Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

The convergence analysis both local under weaker Argyros-type conditions and semilocal under-condition is established using first order Fréchet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Hölder conditions are particular cases of the-condition. Examples can be constructed for which the Lipchitz and Hölder conditions fail but the-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.