Monotone Type Research Articles

Weighted (p,q)-equations with gradient dependent reaction

We consider a weighted (p,q)-equation with a parametric reaction depending on the gradient. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter, the problem has a positive smooth solution. For more information see https://ejde.math.txstate.edu/Volumes/2025/06/abstr.html

Singular double phase problems with convection

We consider a Dirichlet problem driven by the double phase differential operator and a parametric reaction which has the combined effects of a singular term and of a convective perturbation. Using nonlinear operators of monotone type, truncation and comparison techniques, and fixed point theory, we show that for all small values of the parameter, the problem has a bounded positive solution.

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

Pairs of Positive Solutions for a Carrier p(x)-Laplacian Type Equation

The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p(x)-Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč’s version of the celebrated Díaz–Saá’s inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each “positive bump” of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed.

Extremal values-based aggregation functions

We introduce and study aggregation functions based on extremal values, namely extended (l,u)-aggregation functions whose outputs only depend on a fixed number l of extremal lower input values and a fixed number u of extremal upper input values, independently of the arity of the input n-tuples (n≥l+u). We discuss several general properties of (l,u)-aggregation functions and we study special (l,u)-aggregation functions with neutral element, including t-conorms, t-norms and uninorms. We also study (l,u)-aggregation functions defined by means of integrals with respect to discrete fuzzy measures, as well as (l,u)-ordered weighted quasi-arithmetic means based on appropriate weighting vectors. We also stress some generalizations based on recently introduced new types of monotonicity. Some possible applications are sketched, too.

The Convergence and Boundedness of Solutions to SFDEs with the G-Framework

Generally, stochastic functional differential equations (SFDEs) pose a challenge as they often lack explicit exact solutions. Consequently, it becomes necessary to seek certain favorable conditions under which numerical solutions can converge towards the exact solutions. This article aims to delve into the convergence analysis of solutions for stochastic functional differential equations by employing the framework of G-Brownian motion. To establish the goal, we find a set of useful monotone type conditions and work within the space Cr((−∞,0];Rn). The investigation conducted in this article confirms the mean square boundedness of solutions. Furthermore, this study enables us to compute both LG2 and exponential estimates.

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

Positive solutions for singular problems with multivalued convection

We consider an elliptic Dirichlet problem, driven by a general nonlinear, nonhomogeneous differential operator and a reaction that has the competing effects of a parametric singular term and of a multivalued perturbation which is gradient dependent (convection). Using truncations, comparisons and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter λ>0, the problem has a positive and smooth solution.

C-functions in higher-derivative flows across dimensions

In the context of gravitational theories describing renormalization group flows across dimensions via AdS/CFT, we study the role of higher-derivative corrections to Einstein gravity. We use the Null Energy Condition to derive monotonicity properties of candidate holographic central charges formed by combinations of metric functions. We also implement an entropic approach to the characterization of the four-derivative flows using the Jacobson-Myers functional and demonstrate, under reasonable conditions, monotonicity of certain terms in the entanglement entropy via the appropriate generalization of the Ryu-Takayanagi prescription. In particular, we show that any flow from a higher dimensional theory to a holographic CFT2 satisfies a type of monotonicity. We also uncover direct relations between NEC-motivated and entropic central charges.

Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

Abstract We investigate the discrete equations of the form Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . \Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}. Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.

Parameterized transformations and truncation: When is the result a copula?

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Discretization-Free Particle-Based Taxi Dispatch Methods With Network Flow Decomposition

The increasing popularity of on-demand ride-hailing services has introduced research problems, including those related to fleet management. Many existing fleet management approaches discretize a spatiotemporal distribution to reduce the problem complexity. Although spatial discretization greatly simplifies the problems, this approximation results in a suboptimal solution. To address this limitation, we present a novel spatial discretization-free, particle-based formulation of fleet management. Our model requires only temporal discretization. The nodes in our spatiotemporal network model represent the positions of the fleet and ride requests over the time horizon. Under this representation, a taxi dispatch problem can be solved by the minimum-cost maximum-flow (MCMF) algorithm. We show some structural properties that disclose a type of monotonicity by applying the MCMF algorithm to our problem. The obtained dispatch solution maximizes the number of served requests from these structural properties at least at the most current time step, regardless of the demand prediction quality. Moreover, these properties enable the decomposition of the MCMF algorithm without sacrificing optimality, and the decomposed multistage maximum-flow algorithm is significantly faster than the MCMF algorithm. Furthermore, we propose an efficient method that alleviates the dearth of input data for service providers who lack sufficient computing power to calculate substantial travel time information. We demonstrate the performance of our method by using data collected in New York City, USA. Our simulation studies indicate that a service provider can fulfill 74–85% of the ride requests within 2 minutes.

Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid hp-version discontinuous Galerkin finite element method from Congreve et al. [1] for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an hp-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed hp-adaptive two-grid method.

The Stochastic Gierer–Meinhardt System

The Gierer–Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed different activator–inhibitor systems with varying diffusion rates in his pioneering work, which could trigger the emergence of such cell structures. Mathematically, one describes these activator–inhibitor systems as coupled systems of reaction-diffusion equations with different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer–Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer–Meinhardt system with homogeneous Neumann boundary conditions on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer–Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the initial conditions, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.

Probabilistic Transformations of Quantum Resources.

The difficulty in manipulating quantum resources deterministically often necessitates the use of probabilistic protocols, but the characterization of their capabilities and limitations has been lacking. We develop a general approach to this problem by introducing a new resource monotone that obeys a very strong type of monotonicity: it can rule out all transformations, probabilistic or deterministic, between states in any quantum resource theory. This allows us to place fundamental limitations on state transformations and restrict the advantages that probabilistic protocols can provide over deterministic ones, significantly strengthening previous findings and extending recent no-go theorems. We apply our results to obtain a substantial improvement in bounds for the errors and overheads of probabilistic distillation protocols, directly applicable to tasks such as entanglement or magic state distillation, and computable through convex optimization. In broad classes of resources, we strengthen our results to show that the monotone completely governs probabilistic transformations-it serves as a necessary and sufficient condition for state convertibility. This endows the monotone with a direct operational interpretation, as it can exactly quantify the highest fidelity achievable in resource distillation tasks by means of any probabilistic manipulation protocol.

О КОЛЕБАНИЯХ ФУНКЦИОНАЛЬНО-ГРАДИЕНТНЫХ ЭЛЕКТРОУПРУГИХ СТЕРЖНЕЙ

The problems of steady-state vibrations of functionally graded electroelastic rods for two types of polarization were investigated. The functional gradient is characterized by a change along the longitudinal coordinate of the isothermal elastic compliance and the piezoelectric modulus. The linear, quadratic and exponential laws of inhomogeneity are used in the work. To simulate damping, a model of a standard viscoelastic body was used, which is used within the framework of the concept of complex modules. For the numerical solution of the tasks set, the shooting method was applied. In order to verify the computational scheme, an exact solution of the problem is constructed for the case of constant properties. An analysis of the influence of the laws of inhomogeneity was carried out for laws that have the same mean integral values. The amplitude-frequency characteristics of current and conductivity are constructed. Antiresonances are investigated, the presence of two types is established, depending on the laws of inhomogeneity. An asymptotic analysis of the problem is performed for the case of low frequencies. It is shown that in the low-frequency range, the mechanical longitudinal stress depends only on the law of change in the piezoelectric modulus, while the displacement also depends on the law of change in elastic compliance. As a result of computational experiments, the features of the current's amplitude-frequency characteristic (AFC) structure in the vicinity of the second resonance were revealed. The AFC has a different quality factor depending on whether the compliance functions and the piezoelectric modulus have the same or different types of monotonicity. It is revealed that the first resonance from the frequency range under consideration has low sensitivity to the laws of inhomogeneity, and the third resonance is affected sufficiently by the inhomogeneity laws and can be used to determine the type of monotonicity when solving inverse problems on reconstructing properties.

Dirichlet Problems with Unbalanced Growth and Convection

We consider a double phase Dirichlet problem with a gradient dependent reaction term (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a bounded strictly positive solution. Moreover, we show that the set of these solutions is compact in the corresponding generalized Sobolev–Orlicz space.

Generalization of the Hardy-Littlewood theorem on Fourier series

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

Boundary growth of Sobolev functions of monotone type for double phase functionals

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).