Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail. (Received 26 April 2000; in final form 17 November 2000)     

Local convergence theorems of Newton's method for nonlinear equations using outer or generalized inverses

We provide local convergence theorems for Newton's method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations.     

Topological methods for the study of nonlinear mixed stochastic integral equations

This paper is concerned with the questions of existence, uniqueness, and stability of wide sense and random solutions of nonlinear stochastic integral equations of mixed type. The main tools are the theory of admissibility and the fixed-point theorems. The results in this paper improve some earlier works qualitatively by reducing the hypotheses. Also existence-type results, which hold under more general circumstances, and an application to nonlinear stochastic feedback control system are presented.     

Generalized nonlinear ordinary differential equations linear in measures

Summary In this paper one treats the initial value problem for nonlinear measure differential equations. Under various hypotheses one gets existence of global solutions and sometimes uniqueness too. There is also an example giving nonexistence or bifurcation depending on the initial datum and the measure in the equation. The results are generalizations to the nonlinear case of earlier linear results by the author. One main feature is the procedure which shows how the solutions act at points where the involved measures have point masses of arbitrary magnitude.     

Convergence conditions for secant-type methods

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.     

The effect of rounding errors on Newton methods

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newtonlike methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitztype hypotheses on the second Fréchet-derivative instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.     

Gauss-Newton methods for the complementarity problem

Mangasarian has shown that the solution of the complementarity problem is equivalent to the solution of a system of nonlinear equations. In this paper, we propose a damped Gauss-Newton algorithm to solve this system, prove that under appropriate hypotheses one gets rapid local convergence, and present computational experience.The author would like to thank Professor Michael Ferris for pointing out a flaw in one of the proofs in an earlier preprint of this paper (Ref. 1). He is grateful to Professor Olvi Mangasarian for bringing to his attention additional references relevant to the material in this paper, and for his suggestions which resulted in a greatly improved presentation.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

Variational approaches to p-Laplacian discrete problems of Kirchhoff-type

Critical point results for Kirchhoff-type discrete boundary value problems are exploited in order to prove that a suitable class possesses at least one solution under an asymptotical behaviour of the potential of the nonlinear term at zero, and also possesses infinitely many solutions under some hypotheses on the behaviour of the potential of the nonlinear term at infinity. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.     

Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations

The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state.To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold.     

Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise

Non-positivity and oscillations of nonlinear stochastic difference equations are discussed to characterize the long-term behaviour of their solutions in almost sure sense. Moreover, we present results on non-positivity with probability less than one and give numerous examples satisfying our main hypotheses.     

Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization

In this paper, by virtue of the nonlinear scalarization function commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of the unique solution to a parametric vector quasiequilibrium problem is studied based on nonlinear scalarization approach, under three different kinds of monotonicity hypotheses. The globally Lipschitz property of the nonlinear scalarization function is fully employed. Our approach is totally different from the ones used in the literature, and our results not only generalize but also improve the corresponding ones in some related works.     

On a fixed point theorem in Banach algebras with applications

In this article it is shown that some of the hypotheses of a fixed point theorem of the present author [B.C. Dhage, On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 25 (1988) 603–611] involving two operators in a Banach algebra are redundant. Our claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence results.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

一阶中立型时滞差分方程的线性化振动性定理

本文考虑一类非线性中立型时滞差分方程,证明了在一定条件下,若与此方程相关的一个线性差分方程的每个解振动,则该非线性差分方程的每个解也振动。     

随机积分和微分方程解的存在性准则

在[1]中我们已证明了一个一般的随机不动点定理并给出了某些应用,在本文中我们将给出该结果的进一步应用.首先证明了一随机Darbo不动点定理,然后利用此定理在紧性假设下给出了非线性随机Volterra积分方程和非线性随机微分方程Cauchy问题随机解的存在性准则.我们的定理改进和推广了Lakshmikantham[3,4],Vaugham[2],De Blasi和Myjak[5]等人的结果.     

A note on connection properties of fixed point sets of nonexpansive mappings

We consider a discrete-time GALERKIN method for nonlinear evolution equations. We prove convergence properties of this method under various hypotheses. Moreover, we deal with iteration methods reducing the nonlinear GALERKIN equations to linear equations in finite dimensional spaces.     

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT

In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.     

An error analysis for a certain class of iterative methods

We provide local convergence results in affine form for in-exact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first andmth Fréchet-derivative (m ≥ 2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Fréchet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.