Journal of Computational Mathematics and Applied Mathematics

Authors:-Farhana Alam, M. H. Rashid

Abstract:-In the present paper, we are interested to study the following generalized equation by secant-type method 0∈f(x)+g(x)+H(x),                                                                                       (*) where X and Y are real or complex Banach spaces, the function f:X→Y is a Frechet differentiable on neighborhood of a point x ̅ (which is the solution of (*)), g is differentiable at x ̅ but may not differentiable on the neighborhood of x ̅ and H:X⇉Y is a set-valued mapping with closed graph. We prove the existence of the sequence generated by the secant-type method and establish local convergence of the sequence generated by this method for generalized equation (*). Basically, we show the existence of the sequence generated by the secant-type method and establish the local convergence results of the sequence which is linearly convergent when the Frechet derivative of f, denoted by ∇f, is continuous, g admits first order divided deference and the set-valued mapping (f+ g+ H)^-1 is pseudo-Lipschitz. Furthermore, if ∇f satisfies Holder continuity property, g admits first order divided deference satisfying p-Holder continuity property and the set-valued mapping               (f+ g+ H)^-1 is pseudo-Lipschitz, we prove the existence of sequence generated by the secant-type method and prove the local convergence results of the sequence which converges super linearly to the solution of (*). In particular, our results extend and improve the corresponding ones Geoffroy and Pietrus (2004), and fix a gap in the sense of numerical computations in the proof in (Geoffroy and Pietrus (2004), Theorem 3.1).