Monotonicity Assumption Research Articles

On the Existence of Linearly Oscillating Galaxies

We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

INERTIAL RELAXED TSENG METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACE

The research efforts of this paper is to present a new inertial relaxed Tseng extrapolation method with weaker conditions for approximating the solution of a variational inequality problem, where the underlying operator is only required to be pseudomonotone. The strongly pseudomonotonicity and inverse strongly monotonicity assumptions which the existing literature used are successfully weakened. The strong convergence of the proposed method to a minimum-norm solution of a variational inequality problem are established. Furthermore, we present an application and some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature.

Dependence on parameters for nonlinear equations—Abstract principles and applications

We provide parameter‐dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals are also investigated. Applications towards the dependence on parameters for both potential and nonpotenial nonlinear Dirichlet boundary problems are given.

A New Approach to Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces

Very recently, by considering a self-mapping T on a complete metric space satisfying a general contractivity condition of the form ψ(d(Tx,Ty))≤φ(d(x,y)), Proinov proved some fixed-point theorems, which extended and unified many existing results in the literature. Accordingly, inspired by Proinov-type contraction conditions, Roldán López de Hierro et al. introduced a novel family of contractions in fuzzy metric spaces (in the sense of George and Veeramani), whose main advantage is the very weak constraints imposed on the auxiliary functions that appear in the contractivity condition. They also proved the existence and uniqueness of fixed points for the discussed family of fuzzy contractions in the setting of non-Archimedean fuzzy metric spaces. In this paper, we introduce a new family of fuzzy contractions based on Proinov-type contractions for which the involved auxiliary functions are not supposed to satisfy any monotonicity assumptions; further, we establish some new results about the existence and uniqueness of fixed points. Furthermore, we show how the main results in the above-mentioned paper can be deduced from our main statements. In this way, our conclusions provide a positive partial solution to one of the open problems posed by such authors for deleting or weakening the hypothesis of the nondecreasingness character of the auxiliary functions.

Distributed Generalized Nash Equilibrium Seeking for Monotone Generalized Noncooperative Games by a Regularized Penalized Dynamical System

In this work, we study the generalized Nash equilibrium (GNE, see Definition 1) seeking problem for monotone generalized noncooperative games with set constraints and shared affine inequality constraints. A novel projected gradient-based regularized penalized dynamical system is proposed to solve this issue. The idea is to use a differentiable penalty function with a time-varying penalty parameter to deal with the inequality constraints. A time-varying regularization term is used to deal with the ill-poseness caused by the monotonicity assumption and the time-varying penalty term. The proposed dynamical system extends the regularized dynamical system in the literature to the projected gradient-based regularized penalized dynamical system, which can be used to solve generalized noncooperative games with set constraints and coupled constraints. Furthermore, we propose a distributed algorithm by using leader-following consensus, where the players have access to neighboring information only. For both cases, the asymptotic convergence to the least-norm variational equilibrium of the game is proven. Numerical examples show the effectiveness and efficiency of the proposed algorithms.

Distribution-dependent stochastic differential delay equations in finite and infinite dimensions

We prove that distribution-dependent (also called McKean–Vlasov) stochastic delay equations of the form [Formula: see text] have unique (strong) solutions in finite as well as infinite-dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.

Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation

This paper deals with the existence of traveling wave solutions to a delayed temporally discrete non-local reaction diffusion equation model, which has been derived recently for a single species with age structure. When the birth function satisfies monotonic condition, we obtained the traveling wavefront by using upper and lower solution methods together with monotonic iteration techniques. Otherwise, without the monotonicity assumption for birth function, we constructed two auxiliary equations. By means of the traveling wavefronts of the auxiliary equations, using the Schauder’ fixed point theorem, we proved the existence of a traveling wave solution to the equation under consideration with speed c>c*, where c*>0 is some constant. We found that the delayed temporally discrete non-local reaction diffusion equation possesses the dynamical consistency with its time continuous counterpart at least in the sense of the existence of traveling wave solutions.

Periodic Solutions to Klein–Gordon Systems with Linear Couplings

Abstract In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique allows to avoid any monotonicity assumption. We adapt a weak coercivity condition introduced in Castellani and Giuli (J Glob Optim 75:163–176, 2019) for a generalized game which extends an older one proposed by Konnov and Dyabilkin (J Glob Optim 49:575–577, 2011) for equilibrium problems. Our main result encompasses and generalizes several existence results for equilibrium, quasiequilibrium and fixed-point problems.

Master Equation for the Finite State Space Planning Problem

We present results regarding the existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially, in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions.

Is more better? An analysis of toxicity and response outcomes from dose-finding clinical trials in cancer

BackgroundThe overwhelming majority of dose-escalation clinical trials use methods that seek a maximum tolerable dose, including rule-based methods like the 3+3, and model-based methods like CRM and EWOC. These methods assume that the incidences of efficacy and toxicity always increase as dose is increased. This assumption is widely accepted with cytotoxic therapies. In recent decades, however, the search for novel cancer treatments has broadened, increasingly focusing on inhibitors and antibodies. The rationale that higher doses are always associated with superior efficacy is less clear for these types of therapies.MethodsWe extracted dose-level efficacy and toxicity outcomes from 115 manuscripts reporting dose-finding clinical trials in cancer between 2008 and 2014. We analysed the outcomes from each manuscript using flexible non-linear regression models to investigate the evidence supporting the monotonic efficacy and toxicity assumptions.ResultsWe found that the monotonic toxicity assumption was well-supported across most treatment classes and disease areas. In contrast, we found very little evidence supporting the monotonic efficacy assumption.ConclusionsOur conclusion is that dose-escalation trials routinely use methods whose assumptions are violated by the outcomes observed. As a consequence, dose-finding trials risk recommending unjustifiably high doses that may be harmful to patients. We recommend that trialists consider experimental designs that allow toxicity and efficacy outcomes to jointly determine the doses given to patients and recommended for further study.

Reaction-diffusion systems with supercritical nonlinearities revisited

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

Lower and upper solutions method to the fully elastic cantilever beam equation with support

The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, \t\t\t{u(4)(t)=f(t,u(t),u′(t),u″(t),u‴(t)),t∈[0,1],u(0)=u′(0)=0,u″(1)=0,u‴(1)=g(u(1)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \\quad t\\in [0,1], \\\\ u(0)=u'(0)=0, \\\\ u''(1)=0,\\qquad u'''(1)=g(u(1)), \\end{cases} $$\\end{document} where f:[0,1]times mathbb{R}^{4}rightarrow mathbb{R}, g: mathbb{R}rightarrow mathbb{R} are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

Examining the Dimensionality and Monotonicity of an Attitude Dataset based on the Item Response Theory Models

In the current study, the factor structure of an attitude scale was analyzed by using the two different item response theory models that allow modeling non-monotonic item response curves. The current study utilized the two models to examine whether the two-factor solution of factor analysis may be caused by method effect, or by the failure of the analysis in describing and fitting the dataset because of the monotonicity assumption. This study was conducted on a dataset obtained from 355 undergraduate students who were studying at the Middle East Technical University. The data were obtained by carrying out the Attitude Scale Towards Foreign Languages as Medium of Instruction, which was developed by Kartal and Gülleroğlu (2015). The fit of the scale items to the generalized graded unfolding model was examined based on the item response curves, item parameters, item fit statistics and fit graphics. For Mokken scaling, scalability coefficients were calculated, dimensionality analyzes were conducted by using the Automated Item Selection Procedure. The monotonicity assumption was investigated based on the rest-score group methods. The results of the current study revealed that items of the attitude scale fit to the unidimensional models that do not assume monotone increasing item response curves for all items, while the factor analysis suggested a two-factor solution for the data. Researchers are recommended to utilize statistical techniques that can identify any possible violation of the monotonicity assumption and model items having non-monotonic response curves to examine dimensionality of their data.

Generalizations of Efron’s theorem

In this article, we prove two new versions of a theorem proven by Efron in Efron (1965). Efron’s theorem says that if a function ϕ:R2→R is non-decreasing in each argument then we have that the function s↦E[ϕ(X,Y)|X+Y=s] is non-decreasing. We name restricted Efron’s theorem a version of Efron’s theorem where ϕ:R→R only depends on one variable. PFn is the class of functions such as ∀a1≤⋯≤an,b1≤⋯≤bn,det(f(ai−bj))1≤i,j≤n≥0. The first version generalizes the restricted Efron’s theorem for random variables in the PFn class. The second one considers the non-restricted Efron’s theorem with a stronger monotonicity assumption. In the last part, we give a more general result of the second generalization of Efron’s theorem.

Instrumental Variable-based Identification for Causal Effects using Covariate Information

This paper deals with the identification problem of causal effects in randomized trials with noncompliance. In this problem, generally, causal effects are not identifiable and thus have been evaluated under some strict assumptions, or through the bounds. Different from existing studies, we propose novel identification conditions of joint probabilities of potential outcomes, which allow us to derive a consistent estimator of the causal effect. Regarding the identification conditions of joint probabilities of potential outcomes, the assumptions of monotonicity (Pearl, 2009), independence between potential outcomes (Robins & Richardson, 2011), gain equality (Li & Pearl, 2019) and specific functional relationships between cause and effect (Pearl, 2009) have been utilized. In contrast, without such assumptions, the proposed conditions enable us to evaluate joint probabilities of potential outcomes using an instrumental variable and a proxy variable of potential outcomes. The results of this paper extend the range of solvable identification problems in causal inference.

Scrutinizing the Monotonicity Assumption in IV and fuzzy RD designs*

AbstractWhenever treatment effects are heterogeneous, and there is sorting into treatment based on the gain, monotonicity is a condition that both instrumental variable (IV) and fuzzy regression discontinuity (RD) designs must satisfy for their estimate to be interpretable as a local average treatment effect. However, applied economic work often omits a discussion of this important assumption. A possible explanation for this missing step is the lack of a clear framework to think about monotonicity in practice. In this paper, we use an extended Roy model to provide insights into the interpretation of IV and fuzzy RD estimates under various degrees of treatment effect heterogeneity, sorting on gain and violation of monotonicity. We then extend our analysis to two applied settings to illustrate how monotonicity can be investigated using a mix of economic insights, data patterns and formal tests. For both settings, we use a Roy model to interpret the estimate even in the absence of monotonicity. We conclude with a set of recommendations for the applied researcher.

Inertial-Viscosity-Type Algorithms for Solving Generalized Equilibrium and Fixed Point Problems in Hilbert Spaces

In this paper, we introduce a new algorithm of inertial form for solving Split Generalized Equilibrium Problem (SGEP) and Fixed Point Problem (FPP) of multivalued nonexpansive mappings in real Hilbert spaces. Motivated by the viscosity-type method, we incorporate the inertial technique to accelerate the convergence of the proposed method. Here, the viscosity term is a function of the inertial extrapolation sequence and some assumptions on the bifunctions are dispensed with. Under standard and mild assumption of monotonicity and upper hemicontinuity of the SGEP associated mappings, we establish the strong convergence of the scheme which also solves a Variational Inequality Problem (VIP). A numerical example is presented to illustrate the effectiveness and performance of our method as well as comparing it with a related method and conventional inertial-viscosity-type algorithm in the literature.

Limit Shapes for Gibbs Partitions of Sets

This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in which the limit shape is a step function.

A Modified Scaled Spectral-Conjugate Gradient-Based Algorithm for Solving Monotone Operator Equations

This paper proposes a modified scaled spectral-conjugate-based algorithm for finding solutions to monotone operator equations. The algorithm is a modification of the work of Li and Zheng in the sense that the uniformly monotone assumption on the operator is relaxed to just monotone. Furthermore, unlike the work of Li and Zheng, the search directions of the proposed algorithm are shown to be descent and bounded independent of the monotonicity assumption. Moreover, the global convergence is established under some appropriate assumptions. Finally, numerical examples on some test problems are provided to show the efficiency of the proposed algorithm compared to that of Li and Zheng.