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Solving a special case of conservative problems by secant-like methods. (English) Zbl 1080.65044
For nonlinear operator equations in Banach space the authors consider a relaxed form of a general secant method. Convergence is proved and a priori error bounds are obtained by means of the type of recurrence relations used by J. M. Gutiérrez and M. A. Hernandez [Appl. Math. Lett. 10, 63-65 (1997; Zbl 0883.65050)] for Chebyshev’s method. The results are applied to conservative systems of the form \({\ddot x} + \Phi(x(t)) = 0\), \(x(0) = x(1) = 0\).

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Andronow, A.A.; Chaikin, C.E., Theory of oscillations, (1949), Princenton University Press Princenton, NJ · Zbl 0032.36501
[2] Argyros, I.K., On the secant method, Publ. math. debrecen, 43, 223-238, (1993) · Zbl 0796.65075
[3] Argyros, I.K., Some methods for finding error bounds for Newton-like methods under mild differentiability conditions, Acta math. hung., 61, 3-4, 183-194, (1993) · Zbl 0816.65033
[4] Argyros, I.K., Newton-like methods under mild differentiability conditions with error analysis, Bull. austral. math. soc., 37, 131-147, (1998) · Zbl 0629.65061
[5] Balazs, M.; Goldner, G., On existence of divided differences in linear spaces, Rev. anal. numer. theor. approx., 2, 3-6, (1973) · Zbl 0356.65042
[6] Dennis, J.E., Toward a unified convergence theory for Newton-like methods, (), 425-472
[7] Gutiérrez, J.M.; Hernández, M.A., New recurrence relations for Chebyshev method, Appl. math. lett., 10, 63-65, (1997) · Zbl 0883.65050
[8] Hernández, M.A.; Salanova, M.A., Chebyshev method and convexity, Appl. math. comput., 95, 1, 51-62, (1998) · Zbl 0943.65071
[9] Porter, D.; Stirling, D.S.G., Integral equations, (1990), Cambridge University Press Cambridge · Zbl 0908.45001
[10] Potra, F.A., On a modified secant method, Anal. numer. theor. approx., 8, 203-214, (1979) · Zbl 0445.65055
[11] Potra, F.A.; Pták, V., Nondiscrete induction and iterative processes, (1984), Pitman New York · Zbl 0549.41001
[12] Rokne, J., Newton’s method under mild differentiability conditions with error analysis, Numer. math., 18, 401-412, (1972) · Zbl 0221.65084
[13] Schmidt, J.W., Regula-falsi verfahren mit konsistenter steigung und majoranten prinzip, Period. mathemat. hung., 5, 187-193, (1974) · Zbl 0291.65017
[14] Sergeev, A., On the method of chords, Sibirsk. mat. Z̆., 2, 282-289, (1961)
[15] Stoker, J.J., Nonlinear vibrations, (1950), Interscience-Wiley New York · Zbl 0035.39603
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