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Applied Mathematics and Computation 217 (2010) 11–24Contents lists available at ScienceDirect

Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation
William La Cruz
Departamento de Electrónica, Computación y Control, Facultad de Ingeniería, Universidad Central de Venezuela,Caracas 1051-DF, Venezuela

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A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinearintegral equations using symbolic computation are presented. Ó 2010 Elsevier Inc. All rights reserved.

Keywords: Residual method Nonlinear operator equations Nonlinear integral equations Symbolic computation

1. Introduction The mathematical formulation of a nonlinear operator equation reads as follows:

Fy ¼ 0;


where F maps the real Hilbert space H into itself and F is continuous in H.The Hilbert space H is equipped with the inner pffiffiffiffiffiffiffiffiffi product hÁ,Ái and the associated norm k Á k ¼ hÁ; Ái. 0 We are interested in solving (1) with a method that does not use F (the Fréchet derivative), but shares some features in common with the method of minimum residuals. To be precise, we discuss a specific method for solving (1), namely, Residual Algorithm for Nonlinear Operator Equations(RANOE) which is a variation and generalization of the methods SANE [8] and DFSANE [7] for systems of nonlinear equations on Rn . These algorithms have a characteristic in common, that is, they use in a systematic way the residual Fx as search direction. For the new method we also present a convergence analysis. An interesting particular case of (1) is the solution of nonlinear integral equations,that arise in important areas of science and engineering. Different methods for solving integral equations have been developed in the last few years [1,2,4–6, 9–13]. In this work we also describe the application of RANOE to the resolution of nonlinear integral equations. To be precise, we use a symbolic implementation of RANOE for solving nonlinear integral Hammerstein’s equations given bysymmetric and continuous kernels. We show numerical results for some test nonlinear integral equations using the Symbolic Toolbox of Matlab for the implementation of RANOE. 2. The method and its properties The iterations of any method of minimum residuals can be defined by

ykþ1 ¼ yk þ kk dk ;


where y0 2 H is a given initial approximation to the solution of (1), kk are real numbers, and {dk}is a sequence of elements in 0 H [14]. Usually these methods use F to define the numbers kk or the sequence {dk}. Under certain hypothesis, the methods
E-mail address: william.lacruz@ucv.ve 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.045

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12 W. La Cruz / Applied Mathematics and Computation 217 (2010) 11–24

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