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2009 | 7 | 4 | 775-786
Tytuł artykułu

The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
4
Strony
775-786
Opis fizyczny
Daty
wydano
2009-12-01
online
2009-10-31
Twórcy
autor
  • The AGH University of Science and Technology, Krakow, Poland
Bibliografia
  • [1] Babenko K., Numerical analysis, Moscow, Nauka, 1984 (in Russian)
  • [2] Bihun O., Luśtyk 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, 2003, 6, 23–29
  • [3] Bihun O., Luśtyk M., Approximation properties of the Lie-algebraic scheme, Matematychni Studii, 2003, 20, 85–91
  • [4] Bihun O., Prytula M., Modification of the Lie-algebraic scheme and approximation error estimations, Proc. Appl. Math. Mech., 2004, 4, 534–535 http://dx.doi.org/10.1002/pamm.200410248[Crossref]
  • [5] Calogero F., Interpolation, differentiation and solution of eigenvalue problems in more than one dimension, Lettere Al Nuovo Cimento, 1983, 38(13), 453–459 http://dx.doi.org/10.1007/BF02789862[Crossref]
  • [6] Calogero F., Classical many-body problems amenable to exact treatments, Lect. Notes Phys. Monogr., 66, Springer, 2001 [Crossref]
  • [7] Calogero F., Franco E., Numerical tests of a novel technique to compute the eigenvalues of differential operators, Il Nuovo Cimento B, 1985, 89, 161–208 http://dx.doi.org/10.1007/BF02723544[Crossref]
  • [8] Casas F., Solution of linear partial differential equations by Lie algebraic methods, J. Comput. Appl. Math., 1996, 76, 159–170 http://dx.doi.org/10.1016/S0377-0427(96)00099-4[Crossref]
  • [9] Gaevsky H., Greger K., Zakharias K., Nonlinear operator equations and operator differential equations, Mir, Moscow, 1978 (in Russian)
  • [10] Górniewicz L., Topological fixed point theory of multivalued mappings, Kluwer Academic Publishers, 1999
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  • [12] Krasnoselskiy M.A., Vainikkoa G.M., Zabreiko P.P., et al, Approximate solution of operator equations, Nauka, Moscow, 1969 (in Russian)
  • [13] Luśtyk M., Lie-algebraic discrete approximation for nonlinear evolution equations, J. Math. Sci. (N. Y.), 2002, 109(1), 1169–1172 http://dx.doi.org/10.1023/A:1013780223937[Crossref]
  • [14] Luśtyk M., The Lie-algebraic discrete approximation scheme for evolution equations with Dirichlet/Neumann data, Univ. Iagel. Acta Math., 2002, 40, 117–124
  • [15] Michael E., Continuous selections, Ann. of Math. (2), 1956, 63(2), 361–382 http://dx.doi.org/10.2307/1969615[Crossref]
  • [16] Michael E., Continuous selections, Ann. of Math. (2), 1956, 64(3), 562–580 http://dx.doi.org/10.2307/1969603[Crossref]
  • [17] Michael E., Continuous selections, Ann. of Math. (2), 1957, 65(2), 375–390 http://dx.doi.org/10.2307/1969969[Crossref]
  • [18] Mitropolski Yu., Prykarpatsky A.K., Samoylenko V.H., A Lie-algebraic scheme of discrete approximations of dynamical systems of mathematical physics, Ukrainian Mathematical Journal, 1988, 40, 53–458
  • [19] Nirenberg L., Topics in nonlinear functional analysis, American Mathematical Society (AMS), Providence, 2001
  • [20] Prykarpatsky A.K., A Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem, Preprint ICTP, IC/2007/028, Trieste, Italy, 2007 [WoS]
  • [21] Prykarpatsky A.K., An infinite-dimernsional Borsuk-Ulam type generalization of the Leray-Schauder fixed point theorem and some applications, Ukrainian Mathematical Journal, 2008, 60(1), 114–120 http://dx.doi.org/10.1007/s11253-008-0046-3[Crossref][WoS]
  • [22] Rudin W., Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, 1991
  • [23] Samoylenko V.H., 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 of Ukraine, 1988, 144–151 (in Russian)
  • [24] Serre J.-P., Lie algebras and Lie groups, Benjamin, New York, 1966
  • [25] Trenogin V.A., Functional analysis, Nauka, Moscow, 1980 (in Russian)
  • [26] Wei J., Norman E., On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 1964, 15, 27–334 http://dx.doi.org/10.2307/2034065[Crossref]
  • [27] Zeidler E., Nonlinear functinal analysis and its applications, Springer Verlag, Berlin and Heidelberg, 1986
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0038-z
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