Extending the Domain with Application of Four-Step Nonlinear Scheme with Average Lipschitz Conditions

Abstract

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

Keywords

• local convergence
• nonlinear problem
• convergence radius
• Banach space
• generalized Lipschitz conditions
• <i>ϰ</i>-average