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A Steffensen’s type method in Banach spaces with applications on boundary-value problems. (English) Zbl 1139.65040

Summary: A modified Steffensen’s type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
34B15 Nonlinear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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[1] Allgower, E.L.; Böhmer, K.; Potra, F.A.; Rheinboldt, W.C., A mesh-independence principle for operators equations and their applications, SIAM J. numer. anal., 23, 1, 160-169, (1986) · Zbl 0591.65043
[2] Amat, S.; Busquier, S., A modified secant method for semismooth equations, Appl. math. lett., 16, 6, 877-881, (2003) · Zbl 1059.65042
[3] Amat, S.; Busquier, S., Convergence and numerical analysis of a family of two-step Steffensen’s methods, Comput. math. appl., 49, 1, 13-22, (2005) · Zbl 1075.65080
[4] Amat, S.; Busquier, S., On a Steffensen’s type method and its behaviour for semismooth equations, Appl. math. comput., 177, 819-823, (2006) · Zbl 1096.65047
[5] Amat, S.; Busquier, S.; Candela, V.F., A class of quasi-Newton generalized Steffensen’s methods on Banach spaces, J. comput. appl. math., 149, 2, 397-406, (2002) · Zbl 1016.65035
[6] Amat, S.; Busquier, S.; Gutiérrez, J.M., On the local convergence of secant-type methods, Internat. J. comput. math., 81, 9, 1153-1161, (2004) · Zbl 1068.65068
[7] Argyros, I.K., On the newton – kantorovich hypothesis for solving equations, J. comput. appl. math., 169, 2, 315-332, (2004) · Zbl 1055.65066
[8] Argyros, I.K.; Gutiérrez, J.M., A unified approach for enlarging the radius of convergence for Newton’s method and applications, Nonlinear functional anal. appl., 10, 4, 555-563, (2005) · Zbl 1094.65051
[9] Brown, P.N., A local convergence theory for combined inexact-Newton finite-difference projection methods, SIAM J. numer. anal., 24, 2, 402-434, (1987) · Zbl 0618.65037
[10] Gutiérrez, J.M., A new semi-local convergence theorem for Newton’s method, J. comput. appl. math., 79, 131-145, (1997) · Zbl 0872.65045
[11] Hernández, M.A.; Rubio, M.J., A uniparametric family of iterative processes for solving nondifferentiable equations, J. math. anal. appl., 275, 2, 821-834, (2002) · Zbl 1019.65036
[12] Hernández, M.A.; Rubio, M.J., Semilocal convergence of the secant method under mild convergence conditions of differentiability, Comput. math. appl., 44, 3-4, 277-285, (2002) · Zbl 1055.65069
[13] H.B. Keller, Numerical solution of two point boundary value problems, in: Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1990.
[14] Stoer, J.; Burlirsch, R., Introduction to numerical analysis, (1980), Springer New York
[15] Ypma, T.J., Local convergence of inexact Newton method, SIAM J. numer. anal., 21, 3, 583-590, (1984) · Zbl 0566.65037
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