Abstract
The aim of this paper is to present a new improved semilocal and local convergence analysis for two-step secant method to approximate a locally unique solution of a nonlinear equation in Banach spaces. This study is important because starting points play an important role in the convergence of an iterative method. We have used a combination of Lipschitz and center-Lipschitz conditions on the Fréchet derivative instead of only Lipschitz condition. A comparison is established on different types of center conditions and the influence of our approach is shown through the numerical examples. In comparison to some earlier study, it gives an improved domain of convergence along with the precise error bounds. Finally, some numerical examples including nonlinear elliptic differential equations and integral equations validate the efficacy of our approach.
Highlights
Consider the problem to approximate a locally unique solution x∗ of G (x) = 0, (1)where G : D ⊆ X → Y is a nonlinear operator
Mathematical modeling of many problems uses integral equations, boundary value problems, differential equations, and so forth, whose solutions are obtained by solving scalar equations or a system of equations
Many nonlinear differential equations can be solved by transforming them to matrix equations which give a system of nonlinear equations in Rn
Summary
Where G : D ⊆ X → Y is a nonlinear operator. X, Y are Banach spaces and D is an open nonempty convex subset of X. A two-step secant iteration with order of convergence same as (5) with its semilocal and local convergence under combination of Lipschitz and center-Lipschitz continuous divided differences of order one using majorizing sequences for solving (1) is described in Banach space setting in [17]. It is defined for n ≥ 0 by xn+1 = xn − [xn, yn; G]−1 G (xn) , (6). Along with its semilocal and local convergence analysis under weaker Lipschitz continuity condition on divided differences of order one on the involved operator G in Banach space setting.
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