Semilocal Convergence of a Three Step Fifth Order Iterative Method under Höolder Continuity Condition in Banach Spaces

In this paper, we study the semilocal convergence of
a fifth order iterative method using recurrence relation under the
assumption that first order Fréchet derivative satisfies the Hölder
condition. Also, we calculate the R-order of convergence and provide
some a priori error bounds. Based on this, we give existence and
uniqueness region of the solution for a nonlinear Hammerstein
integral equation of the second kind.

Authors:

• Ramandeep Behl
• Prashanth Maroju
• S. S. Motsa

Keywords:

• Hölder continuity condition
• Fréchet derivative
• fifth order convergence
• recurrence relations.

References:

[1] L.B.Rall, Computational Solution of Nonlinear Operator Equations,
Robert E. Krieger, New York, 1969.
[2] L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press,
Oxford, 1982.
[3] Miguel A. Hern´andez, Jose M.Gutierrez, Third-order iterative methods
for operators with bounded second derivative,Journal of Computational
and Applied Mathematics, 82 (1997) 171-183.
[4] P.K.Parida, D.K.Gupta, Recurrence relations for a Newton-like method in
Banach spaces, J. Comput. Appl. Math., 206 (2007) 873-887.
[5] S. Amat, M.A. Hernandez, and N.Romero, Semilocal convergence of
a sixth order iterative method for quadratic equations, J. Appl. Numer.
Math., 62 (2012) 833-841.
[6] M.A. Hernandez, N. Romero, On a characterization of some Newton-like
methods of R-order at least three, J. Comput. Appl. Math., 183 (2005)
53-66.
[7] V.Candela, A.Marquina, Valencia, Recurrence Relation for Rational Cubic
Methods I: The Halley Method, Computing, 44 (1990) 169-184.
[8] V.Candela, A.Marquina, Valencia, Recurrence Relation for Rational Cubic
Methods II: The Chebyshev Method, Computing, 45 (1990) 355-367.
[9] J.A. Ezquerro, M.A. Hernandez, Recurrence relations for Chebyshev-type
methods, Appl. Math. Optim., 41 (2000) 227-236.
[10] J.M.Guti´errez, M.A. Hern´andez, Recurrence Relations for the
Super-Halley Method, Journal of Computer Math.Applications, 36 (1998)
1-8.
[11] M.Prashanth, D.K.Gupta, Recurrence relations for Super-Halley’s
Method with H¨older continuous second derivative in Banach spaces,
Kodai Math. J., 36 (2013), 119–136.
[12] M.Prashanth, D.K.Gupta, Semilocal convergence of a continuation
method with H¨older continuous second derivative in Banach spaces,
Journal of Computational and Applied Mathematics, 236 (2012)
3174–3185.
[13] V.Arroyo, A.Cordero, and J.R.Torregrosa, Semilocal convergence by
using recurrence relations for a fifth-order method in Banach spaces,
Journal of Computational and Applied Mathematics, 273(2015) 205-213.
[14] A.Cordero, M.A.Hernandez, N.Romero and J.R.Torregrosa,
Approximation of artificial satellites preliminary orbits: the efficiency
challenge, Math. Comput. Modelling, 54(2011) 1802-1807.
[15] Lin Zheng, Chuanqing Gu, Recurrence relations for semilocal
convergence of a fifth-order method in Banach spaces, Numer Algor,
59(2012) 623-638.