Lipschitz-type Condition Research Articles

Controllability of discrete-time semilinear Riemann–Liouville-like fractional equations

This article utilizes the Riemann–Liouville-like fractional operator to investigate certain sufficient conditions for the approximate controllability of discrete-time fractional evolution equations. We describe our main results utilizing a sequential approach, the theory of difference equations, and the connection between a class of sequences of operators and C0-semigroups. On the nonlinear term, we impose the Lipschitz-type condition. Finally, a few examples are provided to demonstrate how the outcomes might be used.

Convergence of the Gauss-Newton method for convex composite optimization problems under majorant condition on Riemannian manifolds

In this paper, we consider convex composite optimization problems on Riemannian manifolds, and discuss the semi-local convergence of the Gauss-Newton method with quasi-regular initial point and under the majorant condition. As special cases, we also discuss the convergence of the sequence generated by the Gauss-Newton method under Lipschitz-type condition, or under γ-condition.

Bipartite Tracking Formation Control of Nonlinear Multi-Agent Systems Using Adaptive Output–Feedback Protocols

This paper proposed distributed adaptive formation control for leader-follower bipartite time-varying formation (BTVF) of a nonlinear multi-agent system (MAS). The proposed nonlinear MAS satisfies the one-sided Lipschitz-type condition. In the topological graphs with directed spanning trees, the design of adaptive protocols does not depend on the known communication topology, which can avoid the use of global information. Given limited information, the proposed MAS can achieve desired formation tracking by utilizing an observer protocol and converting the bipartite formation control problem into a stability problem of system errors. The analysis results of the systematic errors by using Lyapunov candidate functions, indicate that the MAS can be globally stable with a certain convergence rate during operation. Finally, numerical simulations are presented to confirm the validity of the proposed approach.

On fixed point approach to equilibrium problem

The equilibrium problem defined by the Nikaidô–Isoda–Fan inequality contains a number of problems such as optimization, variational inequality, Kakutani fixed point, Nash equilibria, and others as special cases. This paper presents a picture for the relationship between the fixed points of the Moreau proximal mapping and the solutions of the equilibrium problem that satisfies some kinds of monotonicity and Lipschitz-type condition.

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

Existence results and two step proximal point algorithm for equilibrium problems on Hadamard manifolds

"In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic upper sign continuity of the equilibrium bifunction and under different kinds of coercivity conditions. We also study the existence of solutions of the equilibrium problems under properly quasimonotonicity of the equilibrium bifunction. We propose a two-step proximal point algorithm for solving equilibrium problems in the setting of Hadamard manifolds. The convergence of the proposed algorithm is studied under the strong pseudomonotonicity and Lipschitz-type condition. The results of this paper either extend or generalize several known results in the literature."

Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces

Abstract In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

Analysis of the Energy Stability for Stabilized Semi-implicit Schemes of the Functionalized Cahn-Hilliard Mass-conserving Gradient Flow Equation

A stabilized semi-implicit scheme was designed in [3] to solve the Functionalized Cahn-Hilliard (CCH) equation, but there is a lack of theoretical analysis of the energy stability. In this paper, we generalize this scheme to solve the general FCH mass-conserving gradient flow (FCH-MCGF) equation and show the theoretical analysis results about the unique solvability and energy stability. We successfully prove that this scheme is uniquely solvable and energy stable in theory by rewriting the double-well potential function to satisfy the Lipschitz-type condition. The range of stabilization parameters is theoretically given as well. In addition, another similar energy stable scheme is proposed, which slightly widens the range of stabilization parameters in theory and has almost the same precision as the previous one. Both the detailed numerical procedure and the selection of stabilization parameters are presented. Finally, several numerical experiments are performed for the FCH-MCGF equation based on these schemes. Specially, the adaptive time step size is considered in the scheme for the simulations of the phase separation in 2D and 3D, since any time step size can be used according to our theoretical results. Numerical results show that these schemes are energy stable and the large time step size indeed can be used in computations. Moreover, by comprehensive comparisons of stability and accuracy among the stabilized semi-implicit scheme, the convex splitting scheme, and the fully implicit scheme, we conclude that the performance of the stabilized semi-implicit scheme is the best, and the convex splitting scheme performs better than the fully implicit scheme.

An Accelerated Popov’s Subgradient Extragradient Method for Strongly Pseudomonotone Equilibrium Problems in a Real Hilbert Space With Applications

In this paper, we introduce a subgradient extragradient method to find the numerical solution of strongly pseudomonotone equilibrium problems with the Lipschitz-type condition on a bifunction in a real Hilbert space. The strong convergence theorem for the proposed method is precisely established on the basis of the standard cost bifunction assumptions. The application of our convergence results is also considered in the context of variational inequalities. For numerical analysis, we consider the well-known Nash-Cournot oligopolistic equilibrium model to support our well-established convergence results.

Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.

New extragradient methods for solving equilibrium problems in Banach spaces

In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the $$\phi $$ -proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means that the Lipschitz-type constant is an input parameter of the algorithm while the next two algorithms are modified such that they can work without any information of the Lipschitz-type constant, and then they can be implemented more easily. Some convergence theorems are proved under mild conditions. Our results extend and enrich existing algorithms for solving equilibrium problem in Banach spaces. The numerical behavior of the new algorithms is also illustrated via several experiments.

On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers

In this paper, a uni-parametric class of third-order iterative algorithms for solving systems of nonlinear equations is proposed. The local convergence of the suggested schemes is studied using generalized Lipschitz-type condition on the first-order Frechet derivative. Furthermore, we analyze the numerical stability of the new methods applying complex dynamics tool. The nonlinear systems related to the equation of molecular interaction, a boundary value problem, the integral equation from Chandrasekhar’s work, etc. are discussed. The most interesting fact about the proposed third-order family is that it generates a super convergent scheme (for $$\gamma =2$$ ) for solving quadratic nonlinear systems (QNS). This particular method produces much better results for QNS in comparison with other third-order schemes. Also, it is observed that the approximate computational order of convergence (ACOC) of this scheme (for $$\gamma =2$$ ) is approximately four (3.99–4.00) while solving QNS.

Lipschitz Type Characterizations for Bergman-Orlicz Spaces and Their Applications

We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.

A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application

In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method.

Extragradient-like method for pseudomontone equilibrium problems on Hadamard manifolds

This paper presents an extragradient-like method for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition on Hadamard manifolds. The algorithm only needs to know the existence of the Lipschitz-type constants of the bifunction, and the stepsize of each iteration is determined by the adjacent iterations. Convergence of the algorithm is analyzed, and its application to variational inequalities is also provided. Finally, several experiments are made to verify the effectiveness of the algorithms.

A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others.

Accelerated hybrid methods for solving pseudomonotone equilibrium problems

In this paper, we introduce some new accelerated hybrid algorithms for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a Hilbert space. The algorithms are constructed around the extragradient method, the inertial technique, the hybrid (or outer approximation) method, and the shrinking projection method. The algorithms are designed to work either with or without the prior knowledge of the Lipschitz-type constants of bifunction. Theorems of strong convergence are established under mild conditions. The results in this paper generalize, extend, and improve some known results in the field. Finally, several of numerical experiments are performed to support the obtained theoretical results.

Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings

The paper considers the problem of finding common solutions of a system of pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. The problem covers various mathematical models of convex feasibility problems and the problems whose constraints are expressed by the intersection of fixed point sets of mappings. The main purpose of the paper is to design and improve computations over each step and weaken several assumptions imposed on bifunctions and mappings. Two parallel algorithms for finding of a particular solution of the problem are proposed in Hilbert spaces where each subproblem in the family can be computed simultaneously. The first one is a modified hybrid method which combines three methods including the generalized gradient-like projection method, the Mann’s iteration and the hybrid (outer approximation) method. This algorithm improves the hybrid extragradient method at each computational step where only one optimization problem is solved for each equilibrium subproblem in the family and the hybrid step does not deal with the feasible set of the considered problem. The strong convergence of the algorithm comes from the hybrid method under the Lipschitz-type condition of bifunctions. The second algorithm is a viscosity-like method with a linesearch procedure that aims to avoid the Lipschitz-type condition imposed on bifunctions. With the incorporated viscosity technique, the algorithm also provides strong convergence. Several numerical experiments are performed to illustrate the efficiency of the proposed algorithms and also to compare them with known parallel hybrid extragradient methods.

Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition of a bifunction. A strong convergence result is provided under some mild conditions, and an iterative sequence is accomplished without previous knowledge of the Lipschitz-type constants of a cost bifunction. A sufficient explanation is that the method operates with a slow-moving stepsize sequence that converges to zero and non-summable. For numerical explanations, we analyze a well-known equilibrium model to support our well-established convergence result, and we can see that the proposed method seems to have a significant consistent improvement over the performance of the existing methods.