Lipschitz Continuous Research Articles

Existence of positive solutions for generalized quasilinear Schrödinger equations with Sobolev critical growth

In this paper, we are devoted to establishing that the existence of positive solutions for a class of generalized quasilinear elliptic equations in R N with Sobolev critical growth, which have appeared from plasma physics, as well as high-power ultrashort laser in matter. To begin, by changing the variable, quasilinear equations are transformed into semilinear equations. The positive solutions to semilinear equations are then presented using the Mountain Pass Theorem for locally Lipschitz functionals and the Concentration-Compactness Principle. Finally, an inverse translation reveals the presence of positive solutions to the original quasilinear equations.

ДЕЯКІ АСИМПТОТИЧНІ ВЛАСТИВОСТІ РОЗВ’ЯЗКІВ ТРИГАРМОНІЙНИХ РІВНЯНЬ

The author considers the optimization problem for the triharmonic equation in the presence of specific boundary conditions. As a result, the triharmonic Poisson integral was constructed in Cartesian coordinates for the upper half-plane. The asymptotic properties of this operator on Lipschitz classes in a uniform metric were studied. An exact equality was found for the upper bound of the deviation of the Lipschitz class functions from the triharmonic Poisson integral defined in Cartesian coordinates for the upper half-plane in the metric space. The results obtained in the article demonstrate the connection between the methods of approximation theory and the principles of optimal decision theory. Keywords: optimization problem, class of Lipschitz functions, uniform metric, triharmonic Poisson integral.

Characterization of Lipschitz Functions on Ball Banach Function Spaces

Characterization of Lipschitz Functions on Ball Banach Function Spaces

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Random dynamics of an SIV epidemic model

Vaccines train the immune system to create antibodies when it has been exposed to a disease. They protect against many different diseases, including Hepatitis B, Influenza, Polio, etc. This paper considers the vaccination process by including individuals who have started the vaccination process in the transmission model. By the non-parametric perturbation technique, we shall formulate the stochastic SIV model. The examination of the deterministic model relies on the global Lipschitz condition, a crucial factor in guaranteeing the uniqueness and stability of solutions to the model’s differential equation. In contrast, our scrutiny of the stochastic model is rooted in the local Lipschitz condition, which proves to be more applicable and realistic in the context of real-world scenarios. Analysis of the stochastic models may be superior to the analysis of the deterministic models. In many cases, surprisingly, analysis of the stochastic models may provide results that are close to the results obtained from the analysis of the deterministic models. However, it comes with a cost, namely, we might not predict the pattern or the random probability distribution of the stochastic system, moreover, the necessary conditions for the extinction of the disease are weaker compared to the deterministic model. The impact of the continuous vaccination strategy on disease spread is demonstrated. Our results reveal that the vaccination process is very helpful and useful in eradicating the disease. Using MATHEMATICA software and MATLAB software, stability regions of the disease-free equilibrium are presented.

New results on dynamic output state feedback stabilization of some class of time-varying nonlinear Caputo derivative systems

In modern control theory applications, input–output feedback often arises and plays a crucial role in stabilizing any target control systems. However, in many practical applications, the knowledge of full information on the state of systems may not be available in general due to measurements. In order to control such types of systems, this paper uses a universal concept of real order calculus to stabilize by the method of dynamic output feedback control. Is it possible to design a dynamic output feedback stabilization for time-varying real order control systems? This work develops a new dynamic output feedback control method and proposes linear homogeneous time-varying control law to stabilize incommensurate nonlinear time-varying real order systems under the action of Caputo derivative with an addendum of ultimate random initial-time. We utilize the ideas of the comparison method and establish order-dependent asymptotic results associated with uniform Lipschitz continuous functions that provide stabilization by means of the dynamic output state feedback control method. We provide three examples that include a practical system and show the importance of some theoretical results in the demonstration to stabilize the problems by means of proposed control method.

Distributed algorithm for nonsmooth multi-coalition games and its application in electricity markets

In this paper, we study distributed nonsmooth multi-coalition games (MCGs), where the players are subject to local convex set constraints and coupled inequality constraints. Different from existing distributed MCGs, our problem involves nonsmooth payoff functions and constraints, relaxes the requirement for communication networks, and only relies on the strict monotonicity of pseudo-gradients. Due to the nonsmoothness of payoff functions and constraints, the weaker communication networks as well as the coexistence of cooperation and competition among players, existing generalized Nash equilibrium (GNE) seeking algorithms cannot solve the problem. Also, they pose obstacles to the algorithm design and analysis, mainly because of the non-global Lipschitz continuity of subgradients and the unconnectedness of subnetworks. To seek the variational GNE (vGNE) of the nonsmooth MCGs, we design a distributed subgradient-based algorithm. We prove that the algorithm converges to the exact vGNE of the nonsmooth MCGs from any initial states. Finally, our result is applied to the electricity market games (EMGs) of smart grids.

Maximal polynomial modulations of singular Radon transforms

We prove L2→Lp estimates on the torus for maximal polynomial modulations of Calderón-Zygmund operators with anisotropic scaling. We obtain improved constants in these estimates. As a corollary, maximal polynomial modulations of a mollified version of the Hilbert transform along the parabola are bounded with only logarithmic dependence of the estimate on the Lipschitz constant of the mollifier.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

An extension of the Cheney-Sharma operator of the first kind

We extend the Cheney-Sharma operators of the first kind using Stancu type technique and we study some approximation properties of the new operator. We calculate the moments, we study local approximation with respect to a K-functional and the preservation of the Lipschitz constant and order.

Event‐triggered adaptive consensus control of nonlinear multi‐agent systems with unknown control directions and actuator saturation

AbstractThis article addresses the event‐triggered adaptive consensus control of nonlinear multi‐agent systems with unknown control direction and actuator saturation. A new robust adaptive control algorithm based on an event‐triggered mechanism is designed. The smooth Lipschitz function approximates the saturated nonlinear function, while the Nussbaum function handles unknown control directions and residual terms. The event‐triggered mechanism is designed to determine the time of communication, significantly reducing the communication burden. An additional estimator is utilized to deal with unknown parameters involved in neighbor dynamics and prevent information exchange to consistency errors between connected subsystems. The results show that all the signals of the closed‐loop system are uniformly bounded, and the consensus tracking error converges to a bounded set. Meanwhile, Zeno's behavior is eliminated. Simulation results confirm the superiority of the proposed method.

Observer-Based Adaptive Consensus Protocol Design for Multi-Agent Systems with One-Sided Lipschitz Nonlinearity

This paper considers the observer-based leader–follower consensus problem of multi-agent systems with one-sided Lipschitz conditions and quadratic inner-boundedness nonlinearity. Based on the relative outputs of neighboring agents, an adaptive protocol over a directed graph is designed via assigning a time-varying coupling weight to each node. Compared with the existing protocols, the proposed protocol can not only be utilized in the unidirectional flow of information, but it does not require any global graph information for its design. It has been shown that the established criteria can not only make the observer error tendency zero, but one can also achieve the leader–follower consensus of such nonlinear multi-agent systems. Furthermore, the gains of observer and protocol parameters can be constructed by solving the linear matrix inequalities (LMIs), which are conveniently checked. The results are illustrated by numerical simulations.

On the Newton-Broyden method for solving systems of nonlinear equations

The Newton-Broyden method for solving systems of nonlinear equations with operator decomposition is studied in this paper. A local convergence analysis is provided under classical Lipschitz conditions and results of numerical experiments are presented. The effectiveness of this method in solving various types of nonlinear systems is demonstrated.

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

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

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

In this paper, we consider a global optimization problem where the objective function is assumed to be Lipschitz-continuous with an unknown Lipschitz constant. Building upon the recently introduced BIRECT (BIsection of RECTangles) algorithm, we propose a new diagonal partitioning and sampling scheme. Our framework, named BIRECT-V (V for vertices), combines bisection with the sampling of two points. In the initial hyper-rectangle, these points are located at 1/3 and 1 along the main diagonal. Unlike most DIRECT-type algorithms, where evaluating the objective function at vertices is not suitable for bisection, our strategy, when combined with bisection, provides more comprehensive information about the objective function. However, the creation of new sampling points may coincide with existing ones at shared vertices, resulting in additional evaluations of the objective function and increasing the number of function evaluations per iteration. To overcome this issue, we propose modifying the original optimization domain to obtain a good approximation of the global solution. Experimental investigations demonstrate that this modification positively impacts the performance of the BIRECT-V algorithm. Our proposal shows promise as a global optimization algorithm compared to the original BIRECT and two popular DIRECT-type algorithms on a set of test problems. It particularly excels at high-dimensional problems

Existence Solution for Fractional Mean-Field Backward Stochastic Differential Equation with Stochastic Linear Growth Coefficients

We deal with fractional mean field backwardWe deal with fractional mean field backward stochastic differential equations with hurst parameter $H\in (\frac{1}{2},1)$ when the coefficient $f$ satisfy a stochastic Lipschitz conditions, we prove the existence and uniqueness of solution and provide a comparison theorem. Via an approximation and comparison theorem, we show the existence of a minimal solution when the drift satisfies a stochastic growth condition.

Existence and convergence of best proximity points for generalized pseudo-contractive and Lipschitzian mappings via an Ishikawa-type iterative scheme

In this article, we prove the existence of the best proximity point for the class of nonself generalized pseudo-contractive and Lipschitzian mappings. Also, we approximate the best proximity point through the proposed Ishikawa’s iteration process for the case of nonself-mappings. Finally, we provide an example to illustrate our main result.

Formation control of multi-agent systems with actuator saturation via neural-based sliding mode estimators

In this paper, the formation control problem for second-order multi-agent systems with model uncertainties and actuator saturation is investigated. An estimator-based robust formation controller is developed to ensure boundedness of the system’s formation tracking error. To estimate uncertain factors without prior knowledge of their Lipschitz constants, two new estimator designs that combine the neural-based estimation process and the sliding mode technique are proposed. Finite-time stability is achieved by introducing a new terminal estimation surface. Regarding the formation controller design, we provide a new framework to study the combined effect of input saturation and input coupling. Both an adaptive compensator and an input regulation algorithm are employed to attenuate state fluctuation led by the reverse effect. Comparative simulations regarding a multi-robot system are conducted to demonstrate the effectiveness of the proposed estimators and the estimator-based controller.

Time-delayed generalized BSDEs

We prove the existence and uniqueness of the solution of a BSDE with time-delayed generators in the small delay setting (or equivalently small Lipschitz constant), which employs the Stieltjes integral with respect to an increasing continuous stochastic process. Moreover, we obtain a result of continuity of the solution with regard to the increasing process, assuming only uniform convergence, but not in variation. We also prove the existence in the case of arbitrary delay by imposing monotonicity and linearity on generators. Lastly, we provide an application of the theoretical framework within an insurance based example.

Convexificators for nonconvex multiobjective optimization problems with uncertain data: robust optimality and duality

In this paper, we investigate robust optimality conditions and duality for a class of nonconvex multiobjective optimization problems with uncertain data in the worst case by the upper semi-regular convexificator. The Fermat principle for a locally Lipschitz function is presented in terms of the upper semi-regular convexificator. We establish robust necessary optimality conditions of the Fritz-John type and KKT type for the uncertain nonconvex multiobjective optimization problems. In addition, robust sufficient optimality conditions as well as saddle point conditions are derived under the generalized ∂ ˆ ∗ -pseudoquasiconvexity and generalized convexity, respectively. The robust duality relations between the original problem and its mixed robust dual problem are obtained under a generalized pseudoconvexity assumption.