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1.
Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples.  相似文献   

2.
In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem.  相似文献   

3.
Let T:D⊂X→XT:DXX be an iteration function in a complete metric space XX. In this paper we present some new general complete convergence theorems for the Picard iteration xn+1=Txnxn+1=Txn with order of convergence at least r≥1r1. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of TT and a convergence function   of TT. We study the convergence of the Picard iteration associated to TT with respect to a function of initial conditions E:D→XE:DX. The initial conditions in our convergence results utilize only information at the starting point x0x0. More precisely, the initial conditions are given in the form E(x0)∈JE(x0)J, where JJ is an interval on R+R+ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ωω-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal αα-theorem of Smale for analytic functions.  相似文献   

4.
In this paper, we study the Sobolev’s spaces on time scales and their properties. As applications, we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order Hamiltonian systems on time scales. By establishing a proper variational setting, three existence results for systems under consideration are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of the existence results.  相似文献   

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