The Banach–Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This article extends the distributed settings; specifically, we assume map which point sought be average individual (not necessarily LC) maps held by a set agents linked communication network. An additional difficulty that LC not assumed come from an underlying optimization problem, prevents exploi...

Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to sequence functions and modify norm in its internal optimization problem H−s norm, some positive integer s, bias it towards low-frequency spectral content residual. We analyze by quantifying improvement over Picard iteration. find that based on H−2 well-suited solve...

Nonlinear diffusion acceleration (NDA) can improve the performance of a neutron transport solver significantly especially for the multigroup eigenvalue problems. The high-order transport equation and the transport-corrected low-order diffusion equation form a nonlinear system in NDA, which can be solved via a Picard iteration. The consistency of the correction of the low-order equation is impor...

In 1988, Parker and Sochacki announced a theorem which proved that the Picard iteration, properly modified, generates the Taylor series solution to any ordinary differential equation (ODE) on n with a polynomial generator. In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. Since the domain of a sol...

In this paper, we present an approximate numerical solution to the well known Michaelis-Menten nonlinear biochemical reaction system using a stochastic technique based on hybrid polynomial basis evolutionary computing. The approximate solution is expanded as a linear combination of polynomial basis with unknown parameters. The system of nonlinear differential equation is transformed into an equ...

In the last three decades many papers have been published on the iterative approximation of fixed points for certain classes of operators, using the Mann and Ishikawa iteration methods, see [4], for a recent survey. These papers were motivated by the fact that, under weaker contractive type conditions, the Picard iteration (or the method of successive approximations), need not converge to the f...

Necessary and sufficient conditions for the convergence of Picard iteration to a fixed point for a continuous mapping in metric spaces are established. As application, we prove the convergence theorem of Ishikawa iteration to a fixed point for a nonexpansive mapping in Banach spaces. 2004 Elsevier Inc. All rights reserved.

The recognition and the calculation of all branches of solutions of the nonlinear boundary value problems is difficult obviously. The complexity of this issue goes back to the being nonlinearity of the problem. Regarding this matter, this paper considers steady state reactive transport model which does not have exact closed-form solution and discovers existence of dual or triple solutions in so...

In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard–Mann hybrid iteration process. We also construct an example of mappings show that it exceeds class mappings. To numerical accuracy our main outcome, process is more effective than all Picard, Mann, Ishikawa iterative processes.

Berinde [V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum {bf 9} (2004), 43-53] introduced almost contraction mappings and proved Banach contraction principle for such mappings. The aim of this paper is to introduce the notion of multivalued almost $Theta$- contraction mappings andto prove some best proximity point results for t...