. Parts 1-3. The Annals of Mathematics, 2nd Ser., 63 (1956), №2, 361-382; 2nd Ser., 64 (1956), №3, 562-580; 2nd Ser., 65 (1957), №2, 375-390. 11. Wei J., Norman E. On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327-334. 12. Kato T. The theory of linear operators, NY, Springer, 1962. 13. Babenko K. Numerical Analysis, Moscow, Nauka, 1984 (in Russian). 14. Krasnoselskiy M.A., Vainikko G.M., Zabreiko P.P. and others. Approximate solution of operator equations, Moscow., Nauka Publ., 1969 (in Russian). 15. Gaevsky~H., Greger K., Zakharias K. Nonlinear operator equations and operator differential equations. Moscow, Mir, 1978, 336p. (in Russian). 16. Prykarpatsky A.K. A Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem. Preprint ICTP, IC/2007/028, Trieste, Italy, 2007. 17. Prykarpatsky A.K. An infinite-dimernsional Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem and some applications, Ukrainian Mathem. Journal, 60 (2008), №1, 114-120. 18. Serre J.-P. Lie algebras and Lie groups. Benjamin, New York, 1966.
Pages	21-32
Volume	32
Issue	1
Year	2009
Journal	Matematychni Studii
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