Uniform Lipschitz Research Articles

Non-smooth integrability theory

We study a method for calculating the utility function from a candidate of a demand function that is not differentiable, but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first concerns the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and that this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is compact under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. We also show a similar result on the topology of pointwise convergence.

Boas Type Results for Two-Sided Quaternion Fourier Transform and Uniform Lipschitz Spaces

Boas Type Results for Two-Sided Quaternion Fourier Transform and Uniform Lipschitz Spaces

MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

In this paper, we calculate the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is the standard [Formula: see text]-transformation with [Formula: see text], [Formula: see text] is a positive function on [Formula: see text] and [Formula: see text] is the usual metric on the torus [Formula: see text]. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let [Formula: see text] be a [Formula: see text] non-singular matrix with real coefficients. Then, [Formula: see text] determines a self-map of the [Formula: see text]-dimensional torus [Formula: see text]. For any [Formula: see text], let [Formula: see text] be a positive function on [Formula: see text] and [Formula: see text] with [Formula: see text]. We obtain the Hausdorff dimension of the fractal set [Formula: see text] where [Formula: see text] is a hyperrectangle and [Formula: see text] is a sequence of Lipschitz vector-valued functions on [Formula: see text] with a uniform Lipschitz constant.

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.

Normal families and quasiregular mappings

AbstractBeardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

Reiterated homogenization of elliptic systems with Neumann boundary conditions

This paper is devoted to the quantitative estimates in reiterated homogenization of second‐order elliptic systems of linear elasticity, where the coefficients oscillate on multiple separated scales. We establish the uniform boundary Lipschitz estimate and the uniform estimate for the problem. The convergence rate in the space is also obtained as a by‐product.

Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Let Ω⊂RN be an open set. In this work we consider solutions of the following gradient elliptic system −div(A(x)∇ui,β)=fi(x,ui,β)+a(x)β|ui,β|γ−1ui,β∑j=1j≠il|uj,β|γ+1,for i=1,…,l. We work in the competitive case, namely β<0. Under suitable assumptions on A, a, fi and on the exponent γ, we prove that uniform L∞–bounds on families of positive solutions {uβ}β<0={(u1,β,…,ul,β)}β<0 imply uniform Lipschitz bounds (which are optimal).One of the main points in the proof are suitable generalizations of Almgren’s and Alt–Caffarelli–Friedman’s monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case A(x)=Id (i.e. the operator is the Laplacian) was treated.

Mean‐square exponential input‐to‐state stability of stochastic delayed recurrent neural networks with local Lipschitz condition

We consider the global existence of solutions and input‐to‐state stability for a class of stochastic delayed recurrent neural networks without uniform Lipschitz condition. Under local Lipschitz condition, we find new sufficient conditions that ensure the solutions of given neural networks exist globally and are mean‐square exponentially input‐to‐state stable. Furthermore, we highlight the advantages of our novel results by comparing with some known results as well as a numerical example.

The Uniform Lipschitz Continuity of Eigenvalues of Sturm–Liouville Problems with Respect to the Weighted Function

The present paper is concerned with the uniform boundedness of the normalized eigenfunctions of Sturm–Liouville problems and shows that the sequence of eigenvalues is uniformly local Lipschitz continuous with respect to the weighted functions.

Boas Type and Titchmarsh Type Theorems for Generalized Fourier-Bessel Transform

Let $$n\in \mathbb Z_{+}=\{0,1,\dots \}$$ , $$\nu >-1/2$$ . For a function f in the space $$L^{1}_{n,\nu }(\mathbb R_{+})$$ of special kind, we consider the generalized Fourier-Bessel transform $$\mathcal F_{n,\nu }(f)$$ . We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of $$\mathcal F_{n,\nu }(f)$$ . Also an analogue of classical Titchmarsh theorem describing Lipschits classes in $$L^{2}$$ space with power weight is established in a more general setting.

Uniform Lipschitz estimates up to the boundary for singular perturbation problem for some nonlinear elliptic PDEs with unbounded ingredients

We prove uniform up to the boundary gradient estimates for one phase nonlinear inhomogeneous singular perturbation problems with unbounded measurable ingredients governed by fully nonlinear elliptic equations. We present similar results for quasilinear PDEs with bounded RHS. Our proof is based on the Lipschitz regularity up to the boundary for free boundary problems (FBP) found in Braga and Moreira (2022).

Hausdorff dimensions of recurrent and shrinking target sets under Lipschitz functions for expanding Markov maps

Let T be an expanding Markov map with a repeller E defined on . This paper concerns the Hausdorff dimension of the sets and where is a sequence of Lipschitz functions with a uniform Lipschitz constant, , f is a positive continuous function on , is the sum and ψ is a positive function defined on . The results can be applied to cookie-cutter dynamical systems and continued fraction dynamical systems, etc.

Diagonalization in a quantum kicked rotor model with non-analytic potential

In this paper we study the lattice quasi-periodic operators with power-law long-range hopping and meromorphic monotone potentials, and diagonalize the operators via a Nash-Moser iteration scheme. As applications, we obtain uniform power-law localization, uniform dynamical localization and Lipschitz continuity of the integrated density of states (IDS) for such operators. Our main motivation comes from investigating quantum suppression of chaos in a quantum kicked rotor model with non-analytical potential.

Parabolic homogenization with an interface

This paper considers a family of second-order parabolic equations in divergence form with rapidly oscillating and time-dependent periodic coefficients and an interface between two periodic structures. Following a framework initiated by Blanc, Le Bris and Lions and a generalized two-scale expansion in divergence form of elliptic homogenization with an interface by Josien, we can determine the effective (or homogenized) equation with the coefficient matrix being piecewise constant and discontinuous across the interface. Moreover, we obtain the $O(\varepsilon)$ convergence rates in $L^{2(d+2)/{d}}_{x,t}$ with $\varepsilon$-smoothing method and the uniform interior Lipschitz estimates via compactness argument.

The Existence, Uniqueness, and Carathéodory’s Successive Approximation of Fractional Neutral Stochastic Differential Equation

The existence, uniqueness, and Carathe´odory’s successive approximation of the fractional neutral stochastic differential equation (FNSDE) in Hilbert space are considered in this paper. First, we give the Carathe´odory’s approximation solution for the FNSDE with variable time delays. We then establish the boundedness and continuity of the mild solution and Carathe´odory’s approximation solution, respectively. We prove that the mean-square error between the exact solution and the approximation solution depends on the supremum of time delay. Next, we give the Carathe´odory’s approximation solution for the general FNSDE without delay. Under uniform Lipschitz condition and linear growth condition, we show that the proof of the convergence of the Carathe´odory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution. Furthermore, under the non-Lipschitz condition, which is weaker than Lipschitz one, we establish the existence and uniqueness theorem of the solution for the FNSDE based on the Carathe´odory’s successive approximation. Finally, a simulation is given to demonstrate the effectiveness of the proposed methods.

Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity

In this manuscript we propose a structural condition on nonseparable Hamiltonians, which we term displacement monotonicity condition, to study second-order mean field games master equations. A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry–Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.

Lipschitz stability analysis of fractional-order impulsive delayed reaction-diffusion neural network models

In this paper, the concept of Lipschitz stability is introduced to impulsive delayed reaction-diffusion neural network models of fractional order. Such networks are an appropriate modeling tool for studying various problems in engineering, biology, neuroscience and medicine. Fractional derivatives of Caputo type are considered in the model. The effects of impulsive perturbations and delays are also under consideration. Lipschitz stability analysis is performed and sufficient conditions for global uniform Lipschitz stability of the model are established. The Lyapunov function approach combined with the comparison principle are employed in the development of the main results. The proposed criteria extend some existing stability results for such models to the Lipschitz stability case. The introduced concept is also very useful in numerous inverse problems.

Zaman Skalasında Fonksiyonel Dinamik Denklemlerin Alternatif Parametreler Değişimi ile h-Kararlılığı

In this paper, we concentrate on nonlinear functional dynamic equations of the form&#x0D; &#x0D; x^∆ (t)=a(t)x(t)+f(t,x(t)), t∈T,&#x0D; &#x0D; on time scales and study h-stability, which implies uniform exponential stability, uniform Lipschitz stability, or uniform stability in particular cases. In our analysis, we use an alternative variation of parameters, which enables us to focus on a larger class of equations since the dynamic equations under the spotlight are not necessarily regressive. Also, we establish a linkage between uniform boundedness and h-stability notions for solutions of dynamic equations under sufficient conditions in addition to our stability results.

Compactness and large-scale regularity for Darcy's law

This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior C1,α and Lipschitz estimates for the velocity as well as the corresponding estimates for the pressure. These estimates, when combined with the classical regularity estimates for the Stokes equations, yield the uniform Lipschitz estimates. As a consequence, we also obtain the uniform Wk,p estimates for 1<p<∞.

Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

Consider the class of optimal partition problems with long range interactions $$\begin{aligned} \inf \left\{ \sum _{i=1}^k \lambda _1(\omega _i):\ (\omega _1,\ldots , \omega _k) \in \mathcal {P}_r(\Omega ) \right\} , \end{aligned}$$ where $$\lambda _1(\cdot )$$ denotes the first Dirichlet eigenvalue, and $$\mathcal {P}_r(\Omega )$$ is the set of open k-partitions of $$\Omega $$ whose elements are at distance at least r: $${{\,\mathrm{dist}\,}}(\omega _i,\omega _j)\ge r$$ for every $$i\ne j$$ . In this paper we prove optimal uniform bounds (as $$r\rightarrow 0^+$$ ) in $$\mathrm {Lip}$$ –norm for the associated $$L^2$$ -normalized eigenfunctions, connecting in particular the nonlocal case $$r>0$$ with the local one $$r \rightarrow 0^+$$ . The proof uses new pointwise estimates for eigenfunctions, a one-phase Alt–Caffarelli–Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas, combined with elliptic and energy estimates. Our result extends to other contexts, such as singularly perturbed harmonic maps with distance constraints.