Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative

The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q, where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

On the Solution of Equations by Extended Discretization

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 ω− 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 ω− 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.

On the Convergence of Newton-like Method for Variational Inclusions under Pseudo-Lipschitz Mapping

In the present paper, we study a Newton-like method for solving the variational inclusion defined by the sums of a Frechet differentiable function, divided difference admissible function and a set-valued mapping with closed graph. Under some suitable assumptions on the Frechet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newton-like method and prove that the sequence generated by this method converges linearly and superlinearly to a solution of the variational inclusion. Specifically, when the Frechet derivative of the differentiable function is continuous, Lipschitz continuous, divided difference admissible function admits first order divided di_erence and the setvalued mapping is pseudo-Lipschitz continuous, we show the linear and superlinear convergence of the method. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 43-53

Local Convergence of Solvers with Eighth Order Having Weak Conditions

In particular, the problem of approximating a solution of an equation is of extreme importance in many disciplines, since numerous problems from diverse disciplines reduce to solving such equations. The solutions are found using iterative schemes since in general to find closed form solution is not possible. That is why it is important to study convergence order of solvers. We extended the applicability of an eighth-order convergent solver for solving Banach space valued equations. Earlier considerations adopting suppositions up to the ninth Fŕechet-derivative, although higher than one derivatives are not appearing on these solvers. But, we only practiced supposition on Lipschitz constants and the first-order Fŕechet-derivative. Hence, we extended the applicability of these solvers and provided the computable convergence radii of them not given in the earlier works. We only showed improvements for a certain class of solvers. But, our technique can be used to extend the applicability of other solvers in the literature in a similar fashion. We used a variety of numerical problems to show that our results are applicable to solve nonlinear problems but not earlier ones.