Abstract
There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results.
Highlights
Solutions for nonlinear equations in Banach spaces are widely studied in numerical analysis and computational mathematics
There are several real life problems that can reduce into nonlinear equations, the solutions of which can be obtained by using iterative methods
The assumption of semilocal convergence [2,3] is based on initial guess and domain estimates, the local convergence [4,5] assumptions are established on the information surrounding the solution, and the global convergence [1,6]
Summary
Solutions for nonlinear equations in Banach spaces are widely studied in numerical analysis and computational mathematics. There are several real life problems that can reduce into nonlinear equations, the solutions of which can be obtained by using iterative methods. The assumption of semilocal convergence [2,3] is based on initial guess and domain estimates, the local convergence [4,5] assumptions are established on the information surrounding the solution, and the global convergence [1,6]. The difference between these convergence approaches is that the condition at initial point x0 is imposed in results of semilocal convergence and the condition on solution x ∗ is imposed in results of local convergence. The semilocal convergence of Newton’s method is established by Kantorovich [4] under the different assumptions. There is a lot of literature available on higher order iterative methods to discuss the local and semilocal convergence (for reference please see [2,4,7,8,9,10,11])
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