Generalized Lipschitz Condition Research Articles

Combined Newton‐Kurchatov method for solving nonlinear operator equations

AbstractWe investigate local and semi‐local convergence of the combined Newton‐Kurchatov method under the classical and generalized Lipschitz conditions for solving nonlinear equations. The convergence order of the method is examined and the uniqueness ball for the solution of the nonlinear equation is proved. Numerical experiments are conducted on test problems. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Local convergence of Newton’s method on the Heisenberg group

In the present paper, we study Newton’s method on the Heisenberg group for solving the equation f(x)=0, where f is a mapping from Heisenberg group to its Lie algebra. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton’s method and the estimation of the uniqueness ball of the zero point of f. Some applications to special cases including Kantorovich’s condition and γ-condition are provided. The determination of an approximate zero point of an analytic mapping is also presented. Concrete examples are given to illustrate applications of our results.

On Convergence of the Accelerated Newton Method Under Generalized Lipschitz Conditions

We study the problem of local convergence of the accelerated Newton method for the solution of nonlinear functional equations under generalized Lipschitz conditions for the first- and second-order Frechet derivatives. We show that the accelerated method is characterized by the quadratic order of convergence and compare it with the classical Newton method.

On Continuous Selection Sets of Non-Lipschitzian Quantum Stochastic Evolution Inclusions

We establish existence of a continuous selection of multifunctions associated with quantum stochastic evolution inclusions under a general Lipschitz condition. The coefficients here are multifunctions but not necessarily Lipschitz.

On the Behavior of the Four Order Iteration in Euler's Family Near a Zero

The aim of this paper is to study the local convergence of the four order iteration of Euler's family for solving nonlinear operatorequations. We get the optimal radius of the local convergence ball of the method for operators satisfying the weak third order generalized Lipschitz condition with L−average. We also show that the local convergence of the method is determined by a period 2 orbit of the method itself applied to a real function. AMS subject classifications: 65H05, 65L20, 65N12, 39B42

The majorant method in the theory of Newton–Kantorovich approximations and generalized Lipschitz conditions

We provide a semilocal as well as a local convergence analysis for Newton’s and modified Newton’s methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. We use more precise majorizing sequences than in earlier studies such as Appell et al. (1997), Appell et al. (1991), Argyros (2004), Argyros and Hilout (2009), Kantorovich and Akilov (1982), Ortega and Rheinboldt (1970) and generalized Lipschitz continuity conditions. Our sufficient convergence conditions are weaker than before and our convergence analysis is tighter. Special cases and numerical examples are also given in this study.

On the Convergence Analysis of a Two‐Step Modification of the Gauss‐Newton Method

AbstractWe investigate the convergence of a two‐step modification of the Gauss‐Newton method applying the generalized Lipschitz condition for the first and second order derivatives. The convergence order as well as the convergence radius of the method are studied and the uniqueness ball of solution of the nonlinear least squares problem is examined. Finally, we carry out numerical experiments on a set of well‐known test problems. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive State Estimation for a Class of Nonlinear Stochastic Systems under Generalized Lipschitz Condition

This paper is concerned with adaptive observer design problem for a class of nonlinear stochastic systems. Unknown constant parameters are assumed to be norm bounded. In order to better use the structural knowledge of the nonlinear part, a generalized Lipschitz condition is introduced to the adaptive observer design for a class of nonlinear stochastic systems for the first time. Based on a Lyapunov-Krasovskii functional approach and stochastic Lyapunov stability theory, we present a new adaptive observer design condition with ultimately exponentially bounded in sense of mean square for errors systems in terms of linear matrix inequality (LMI). A numerical example is exploited to show the validity and feasibility of the results.

A general Lipschitz uniqueness criterion for scalar ordinary differential equations

The classical Lipschitz-type criteria guarantee unique solvability of the scalar initial value problem ˙ x = f(t, x), x(t0) = x0, by putting restrictions onj f(t, x) f(t, y)j in dependence ofjx yj. Geometrically it means that the field differences are estimated in the direction of the x-axis. In 1989, Stettner and the second author could establish a generalized Lipschitz condition in both arguments by showing that the field differences can be measured in a suitably chosen direction v = (dt, dx), provided that it does not coincide with the directional vector (1, f(t0, x0)). Considering the vector v depending on t, a new general uniqueness result is derived and a short proof based on the implicit function theorem is developed. The advantage of the new criterion is shown by an example. A comparison with known results is given as well.

Error estimates of cubature formulas exact for Haar polynomials in classes of two-variable functions satisfying a general Lipschitz condition

Upper error estimates are obtained for cubature formulas with the Haar d-property in the classes Lip(L1, L2) of two-variable functions satisfying a general Lipschitz condition. It is shown that the error of minimal cubature formulas possessing the Haar d-property have the best order of convergence to zero in the indicated classes.

Adaptive state observer for Lipschitz nonlinear systems

The problem of adaptive observer synthesis for Lipschitz nonlinear systems is addressed. In the case of known parameters, the convergence property of state observer is first discussed. Based on a generalized Lipschitz condition, the sufficient conditions to ensure the stability of observer error dynamics are presented in terms of some LMIs. In the case of system dynamics with some unknown parameters, the proposed conditions along with an equality constraint are then employed to guarantee the convergence property of an adaptive state observer. Moreover, an adaptive observer form is derived which can be utilized in designing the reduced order state observer. The simulation results are finally given to exhibit the effectiveness of proposed synthesis approaches in dealing with the practical systems.

Existence, uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition

In this paper, we prove the existence, uniqueness and the stability of solutions for some nonlinear functional-integral equations by using generalized Lipschitz condition. We prove a fixed point theorem to obtain the mentioned aims in Banach space X:= C([a, b],R). As application we study some Volterra integral equations with linear, nonlinear and singular kernel.

Random fuzzy differential equations under generalized Lipschitz condition

We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.

Convergence of inexact difference methods under the generalized Lipschitz conditions

We study the convergence of the inexact chord method and Steffensen method for the solution of systems of nonlinear equations under the generalized Lipschitz conditions for first-order divided differences. We consider methods with a check of the relative discrepancy. The results obtained easily provide an estimate of the convergence sphere for inexact methods. For special cases, these results coincide with the known ones.

Convergence of Newton’s Method for Sections on Riemannian Manifolds

The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).

On a two-step iterative process under generalized Lipschitz conditions for first-order divided differences

We investigate a Newton-type two-step iterative method, using the approximation of the Fréchet derivative of a nonlinear operator by divided differences. We study the local convergence of this method provided that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and rate of convergence of this method are determined, and the domain of uniqueness of solution of the equation is found.

On an one‐step modification of Gauss‐Newton method under generalized Lipschitz conditions for solving the nonlinear least squares problem

AbstractWe investigate convergence of the one‐step modification of Gauss‐Newton method using the divided differences and the weak generalized Lipschitz condition for the divided differences. Convergence order of the method was examined and the uniqueness ball for the solution of the nonlinear least squares problem was proved. Numerical experiments were also provided. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Steffensen method under the generalized Lipschitz conditions for the divided difference operator

AbstractConvergence of the Steffensen method for solving nonlinear operator equations in the Banach spaces under the generalized Lipschitz condition for the first–order divided differences is investigated. The conditions and quadratic speed of convergence of this method are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Secant method under generalized Lipschitz conditions for the divided difference operator

AbstractIn this work we introduce for the first time the generalized Lipschitz conditions for the divided difference operator. A positive integrable function, partial case of which is usual Lipschitz constant, is suggested. Under the given conditions the convergence of the Secant method for solving the operator equations in Banach spaces is investigated, the uniqueness ball for the solution is obtained. As a partial case the known results for the Lipschitz constants are received. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Further results on the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations

Under a more general Lipschitz condition on the coefficients than our consideration in [E.O. Ayoola, Existence and stability results for strong solutions of quantum stochastic differential equations, Stochastic Anal. Appl. 20 (2) (2002) 263–281], we establish the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations (QSDE). This enables us to exhibit a class of Lipschitzian QSDE whose coefficients are continuous on the locally convex space of solution.