Generalized Lipschitz Condition Research Articles

Generalization and Extension of Liu,H. and Xu,S. and, Jiandong Wang Results in Cone Metric Spaces Over Banach Algebras for Generalized Lipschitz Conditions

This article aims to present and establish some fixed-point results for a mapping satisfying generalized Lipschitz conditions and appealing to the normality of the cone. Our results extend and generalize several known results from the existing literature.

Existence and asymptotic behavior of square-mean $ S $-asymptotically periodic solutions of fractional stochastic evolution equations

This paper investigates the abstract fractional stochastic evolution equations. A new existence result of the square-mean $ S $-asymptotically periodic mild solutions are obtained under the assumption that the nonlinear terms only satisfy some growth conditions. Moreover, the uniqueness and asymptotic stability results of the square-mean $ S $-asymptotically periodic solution are presented when the nonlinear functions satisfy the general Lipschitz condition. Finally, two examples are given to illustrate our main results.

Generalized Lipschitz conditions for absolute convergence of weighted Jacobi–Dunkl series

Generalized Lipschitz conditions for absolute convergence of weighted Jacobi–Dunkl series

An adaptive regularization method in Banach spaces

This paper considers optimization of nonconvex functionals in smooth infinite dimensional spaces. It is first proved that functionals in a class containing multivariate polynomials augmented with a sufficiently smooth regularization can be minimized by a simple linesearch-based algorithm. Sufficient smoothness depends on gradients satisfying a novel two-terms generalized Lipschitz condition. A first-order adaptive regularization method applicable to functionals with β-Hölder continuous derivatives is then proposed, that uses the linesearch approach to compute a suitable trial step. It is shown to find an ϵ-approximate first-order point in at most evaluations of the functional and its first p derivatives.

Ultimately Exponentially Bounded Estimates for a Class of Nonlinear Discrete−Time Stochastic Systems

In this paper, the ultimately exponentially bounded estimate problem of nonlinear stochastic discrete−time systems under generalized Lipschitz conditions is considered. A new sufficient condition making the estimation error system uniformly exponentially bounded in the mean square sense is given. The gain matrix can be obtained by solving matrix inequality. In the last section, numerical examples are provided verify the effectiveness of the conclusions.

On the Convergence of Two-Step Kurchatov-Type Methods under Generalized Continuity Conditions for Solving Nonlinear Equations

The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one.

Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces

In the given study, we investigate the three-step NTS’s ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator’s first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the κ-average condition. Furthermore, several theorems on the convergence of the same method in Banach spaces are developed with the conditions that the derivative of the operators must satisfy the radius or center-Lipschitz condition with a weak κ-average and that κ is a positive integrable but not necessarily non-decreasing function. This novel approach allows for a more precise convergence analysis even without the requirement for new circumstances. As a result, we broaden the applicability of iterative approaches. The theoretical results are supported further by illuminating examples. The convergence theorem investigates the location of the solution ϵ* and the existence of it. In the end, we achieve weaker sufficient convergence criteria and more specific knowledge on the position of the ϵ* than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ(u). Numerical tests are carried out to corroborate the hypotheses established in this work.

Lyapunov characterizations on input-to-state stability of nonlinear systems with infinite delays

This paper addresses Lyapunov characterizations on input-to-state stability (ISS) of time-varying nonlinear systems with infinite delays. With novel ISS definitions in the case of nonlinear systems with infinite delays, we present several results on their ISS Lyapunov characterizations in the form of both ISS Lyapunov theorems and converse ISS Lyapunov theorems. It is shown that an infinite-delayed system is (locally) ISS if it has a (local) ISS Lyapunov functional, and conversely, there exists a (local) ISS Lyapunov functional if it is (locally) ISS. To prove the converse ISS Lyapunov theorems, we establish a key technical lemma bridging ISS/LISS and robust asymptotic stability of systems with infinite delays and two converse Lyapunov theorems concerning robust asymptotic stability of systems with infinite delays. Two distinctive advantages of this work are that a large class of infinite dimensional spaces are allowed and the results are established based on a more general Lipschitz condition, i.e., the right hand side Lipschitz (RS-L) condition. An example is provided for illustration of the obtained results.

Extended three step sixth order Jarratt-like methods under generalized conditions for nonlinear equations

The convergence balls as well as the dynamical characteristics of two sixth order Jarratt-like methods (JLM1 and JLM2) are compared. First, the ball analysis theorems for these algorithms are proved by applying generalized Lipschitz conditions on derivative of the first order. As a result, significant information on the radii of convergence and the regions of uniqueness for the solution are found along with calculable error distances. Also, the scope of utilization of these algorithms is extended. Then, we compare the dynamical properties, using the attraction basin approach, of these iterative schemes. At the end, standard application problems are considered to demonstrate the efficacy of our theoretical findings on ball convergence. For these problems, the convergence balls are computed and compared. From these comparisons, it is confirmed that JLM1 has the bigger convergence balls than JLM2. Also, the attraction basins for JLM1 are larger in comparison to JLM2. Thus, for numerical applications, JLM1 is better than JLM2.

Motorcycle State Estimation and Tire Cornering Stiffness Identification Applied to Road Safety: Using Observer-Based Identifiers

This paper deals with an observer-based identification framework to estimate both lateral dynamics states and tires’ cornering parameters in the perspective of designing advanced rider assistance systems for powered two-wheeled vehicle. An adaptive observer is proposed to reconstruct the state variables regardless of the forward velocity variations and to estimate the real unknown tires’ parameters. The stability and convergence analysis of the proposed observer is based on the Lyapunov theory, the persistency of excitation and the general Lipschitz condition. To enable this observer design, the linear parameter varying observer is transformed into Takagi-Sugeno exact form where the sufficient conditions are given in terms of linear matrix inequalities. Finally, an evaluation framework is proposed to provide a critical overview of the method’s effectiveness. The proposed adaptive law is compared to direct estimation and dynamic inversion estimation methods. Co-simulation scenarios are performed by using both BikeSim© motorcycle simulator and real data-log obtained from an instrumented electrical scooter.

New Lipschitz–type conditions for uniqueness of solutions of ordinary differential equations

We present some generalized Lipschitz conditions which imply uniqueness of solutions for scalar ODEs. We illustrate the applicability of our results with examples not covered by earlier Lipschitz–type uniqueness tests.

Finite-time stability of Hadamard fractional differential equations in weighted Banach spaces

The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definitions of finite-time stability of HFDEs in compatible Banach spaces are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria of finite-time stability for linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results.

Decrease the order of nonlinear predictors based on generalized-Lipschitz condition

In this paper, a low order asymptotic predictor is introduced for nonlinear systems based on measurable outputs. Predictors play basic role in the control of dead-time systems having no delay-free input. Generalized Lipschitz Condition is employed here to develop Sequential Sub-Predictors (SSP) for complex nonlinear systems. Compared with existing synthesis methods which involve Lipschitz Condition, Generalized Lipschitz condition relaxes the conservative of the stability results using some modified matrix inequalities. The proposed idea can be more attractive for unstable systems due to reduce significantly the order of the predictors. Also using the proposed method, the effect of the external disturbance can be minimized based on H∞ index. Afterward, a SSP-based controller is presented to illustrate the effectiveness of proposed method to stabilize nonlinear systems with input-delay. Finally, the predictor capability is investigated by simulation examples.

On semilocal convergence analysis for two-step Newton method under generalized Lipschitz conditions in Banach spaces

In the present paper, we consider the semilocal convergence issue of two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. As applications of our convergence results, a nonsymmetric algebraic Riccati equation arising from transport theory and a two-dimensional nonlinear convection-diffusion equation are provided.

Local convergence of the Gauss-Newton-Kurchatov method under generalized Lipschitz conditions

We investigate the local convergence of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems. This method is a combination of Gauss-Newton and Kurchatov methods and it is used for problems with the decomposition of the operator. The convergence analysis of the method is performed under the generalized Lipshitz conditions. The conditions of convergence, radius and the convergence order of the considered method are established. Given numerical examples confirm the theoretical results.

Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

Integrability of the Fourier–Jacobi transform of functions satisfying Lipschitz and Dini–Lipschitz-type estimates

ABSTRACT The purpose of this work is to prove analogues of the classical Titchmarsh theorem and Younis' theorem associated with the Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition of a certain order in suitable weighted spaces , . For this purpose, we use a generalized translation operator defined by Flensted-Jensen and Koornwinder.

On Goursat problem for fuzzy random partial differential equations under generalized Lipschitz conditions

Fuzzy random partial differential equations (PDEs) present a connection between random dynamical systems with nonstatistical inexactness data. These blended models could be efficiently used in modeling dynamical systems working in vagueness and ambiguity environments such as fuzzy random adaptive control, fuzzy random financial prediction, fuzzy random biological modeling, etc. In this article, we study Goursat problem for fuzzy random wave equations in the framework of generalized complete metric spaces in the sense of Luxemburg. We consider equations under generalized Hukuhara differentiability. The force functions are constrained by generalized Lipschitz conditions, that makes the range of PDEs types wider than using unbounded and locally Lipschitz conditions. The existence, uniqueness and boundedness of fuzzy solutions are investigated by employing Picard successive approximation method and Luxemburg fixed point theorem. Some illustrated examples are given to demonstrate for theoretical results.

On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.