A Newton-like Midpoint Method for Solving Equations in Banach Space

The present paper includes the local and semilocal convergence analysis of a fourth-order method based on quadrature formula in Banach spaces. weaker hypotheses used are only first Fréchet derivative. new approach provides residual errors, number iterations, radii, expected order convergence, estimates uniqueness solution. Such not provided approaches using Taylor expansions involving higher-order derivatives, which may exist or be very expensive impossible to compute. Numerical examples, including nonlinear integral equation partial differential equation, validate theoretical results.