Third-order iterative methods with applications to Hammerstein equations: A unified approach

AbstractThe geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in [S. Amat, S. Busquier, J.M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157(1) (2003) 197–205]. This family includes, as particular cases, some of the most famous third-order iterative methods: Chebyshev methods, Halley methods, super-Halley methods, C-methods and Newton-type two-step methods. The aim of the present paper is to analyze the convergence of this family for equations defined between two Banach spaces by using a technique developed in [J.A. Ezquerro, M.A. Hernández, Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57(3) (2007) 354–360]. This technique allows us to obtain a general semilocal convergence result for these methods, where the usual conditions on the second derivative are relaxed. On the other hand, the main practical difficulty related to the classical third-order iterative methods is the evaluation of bilinear operators, typically second-order Fréchet derivatives. However, in some cases, the second derivative is easy to evaluate. A clear example is provided by the approximation of Hammerstein equations, where it is diagonal by blocks. We finish the paper by applying our methods to some nonlinear integral equations of this type

Hermite collocation method for solving Hammerstein integral equations

In this paper, we are presenting Hermite collocation method to solve numer- ically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to …nd ordinary derivatives. This method is based on replace- ment of the unknown function by truncated series of well known Hermite expan-sion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coe¢ cients. Thus, by solving the matrix equation, Hermite coe¢ cients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique

A numerical approximation for solutions of Hammerstein integral equations in Lp spaces

In this work, we give conditions that guarantee the existence and the uniqueness of the solution of the Hammerstein integral equation in the Lp space and under such assumptions the successive approximation converges almost everywhere to the solution of the equation. Finally, we treat numerical exemples to confirm these results

A numerical method for functional Hammerstein integro-differential equations

In this paper, a numerical method is presented to solve functional Hammerstein integro-differential equations. The presented method combines the successive approximations method with trapezoidal quadrature rule and natural cubic spline interpolation to solve the mentioned equations. The existence and uniqueness of the problem is also investigated. The convergence and numerical stability of the problem are proved, and finally, the accuracy of the method is verified by presenting some numerical computations

Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition

The semilocal and local convergence in Banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the Frechet derivative satisfies the Holder condition. The Holder condition generalizes the Lipschtiz condition. The importance of our work lies in the fact that many examples are available which fail to satisfy the Lipschtiz condition but satisfy the Holder condition. The existence and uniqueness theorem is established with error bounds for the solution. The convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained. These examples include nonlinear Hammerstein and Fredholm integral equations and a boundary value problem. It is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions. (C) 2015 Elsevier Inc. All rights reserved.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition. Applied Mathematics and Computation. 276:266-277. doi:10.1016/j.amc.2015.11.062S26627727

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

[EN] The convergence analysis both local under weaker Argyros-type conditions and semilocal under. omega-condition is established using first order Frechet 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 Holder conditions are particular cases of the omega-condition. Examples can be constructed for which the Lipchitz and Holder conditions fail but the omega-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.Singh, S.; Gupta, D.; Singh, R.; Singh, M.; Martínez Molada, E. (2018). Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces. International Journal of Computational Methods. 15(6):1-18. https://doi.org/10.1142/S0219876218500482S11815