×

Approximative methods for nonlinear equations (two approaches to the convergence problem). (English) Zbl 0401.65034


MSC:

65J15 Numerical solutions to equations with nonlinear operators
65N06 Finite difference methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Vainikko, G.; Tamme, E., Convergence of difference method in the periodic solution problem for equations of elliptic type, Ž. vyčisl. Mat. mat. Fiz., 16, 652-664 (1976) · Zbl 0349.65057
[2] Tamme, E., On regular convergence of difference approximations for Dirichlet problem, Izv. Akad. Nauk Estonsk. SSR, Ser. Fiz. Mat., 26, 3-8 (1977)
[3] VAINIKKOProceedings of Int. Summer-School on Finite Difference and Finite Elements Methods in Geophysics(to appear); VAINIKKOProceedings of Int. Summer-School on Finite Difference and Finite Elements Methods in Geophysics(to appear)
[4] Diestel, J., Geometry of Banach Spaces—Selected Topics (1975), Springer: Springer Berlin · Zbl 0307.46009
[5] Richtmyer, R.; Morton, K., Difference Methods for Initial-Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502
[6] Gudovič, N. N., An abstract scheme of difference method, Z. vycisl. Mat. mat. Fiz., 6, 916-920 (1966)
[7] Krein, S. G., Linear Differential Equations in a Banach Space (1967), Nauka: Nauka Moscow · Zbl 0636.34056
[8] Deimling, K., Nichtlineare Gleichungen und Abbildungsgrade (1974), Springer: Springer Berlin · Zbl 0281.47033
[9] Krasnoselskiǐ, M. A., Topological Methods in the Theory of Integral Equations (1956), Gostehizdat: Gostehizdat Moscow
[10] Krasnoselskiǐ, M. A.; Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis (1975), Nauka: Nauka Moscow
[11] Dugundji, J., An extension of Tietze’s theorem, Pacif. J. Math., 1, 353-367 (1951) · Zbl 0043.38105
[12] Sadovskiǐ, B. N., Limit-compact and condensing operators, Usp. mat. Nauk, 27, 81-146 (1972) · Zbl 0232.47067
[13] Szegö, G., Orthogonal Polynomials (1959), A.M.S: A.M.S New York · JFM 65.0278.03
[14] Kerge, R. M., On error estimation of subregion method, Z. vycisl. Mat mat. Fiz., 18, No. 3 (1978) · Zbl 0395.65070
[15] Zygmund, A., Trigonometrical Series (1952), New York · JFM 61.0263.03
[16] Vainikko, G., On convergence speed of method of moments for ordinary differential equations, Sib. Mat. Ž., 9, 21-28 (1968) · Zbl 0165.41101
[17] Gavurin, M. K., Lectures on Numerical Methods (1971), Nauka: Nauka Moscow · Zbl 0226.65046
[18] Stummel, F., Diskrete Konvergenz linearer Operatoren I, Math. Ann., 190, 45-92 (1970) · Zbl 0203.45301
[19] Stummel, F., Diskrete Konvergenz linearer Operatoren II, Math. Z., 120, 231-264 (1971) · Zbl 0209.15502
[20] Stummel, F., Diskrete Konvergenz linearer Operatoren III. Proc. Oberwolfach Conference in linear Operators and Approximations, Int. Series of Numerical Mathematics, 20 (1972), Basel
[21] Stummel, F., Discrete convergence of mappings, Top. Numerical Analysis, 285-310 (1973), London
[22] Stummel, F., Perturbations of nonlinear integral equations, Proc. R. Soc. Edinb., A 74, 55-70 (1976) · Zbl 0333.65056
[23] Stummel, F., Stability and discrete convergence of differentiable mappings, Rev. roum. Math. pures appl., 21, 63-96 (1976) · Zbl 0336.65029
[24] Stummel, F.; Reinhardt, J., Discrete convergence of continuous mappings in metric spaces, Lect. Notes Math., 333, 218-242 (1973)
[25] Kantorovič, L. V., Functional analysis and applied mathematics, Usp. mat. Nauk, 3, 89-187 (1948)
[26] Kantorovič, L. V.; Akilov, G. P., Functional Analysis in Normed Spaces (1959), Fizmatgiz: Fizmatgiz Moscow · Zbl 0127.06102
[27] Mihlin, S. G., Variational Methods in Mathematical Physics (1970), Nauka: Nauka Moscow
[28] Krasnoselskiǐ, M. A.; Vainikko, G. M.; Zabreiko, P. P.; Rutizkiǐ, J. B.; Ctecenko, V. J., Approximate Solution of Operator Equations (1969), Nauka: Nauka Moscow
[29] Aubin, P. J., Approximations des espaces de distributions et des opérateurs différentiels, Bull. Soc. math. Fr., 12, 1-139 (1967) · Zbl 0157.21901
[30] Aubin, P. J., Approximation of Elliptic Boundary-Value Problems (1972), Wiley-Interscience: Wiley-Interscience New York
[31] Vainberg, M. M., Variational Method and the Method of Monotonous Operators (1972), Nauka: Nauka Moscow
[32] Gajewski, H.; Gröger, K.; Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (1974), Akademie-Verlag: Akademie-Verlag Berlin · Zbl 0289.47029
[33] Varga, R., Functional analysis and approximation theory in numerical analysis (1971), SIAM: SIAM Philadelphia, PA · Zbl 0226.65064
[34] Vainikko, G., Funktionalanalysis der Diskretisierungsverfahren, (Skripte TH Karl-Marx-Stadt (1976), Teubner-Texte zur Mathematik: Teubner-Texte zur Mathematik Leipzig) · Zbl 0152.14505
[35] Vainikko, G., Analysis of Discretization Methods (1976), University of Tartu: University of Tartu Tartu, (a special course)
[36] Vainikko, G., Discretely compact sequences, Z. vycisl. Mat. mat. Fiz., 14, 572-583 (1974) · Zbl 0287.46024
[37] Vainikko, G., The principle of compact approximation in the theory of approximative methods, Z. vycisl. Mat. mat. Fiz., 9, 739-761 (1969) · Zbl 0197.10802
[38] Vainikko, G., Über Konvergenzbegriffe für lineare Operatoren in der Numerischen Mathematik, Math. Nachr., 78, 165-183 (1977) · Zbl 0369.65016
[39] Gingorieff, R. D., Zur Theorie approximationsregulären Operatoren I, Math. Nachr., 55, 233-249 (1973)
[40] Petryshyn, W. V., On projectional-solvability and the Fredholm alternative for equations involving linear \(A\)-proper operators, Archs rat. mech. Analysis, 30, 270-284 (1968) · Zbl 0176.45902
[41] Karma, O., On approximation of operator functions and convergence of approximate eigenvalues, Dissertation (1971), Tartu · Zbl 0318.41012
[42] Grigorieff, R. D., Über die Fredholm-Alternative bei linearen approximationsregulären Operatoren, Appl. Analysis, 2, 217-227 (1972) · Zbl 0244.47005
[43] Grigorieff, R. D., Zur Theorie approximationsregulären Operatoren II, Math. Nachr., 55, 251-263 (1973)
[44] Vainikko, G.; Piskarev, S., On regularly consistent operators, Izv. Vysš. Učebn. Zaved. Matematika, No. 10, 25-36 (1977) · Zbl 0388.47007
[45] Petryshyn, W. V., Projection methods in nonlinear numerical functional analysis, J. Math. Mech., 17, 353-372 (1967) · Zbl 0162.20202
[46] Sobolev, S. L., Some remarks on numerical solution of integral equations, Izv. Akad. Nauk SSSR Ser. Mat., 20, 413-436 (1956)
[47] Anselone, P. M., Collectively compact approximation of integral operator with discontinuous kernels, J. math. Analysis Applic., 22, 582-590 (1968) · Zbl 0157.45201
[48] Anselone, P. M., Collectively Compact Approximation Theory (1971), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0157.45201
[49] Anselone, P. M.; Moore, R. H., Approximate solution of integral and operator equations, J. math. Analysis Applic., 9, 268-277 (1964) · Zbl 0149.11502
[50] Vainikko, G., Compact approximation of linear compact operators by operators in factor spaces, Uchen. Zap. Tartu. gos. Univ., 220, 190-204 (1968)
[51] Vainikko, G., Compact approximation of operators and approximative solution of operator equations, Dokl. Akad. Nauk. SSSR, 189, 237-240 (1969)
[52] Vainikko, G., Compact Approximation of Operators and Approximate Solution of Equations (1970), University of Tartu: University of Tartu Tartu
[53] Vainikko, G., On approximation of fixed points of compact operators, Učen. Zap. Tartu. gos. Univ., 342, 225-236 (1974)
[54] Wolf, R., Über lineare approximationsreguläre Operatoren, Math. Nachr., 59, 325-341 (1974) · Zbl 0295.47018
[55] Vainikko, G.; Karma, O., On convergence of approximative methods for linear and nonlinear operator equations, Zh. vyčisl. Mat. mat. Fiz., 14, 828-837 (1974)
[56] Vainikko, G., Über die Konvergenz und Divirgenz von Näherungsmethoden bei Eigenwertproblemen, Math. Nachr., 78, 145-164 (1977) · Zbl 0369.65015
[57] Karma, O., On compact approximation of operator functions, Učen Zap. Tartugos. Univ., 277, 194-204 (1971)
[58] Karma, O., Asymptotic error estimations for characteristic values of holomorphic Fredholm operator functions, Zh. vycisl. Mat. mat. Fiz., 11, 559-568 (1971) · Zbl 0222.47007
[59] Karma, O., On approximation of operator functions and convergence of approximate eigenvalues, Trudy vycisl. Centra Tartu. Univ., 24, 3-143 (1971)
[60] Vainikko, G.; Karma, O., On convergence speed of approximative methods in eigenvalue problems with nonlinear depending on a parameter, Zh. vyčisl. Mat. mat. Fiz., 14, 1393-1408 (1974)
[61] Grigorieff, R. D., Diskrete Approximation von Eigenwertproblemen I: Qualitative Konvergenz, Num. Math., 24, 355-374 (1975) · Zbl 0391.65020
[62] Grigorieff, R. D., Diskrete Approximation von Eigenwertproblemen II: Konvergenzordnung, Num. Math., 24, 415-433 (1975) · Zbl 0391.65021
[63] Grigorieff, R. D., Diskrete Approximation von Eigenwertproblemen III: Asymptotische Entwicklungen, Num. Math., 25, 79-97 (1975) · Zbl 0391.65022
[64] Grigorieff, R. D.; Jeggle, H., Approximation von Eigenwertproblem bei nichtlinearer Parameterabhängigkeit, Manuscr. math., 10, 245-271 (1973) · Zbl 0267.65045
[65] Jeggle, H., Über die Approximation von linearen Gleichungen zweiter Art und Eigenwertproblemen in Banach-Räumen, Math. Z., 124, 319-342 (1972) · Zbl 0215.48901
[66] Chatelin, F., Approximation du spectre d’un opérateur linéaire: Application aux opérateurs différentiels elliptiques non autoadjoint, Num. Math., 20, 193-204 (1973) · Zbl 0268.65070
[67] Vainikko, G., Perturbed Galerkin method and a general theory of approximative methods fornonlinear equations, Zh. vycisl. Mat. mat. Fiz., 7, 723-751 (1967)
[68] Vainikko, G., On connection between the methods of mechanical quadrature and finite difference, Zh. vycisl. Mat. mat. Fiz., 9, 259-270 (1969)
[69] Vainikko, G., On difference method for ordinary differential equations, Zh. vyčisl. Mat. mat. Fiz., 9, 1057-1974 (1969)
[70] Vainikko, G., On approximation of linear and nonlinear operators and approximative solution of operator equations, (Dissertation (1969), Tartu-Voronež)
[71] Keller, H. B., Approximation methods for nonlinear problems with application to two-point boundary value problems, Maths Comput., 29, 464-474 (1975) · Zbl 0308.65039
[72] Grigorieff, R. D., Über diskrete Approximationen nichtlinearer Gleichungen 1. Art, Math. Nachr., 69, 253-272 (1975) · Zbl 0369.65014
[73] Vainikko, G., Über die Invarianz der Rotation bei Approximation der Vektorfelder, (Proceedings of Int. Summer-School on Theory of Nonlinear Operators (1977), Akademie-Verlag: Akademie-Verlag Berlin) · Zbl 0373.47036
[74] Bobylev, N. A., On the theory of factor-methods to solve nonlinear problems, Dokl. Akad. Nauk SSSR, 199, 9-12 (1971)
[75] Potapov, A. S., On the theory of rotation of limit-compact vector fields, Commentat. math. Univ. Carol., 15, 693-716 (1974) · Zbl 0294.47038
[76] Skrypnik, I. V., Nonlinear Elliptic Equations of Higher Order (1973), Naukova Dumka: Naukova Dumka Kiev · Zbl 0296.35032
[77] Browder, F. E.; Petryshyn, W. V., The topological degree and Galerkin approximations for noncompact operators in Banach spaces, Bull. Am. math. Soc., 74, 641-646 (1968) · Zbl 0164.17003
[78] Petry, W., Existence theorems for a class of nonlinear operator equations, J. Math. Analysis Applic., 43, 250-260 (1973) · Zbl 0264.47042
[79] Vainikko, G., On convergence of quadrature formulae method for integral equations with discontinuous kernels, Sib. Mat. Zh., 12, 40-53 (1971)
[80] Baluev, A. N., On approximate solution of nonlinear integral equations, Učen. Zap. Leningr. gas. Univ., 33, 28-31 (1958)
[81] Jeggle, H., Uniformly convergent approximations for integral equations on non-compact manifolds, Appl. Analysis, 5, 227-248 (1976) · Zbl 0335.45015
[82] Vainikko, G.; Pedas, A., On solution of integral equations with logarithmical singularities by quadrature formulae methods, Učen. Zap. Tartu. gos. Univ., 281, 201-210 (1971)
[83] Karpilovskaja, E. B., On convergence of interpolation method for ordinary differential equations, Usp. mat. Nauk, 8, 111-118 (1953)
[84] Karpilovskaja, E. B., On convergence of collocation method, Dokl. Akad. Nauk SSSR, 151, 766-769 (1963) · Zbl 0192.21704
[85] Karpilovskaja, E. B., On convergence of collocation methods for certain boundary value problems of mathematical physics, Sib. mat. Zh., 4, 632-640 (1963) · Zbl 0192.21704
[86] Karpilovskaja, E. B., On convergence of subregion method for integro-differential equations, Zh. vyčisl. Mat. mat. Fiz., 5, 124-132 (1965)
[87] Petersen, I., On convergence of approximative methods of interpolation type for ordinary differential equations, Izv. Akad. Nauk Estonsk. SSR Ser. Fiz. Mat. Tehn., 10, 3-12 (1961)
[88] Vainikko, G., On convergence and stability of collocation method, Differencial’nye Uravnenija, 1, 244-254 (1965)
[89] James, R. L., Uniform convergence of positive operators, Math. Z., 120, 124-142 (1971) · Zbl 0202.13104
[90] Vainikko, G., On convergence of collocation method for multidimensional integral equations, Učen. Zap. Tartu. gos. Univ., 253, 244-257 (1970)
[91] Vainikko, G., On stability problem of collocation method, Učen. Zap. Tartu. gos. Univ., 281, 190-196 (1971)
[92] Vainikko, G., On convergence of collocation method for nonlinear differential equations, Zh. vycisl. Mat. mat. Fiz., 6, 35-42 (1966) · Zbl 0154.17301
[93] Urabe, M., Numerical solution of multi-point boundary value problems in Chebyshev series. Theory of the method, Publs. Res. Inst. Math. Sci., B, no. 9, 341-366 (1967) · Zbl 0168.14005
[94] Voss, H., Projektionsverfahren für Randwertaufgaben mit nichtlinearen Randwertbedingungen, Num. Math., 24, 317-329 (1975) · Zbl 0295.65040
[95] Fujii, M., Numerical solution of boundary value problems with nonlinear boundary conditions in Chebyshev series, Bull. Fukuoka Univ. Educ. Nat. Sci., 25, 27-45 (1975)
[96] Lucas, T. R.; Reddien, G. W., Some collocation methods for nonlinear boundary value problems, SIAM J. Num. Analysis, 9, 341-356 (1972) · Zbl 0266.34024
[97] Russell, R. D.; Shampine, L. F., A collocation method for boundary value problems, Num. Math., 19, 1-28 (1972) · Zbl 0221.65129
[98] Russell, R. D., Collocation for systems of boundary value problems, Num. Math., 23, 119-133 (1974) · Zbl 0279.65070
[99] DeBoor, C.; Swartz, B., Collocation at Gaussian points, SIAM J. Num. Analysis, 10, 582-606 (1973) · Zbl 0232.65065
[100] Wittenbrink, K. A., High order projection methods of moment- and collocation-type for nonlinear boundary value problems, Computing, 11, 255-274 (1973) · Zbl 0288.65047
[101] Urabe, M., Galerkins procedure for nonlinear periodic systems, Archs rat. mech. Analysis, 20, 120-152 (1965) · Zbl 0133.35502
[102] Samoilenko, A. M.; Ronto, N. I., Numerical-Analytical Methods to Investigate Periodic Solutions (1976), Visa skola: Visa skola Kiev
[103] Schneider, K. R., Näherungsverfahren zur Ermittlung periodischer Lösungen von Systemen nichtlinearer periodischer Differentialgleichungen, Computing, 10, 63-82 (1972) · Zbl 0302.65065
[104] Nitzsch, J., Analytical and numerical treatment of nonlinear oscillation problems, Berlin 1975. Berlin 1975, Proceedings of VII ICNO (1975)
[105] Vainikko, G., On convergence of difference method in the periodic solution problem for ordinary differential equations, Zh. vyčisl. Mat. mat. Fiz., 15, 87-100 (1975) · Zbl 0311.65053
[106] Vainikko, G., On convergence of quadrature-difference methods for linear inegro-differential equations, Zh. vyčisl. Mat. mat. Fiz., 11, 770-776 (1971) · Zbl 0221.45013
[107] Jokk, H., Results on convergence considerations of difference method for nonlinear differential equations of second order on non-uniform grid, Izv. Akad. Nauk Estonsk. SSR Ser. Fiz.-Mat., 23, 86-88 (1974)
[108] Saarniit, I., Construction of a \(h^2\)-scheme for a linear boundary value problem for second order differential equation with deviating argument, Zh. vycisl. Mat. mat. Fiz., 12, 105-111 (1972)
[109] Saarniit, I., On solving by finite difference method of a boundary value problem for a differential equation with deviating argument, Zh. vycisl. Mat. mat. Fiz., 16, 372-384 (1976) · Zbl 0355.65061
[110] Vainikko, G.; Miidla, P., On convergence of approximative methods to find autooscillations, (Proc. of VII Int. Confer. on Nonlin. Oscillations, Berlin, 1975 (1977), Akademie-Verlag: Akademie-Verlag Berlin), 347-353
[111] Strygin, V. V., Application of Bubnov-Galerkin method to find autooscillations, Prikl. Mat. Meh., 37, 1015-1019 (1973)
[112] Strygin, V. V.; Cygankov, A. I., Application of collocation and difference method to find autooscillations of differential-difference equations, Zh. vycisl. Mat. mat. Fiz., 14, 691-698 (1974) · Zbl 0294.65043
[113] VAINIKKOAZAKOVA; VAINIKKOAZAKOVA
[114] Mysovskih, I. P., On convergence of mechanical cubature method for solving the integral equations, (Metody vyčisl., 4 (1967), Leningr. Univ), 63-72
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.