On the Solution of Equations by Extended Discretization

• Main Authors: Gus I. Argyros, Michael I. Argyros, Samundra Regmi, Ioannis K. Argyros, Santhosh George
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Computation
• Subjects: banach space; lipschitz condition; hölder condition; newton’s method; discretization
• Online Access: https://www.mdpi.com/2079-3197/8/3/69

The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using <inline-formula><math display="inline"><semantics><mrow><mi>ω</mi><mo>−</mo></mrow></semantics></math></inline-formula> continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize <inline-formula><math display="inline"><semantics><mrow><mi>ω</mi><mo>−</mo></mrow></semantics></math></inline-formula> continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved.