Monotonicity Condition Research Articles

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds.

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399-447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature boundedbelow.

Linear-quadratic mean field games of controls with non-monotone data

In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding N N -player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from the LQ mean field game of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the N N -player game directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the N N -player game to the mean field game and the propagation of chaos property for the related optimal trajectories.

A theoretical view of existence results by using fixed point theory for quasi-variational inequalities

In this paper, we present new existence results for the quasi-variational inequality problem (QV I) in reflexive Banach spaces using the fixed point method with quasi-monotonicity and local upper sign-continuity assumptions. These results improve upon previous ones which only required a weaker monotonicity condition and did not impose compactness on the involved set.

Addressing impulsive fractional integro-differential equations with Caputo-Fabrizio via monotone iterative technique in Banach spaces

In this paper, we investigate the existence of solutions for initial value problem of the impulsive integro-differential for fractional differential equations involving a Caputo-Fabrizio fractional derivative of order r ? (0, 1) in Banach spaces. Under some monotonicity conditions and the noncompactness measure condition of nonlinearity functions. We obtain the existence of extremal solutions between lower and upper solutions.

Group invariant solutions for the planar Schrödinger-Poisson equations

<abstract><p>This paper is concerned with the following planar Schrödinger-Poisson equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\Delta{u}+V(x)u+\left(\ln{|\cdot|}\ast |u|^p\right)|u|^{p-2}u = f(x,u),\; \; \; x\in\mathbb{R}^{2}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ p\geq2 $ is a constant, and $ V(x) $ and $ f(x, u) $ are continuous, mirror symmetric or rotationally periodic functions. The nonlinear term $ f(x, u) $ satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense. We adopted a version of mountain pass theorem by constructing a Cerami sequence, which in turn leads to a ground state solution. Our method has two new insights. First, we observed that the integral $ \int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\ln{(|x-y|)}|u(x)|^{p}|u(y)|^pdxdy $ is always negative if $ u $ belongs to a suitable space. Second, we built a new Moser type function to ensure the boundedness of the Cerami sequence, which further guarantees its compactness. In particular, by replacing the monotonicity condition with the Ambrosetti–Rabinowitz condition, our approach works also for the subcritical growth case.</p></abstract>

On a class of three coupled fractional Schrödinger systems with general nonlinearities

<abstract><p>In this paper, a class of systems of three-component coupled nonlinear fractional Schrödinger equations with general nonlinearities is investigated. Without any monotonicity condition and the Ambrosetti-Rabinowitz growth condition, we obtain some novel existence results of least energy solutions by using variational arguments and a Pohozaev manifold method.</p></abstract>

Monotonicity and discretization of Hammerstein integrodifference equations

<p style='text-indent:20px;'>The paper provides sufficient conditions for monotonicity, subhomogeneity and concavity of vector-valued Hammerstein integral operators over compact domains, as well as for the persistence of these properties under numerical discretizations of degenerate kernel type. This has immediate consequences on the dynamics of Hammerstein integrodifference equations and allows to deduce a local-global stability principle.</p>

ЕФЕКТИВНИЙ ЧИСЕЛЬНИЙ МЕТОД ДЛЯ РОЗВ'ЯЗАННЯ РІВНЯННЯ РІЧАРДСА-КЛЮТА ЗА УМОВ МОНОТОННОСТІ РОЗВ'ЯЗКУ

The article contains new numerical methods for solving the Richards-Klute equation under the condition of monotonicity of the solution.The article contains new numerical methods for solving the Richards-Klute equation under the condition of monotonicity of the solution. Modified numerical methods with tracking of the full saturation zone are proposed to improve efficiency of the finding an approximate solution of the equation process. A comparative analysis of the proposed methods in one- and two-dimensional cases was carried out. The analysis of the estimation of the efficiency improvement of the solving the Richards-Klute equation process was carried out.

Nonstationary homoclinic solutions for infinite-dimensional fractional reaction-diffusion system with two types of superlinear nonlinearity

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |u(t, x)|+|v(t, x)|\rightarrow 0, \ \ \text{as}\ \ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $\end{document} where $ 0<s<1, \ z = (u, v): \mathbb{R}\times \mathbb{R}^N\rightarrow \mathbb{R}^{2} $, which originate from a wide variety of fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of superlinear nonlinearity. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Nevertheless, we need to overcome some difficulties: one is that the associated functional is strongly indefinite, the second is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity and in verifying the link geometry and showing the boundedness of Cerami sequences.

АЛГОРИТМ ЕКСТРАПОЛЯЦІЇ З МИНУЛОГО ДЛЯ ВАРІАЦІЙНИХ НЕРІВНОСТЕЙ В ГІЛЬБЕРТОВОМУ ПРОСТОРІ

The article considers variational inequalities with operators acting in a Hilbert space. For these problems, variants of the Extrapolation from the Past method have been proposed and studied. A sub-linear efficiency estimate for the gap function is proved. The strong convergence of the Extrapolation from the Past method for variational inequalities with uniformly monotone operators is proved. The linear rate of convergence of the Extrapolation from the Past method for variational inequalities with operators satisfying the generalized strong monotonicity condition is proved. An adaptive version of the algorithm is proposed. Regularized variants of the algorithm are proposed and theorems on their strong convergence are proved.

Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

This paper introduces a new class of singularly perturbed systems in which the small, but constant, perturbation coefficient in standard singular perturbation theory is replaced by a state-dependent function. This generalization is aimed at broadening the applicability of singular perturbation theory in practice. For this class of singularly perturbed systems, it is assumed that the boundary-layer subsystem is globally asymptotically stable (GAS) at the origin and the reduced subsystem is input-to-state stable (ISS) with respect to the state of the boundary-layer subsystem. Under a mild monotonicity condition, sufficient conditions on the perturbation functions are given under which the singularly perturbed system is GAS at the origin. ISS and nonlinear small-gain techniques are exploited in the stability analysis. The efficacy of the proposed theoretical result is validated via its applications to tackling integral control and feedback optimization problems.

Groundstates and infinitely many solutions for the Schrödinger-Poisson equation with magnetic field

<p style='text-indent:20px;'>In this paper, we investigate the nonlinear Schrödinger-Poisson equation with magnetic field. By combining non-Nehari manifold method and some new energy estimate inequalities, we obtain the existence of a ground state solution, where the strict monotonicity condition and the Ambrosetti-Rabinowitz growth condition are not needed. Moreover, when both the potential and the nonlinearity are sign-changing, by applying the Fountain Theorem and some analytical techniques, we prove the existence of infinitely many solutions. Our results extend and improve the present ones in the literature.</p>

An exit contract optimization problem

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents’ exit times. Under a technical monotonicity condition, and by using Bank-El Karoui’s representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimization problem over the class of all Markovian and/or continuous exit contracts.

Existence and uniqueness for variational data assimilation in continuous time

<p style='text-indent:20px;'>A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.</p>

Generalized BSDEs for time inhomogeneous Lévy processes under non-deterministic Lipschitz coefficient

In this paper, we study the generalized backward stochastic differential equations driven by inhomogeneous Lévy processes (GBSDELs in short). We establish the existence and uniqueness of solution by using Picard's iteration setting under non-deterministic Lipschitz and monotone condition.

Existence and multiplicity results for first-order Stieltjes differential equations

Abstract In this article, we establish existence and multiplicity results for first-order Stieltjes differential equations satisfying a periodic boundary condition or an initial value condition. No monotonicity condition involving the right-hand side f f is imposed at the discontinuity points of the derivator g g . Our results rely on the fixed point index theory and new notions of strict lower and upper solutions. An application to a population model with an extreme event is presented to study the persistence of a species.

Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model

Abstract The stochastic multi-group susceptible–infected–recovered (SIR) epidemic model is nonlinear, and so analytical solutions are generally difficult to obtain. Hence, it is often necessary to find numerical solutions, but most existing numerical methods fail to preserve the nonnegativity or positivity of solutions. Therefore, an appropriate numerical method for studying the dynamic behavior of epidemic diseases through SIR models is urgently required. In this paper, based on the Euler–Maruyama scheme and a logarithmic transformation, we propose a novel explicit positivity-preserving numerical scheme for a stochastic multi-group SIR epidemic model whose coefficients violate the global monotonicity condition. This scheme not only results in numerical solutions that preserve the domain of the stochastic multi-group SIR epidemic model, but also achieves the “ order - 1 2 {\mathrm{order}-\frac{1}{2}} ” strong convergence rate. Taking a two-group SIR epidemic model as an example, some numerical simulations are performed to illustrate the performance of the proposed scheme.

Image restorations using a modified relaxed inertial technique for generalized split feasibility problems

In this article, we propose a new efficient self‐adaptive method and prove that it converges strongly to a minimum‐norm solution of a generalized split feasibility problem in real Hilbert spaces. The proposed method is derived from a definite discrete dynamical system in time, which combines both the relaxation and inertial techniques for the purpose of increasing the rate of convergence of the iterative scheme. Furthermore, the method requires the monotonicity and Lipschitz continuity condition of the underlying single‐valued cost operator , and it employs some simple self‐adaptive stepsizes that are generated at each iteration by some easy computations. As a by‐product, we obtain methods for solving other classes of generalized split feasibility problems in real Hilbert spaces. Two major merits of our scheme in solving image restoration problems over related schemes are the higher signal‐to‐noise ratio value and lower CPU time for generating recovered images. Finally, we analyze our methods with different related strong convergent methods in the literature.

A Label-State Formulation of Stochastic Graphon Games and Approximate Equilibria on Large Networks

This paper studies stochastic games on large graphs and their graphon limits. We propose a new formulation of graphon games based on a single typical player’s label-state distribution. In contrast, other recently proposed models of graphon games work directly with a continuum of players, which involves serious measure-theoretic technicalities. In fact, by viewing the label as a component of the state process, we show in our formulation that graphon games are a special case of mean field games, albeit with certain inevitable degeneracies and discontinuities that make most existing results on mean field games inapplicable. Nonetheless, we prove the existence of Markovian graphon equilibria under fairly general assumptions as well as uniqueness under a monotonicity condition. Most importantly, we show how our notion of graphon equilibrium can be used to construct approximate equilibria for large finite games set on any (weighted, directed) graph that converges in cut norm. The lack of players’ exchangeability necessitates a careful definition of approximate equilibrium, allowing heterogeneity among the players’ approximation errors, and we show how various regularity properties of the model inputs and underlying graphon lead naturally to different strengths of approximation. Funding: D. Lacker was partially supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0291] and the National Science Foundation [Award DMS-2045328].