Monotonicity Condition Research Articles

WELL-POSEDNESS RESULTS FOR GENERAL REACTION–DIFFUSION TRANSPORT OF OXYGEN IN ENCAPSULATED CELLS

Abstract We provide well-posedness results for nonlinear parabolic partial differential equations (PDEs) given by reaction–diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption, monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us from applying classical results on strong solutions. However, by directly applying a Galerkin method, we obtain uniqueness and existence of the strong form solution. These results provide the basis to study the dynamics of cells in critical states.

Rank-preserving multidimensional mechanisms: An equivalence between identical-object and heterogeneous-object models

We show that the mechanism-design problem for a monopolist selling multiple, heterogeneous objects to a buyer with ex ante symmetric and additive values is equivalent to the mechanism-design problem for a monopolist selling identical objects to a buyer with decreasing marginal values. We derive three new results for the identical-objects model: (i) a new condition for revenue monotonicity of stochastic mechanisms, (ii) a sufficient condition on priors, such that prices in optimal deterministic mechanism are not increasing, and (iii) a simplification of incentive constraints for deterministic mechanisms. We use the equivalence to establish corresponding results in the heterogeneous-objects model.

On Banach Contraction Principle for Generalized Fuzzy Variational Inequality Problems

The main objective of the paper is to study the existence theorem of the newly defined generalized fuzzy ψ-variational inequality problem, generalized fuzzy ψ-complementarity problem. Uniqueness of the problems are also studied. Existence of these problems are established with the help of unique point condition of η. The existence of the solution of generalized fuzzy ψ-variational inequality problems is studied with the help of Banach contraction principle. The existence of the solution of generalized fuzzy ψ-complementarity problem is established in η-invex space.Conclusion: Equality of the solution sets of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is shown using unique point condition of the bifunction η. Unique of the solution of generalized fuzzy ψ-variational inequality problem and generalized fuzzy ψ-complementarity problem is studied using strictly monotonicity condition. Existence theorems of the solution of generalized fuzzy ψ-variational inequality problem are shown using Banach contraction principle in the presence of Lipschitz continuous, strongly monotone mappings T_ψ and weakly monotonicity of η. Existence theorems of the solution of generalized fuzzy ψ-complementarity problem in the presence of condition C_0.

Generalized Nash Equilibrium Seeking for Noncooperative Game With Different Monotonicities by Adaptive Neurodynamic Algorithm.

This article proposes a novel adaptive neurodynamic algorithm (ANA) to seek generalized Nash equilibrium (GNE) of the noncooperative constrained game with different monotone conditions. In the ANA, the adaptive penalty term, which acts as trajectory-dependent penalty parameters, evolves based on the degree of constraints violation until the trajectory enters the action set of noncooperative game. It is shown that the trajectory of the ANA enters the action set in finite time benefited from the adaptive penalty term. Moreover, it is proven that the trajectory exponentially (or polynomially) converges to the unique GNE when the pseudo-gradient of cost function in noncooperative game satisfies strong (or "generalized" strong) monotonicity. To the best of our knowledge, this is the first time to study the polynomial convergence of GNE seeking algorithm. Furthermore, when the pseudo-gradient mentioned above satisfies monotonicity in general, based on Tikhonov regularization method, a new ANA for finding its ε -generalized Nash equilibrium ( ε -GNE) is proposed, and the related exponential convergence of the algorithm is established. Finally, the river basin pollution game and 5G base station location game are given as examples to showcase the algorithm's effectiveness.

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.

Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\in \\mathbb {N}$$\\end{document} be the number of urns/colors. At each time n, we distribute σn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _n$$\\end{document} balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} assuming some monotonicity and growth condition. The class R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {R}$$\\end{document} includes convex functions and the classical case f(x)=xα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x)=x^{\\alpha }$$\\end{document}, α>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >1$$\\end{document}. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.

Estimation of random functions proxying for unobservables

This article considers the model Y = M ( X , U ) where U is an unobservable continuously distributed scalar and M is monotonic with respect to U. It is assumed there is an observable scalar W satisfying the restriction T F W | X , U = T F W | U almost surely where T is a known functional and F A|B denotes the distribution of A|B. This article shows that M can be identified under a mild monotonicity condition. This result requires neither statistical independence between X and U nor X to be continuously distributed. The estimation problem is treated when T F A | B ≡ E [ A | B ] . The proposed pointwise estimator of M is asymptotically normally distributed under weak technical conditions. Furthermore, the rate of convergence in probability is equal to n − r / ( 2 r + d + 1 ) where d denotes the dimension of the continuously distributed components of X and r is a positive integer which relates to the smoothness of certain functions. A Monte Carlo experiment is conducted and reveals the benefits of the estimator in the presence of endogeneity. We apply our estimator to estimate the returns to schooling.

On a hemivariational inequality with non-monotone operator

We investigate the existence of generalized variational solutions to abstract hemivariational inequality driven by a coercive and locally Lipschitz operator with no further monotonicity conditions imposed on it. A finite dimensional Ritz type approximation scheme is derived. Applications to competitive systems driven by the generalized ( p , q ) - Laplacian with convection are given.

A fixed-time stable forward–backward dynamical system for solving generalized monotone inclusions

AbstractWe propose a forward–backward splitting dynamical system for solving inclusion problems of the form $$0\in A(x)+B(x)$$ 0 ∈ A ( x ) + B ( x ) in Hilbert spaces, where A is a maximal operator and B is a single-valued operator. Involved operators are assumed to satisfy a generalized monotonicity condition, which is weaker than the standard monotone assumptions. Under mild conditions on parameters, we establish the fixed-time stability of the proposed dynamical system. In addition, we consider an explicit forward Euler discretization of the dynamical system leading to a new forward backward algorithm for which we present the convergence analysis. Applications to other optimization problems such as constrained optimization problems, mixed variational inequalities, and variational inequalities are presented and some numerical examples are given to illustrate the theoretical results.

A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.

Coupled wave equations with monotone nonlinearity: existence of solution and controllability results

In this manuscript, we investigate a coupled semilinear system characterised by two wave equations in which the nonlinear function satisfies the monotonicity condition. Initially, the existence of a solution is established through the application of Schauder's fixed point theorem. Thereafter, the results pertaining to the exact controllability are demonstrated for two different cases by establishing a relation between the reachable set of the semilinear system and its linear counterpart. Additionally, an illustrative example is provided to showcase the practical relevance of our theoretical findings.

Population Lorenz-monotonic allocation schemes for TU-games

Sprumont (Games Econ Behav 2:378–394, 1990) introduces population monotonic allocation schemes (PMAS) and proves that every assignment game with at least two sellers and two buyers, where each buyer-seller pair derives a positive gain from trade, lacks a PMAS. In particular glove games lacks PMAS. We propose a new cooperative TU-game concept, population Lorenz-monotonic allocation schemes (PLMAS), which relaxes some population monotonicity conditions by requiring that the payoff vector of any coalition is Lorenz dominated by the corresponding restricted payoff vector of larger coalitions. We show that every TU-game having a PLMAS is totally balanced, but the converse is not true in general. We obtain a class of games having a PLMAS, but no PMAS in general. Furthermore, we prove the existence of PLMAS for every glove game and for every assignment game with at most five players.Additionally, we also introduce two new notions, PLMAS-extendability and PLMAS-exactness, and discuss their relationships with the convexity of the game.

Semihypergroup actions by using the generalized permutations

An ordered hypergroupoid is a hypergroupoid together with a partial order such that satisfies the monotone condition. In this paper, we introduce the notion of semihypergroup actions on ordered hypergroupoids. Some results in this respect are investigated. In particular, we prove that if $S$ is a commutative semihypergroup acting on an ordered hypergroupoid $H$, then there exists a commutative semihypergroup $\widetilde{S}$ acting on the ordered hypergroupoid $\widetilde{H}:=(H \times S)/ \bar{\rho}$ in such a way that $H$ is embedded in $\widetilde{H}$.

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

Generalized convergence of solutions for nonlinear Hamilton–Jacobi equations with state-constraint

For a continuous Hamiltonian H:(x,p,u)∈T⁎Rn×R→R, we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint{H(x,Du,λu)≤Cλ,x∈Ωλ⊂Rn,H(x,Du,λu)≥Cλ,x∈Ω‾λ⊂Rn, as λ→0+. When H satisfies certain convex, coercive and monotone conditions, the domain Ωλ:=(1+r(λ))Ω keeps bounded, star-shaped for all λ>0 with limλ→0+⁡r(λ)=0, and limλ→0+⁡Cλ=c(H) equals the ergodic constant of H(⋅,⋅,0), we prove the convergence of solutions uλ to a specific solution of the critical equation{H(x,Du,0)≤c(H),x∈Ω⊂Rn,H(x,Du,0)≥c(H),x∈Ω‾⊂Rn. We also discuss the generalization of such a convergence for equations with more general Cλ and Ωλ.

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

Iterative methods for solving constraint nonlinear monotone equations have been developed and improved by many researchers. The aim of this research is to present a modified three-term conjugate descent (TTCD) derivative-free method for constrained nonlinear monotone equations. The proposed algorithm requires low storage memory; therefore, it has the capability to solve large-scale nonlinear equations. The algorithm generates a descent and bounded search direction dk at every iteration independent of the line search. The method is shown to be globally convergent under monotonicity and Lipschitz continuity conditions. Numerical results show that the suggested method can serve as an alternative to find the approximate solutions of nonlinear monotone equations. Furthermore, the method is promising for the reconstruction of sparse signal problems.

Perturbation analysis for continuous-time Markov chains in a weak sense

Abstract By the technique of augmented truncations, we obtain the perturbation bounds on the distance of the finite-time state distributions of two continuous-time Markov chains (CTMCs) in a type of weaker norm than the V-norm. We derive the estimates for strongly and exponentially ergodic CTMCs. In particular, we apply these results to get the bounds for CTMCs satisfying Doeblin or stochastically monotone conditions. Some examples are presented to illustrate the limitation of the V-norm in perturbation analysis and to show the quality of the weak norm.

Nonlinear and nonseparable structural functions in regression discontinuity designs with a continuous treatment

Many empirical examples of regression discontinuity (RD) designs concern a continuous treatment variable, but the theoretical aspects of such models are less studied. This study examines the identification and estimation of the structural function in fuzzy RD designs with a continuous treatment variable. The structural function fully describes the causal impact of the treatment on the outcome. We show that the nonlinear and nonseparable structural function can be nonparametrically identified at the RD cutoff under shape restrictions, including monotonicity and smoothness conditions. Based on the nonparametric identification equation, we propose a three-step semiparametric estimation procedure and establish the asymptotic normality of the estimator. The semiparametric estimator achieves the same convergence rate as in the case of a binary treatment variable. As an application of the method, we estimate the causal effect of sleep time on health status by using the discontinuity in natural light timing at time zone boundaries.

Negative Weights Are No Concern in Design-Based Specifications

Recent work shows that popular partially-linear regression specifications can put negative weights on some treatment effects, potentially producing incorrectly-signed estimands. We show this is not an issue in design-based specifications, in which low-dimensional controls span the conditional expectation of the treatment. Specifically, the estimands of such specifications are convex averages of causal effects with ex-ante weights that average the potentially negative ex-post weights across possible treatment realizations. This result extends to design-based instrumental variable estimands under a first-stage monotonicity condition and applies to formula treatments and instruments such as shift-share instruments.

On an Anisotropic Logistic Equation

We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution uλ* and establish the monotonicity of the map λ⟼uλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)>0 (the principal eigenvalue of (−Δq,W01,q(Ω))).