Weak Monotonicity Condition Research Articles

The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Large deviation principles for SDEs under locally weak monotonicity conditions

Large deviation principles for SDEs under locally weak monotonicity conditions

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature.

Existence of solutions for a quasilinear elliptic system with variable exponent

We consider the following quasilinear elliptic system in a Sobolev space with variable exponent: [-text{div}(a(|Du|)Du)=f,] where $a$ is a $C^1$-function and $fin W^{-1,p'(x)}(Omega;R^m)$. We use the theory of Young measures and weak monotonicity conditions to obtain the existence of solutions.

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted Lp-path spaces is proved. In particular, as special cases the classical Caputo derivative and other fractional derivatives appearing in applications are included. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of typeddt(k⁎u)(t)+A(t,u(t))=f(t),0<t<T, with (in general nonlinear) operators A(t,⋅) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lims→∞k(s)=0. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators and the time-fractional (stochastic) p-Laplace equation are covered.

Existence of Weak Solutions to an Elliptic-Parabolic Equation with Variable Order of Nonlinearity

We consider an equation with variable nonlinearity of the form |u|p(x), in which the parabolic term can vanish, i.e., in the corresponding domain the parabolic equation becomes “elliptic.” Under the weak monotonicity conditions (nonstrict inequality) we prove the existence of a solution to the first mixed problem in a cylinder with a bounded base.

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

The aim of this paper is to investigate strong convergence of modified truncated Euler–Maruyama method for neutral stochastic differential delay equations introduced in Lan (2018). Strong convergence rates of the given numerical scheme to the exact solutions at fixed time T are obtained under local Lipschitz and Khasminskii-type conditions. Moreover, convergence rates over a time interval [0,T] are also obtained under additional polynomial growth condition on g without the weak monotonicity condition (which is usually the standard assumption to obtain the convergence rate). Two examples are presented to interpret our conclusions.

Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

In this paper we are concerned with the fractional Schrodinger equation \begin{document}$ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $\end{document} , \begin{document}$ x\in {{\mathbb{R}}^{N}} $\end{document} , where \begin{document}$ f $\end{document} is superlinear, subcritical growth and \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} is nondecreasing. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are periodic in \begin{document}$ x_{1},\ldots, x_{N} $\end{document} , we show the existence of ground states and the infinitely many solutions if \begin{document}$ f $\end{document} is odd in \begin{document}$ u $\end{document} . When \begin{document}$ V $\end{document} is coercive or \begin{document}$ V $\end{document} has a bounded potential well and \begin{document}$ f(x, u) = f(u) $\end{document} , the ground states are obtained. When \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} are asymptotically periodic in \begin{document}$ x $\end{document} , we also obtain the ground states solutions. In the previous research, \begin{document}$ u\mapsto\frac{f(x, u)}{\vert u\vert} $\end{document} was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.

On Wold’s approach to representation of preferences

This paper presents easily verifiable sufficient conditions on sequence spaces that guarantee representation of preference orders. Our approach involves identifying a suitable subset of the set of alternatives, such that (a) the preference order is representable on this subset, and (b) the subset has the property that for each alternative, there is some element in this subset which is indifferent to it. We follow Wold in choosing this subset to be the diagonal. Our first result uses a weak monotonicity condition (on the diagonal), and a substitution condition, and may be identified as the essence of Wold’s contribution. In the second result, we show that one can obtain a Wold-type representation result when weak monotonicity is replaced by a weak continuity condition. We use the countable order-dense characterization of representability in the proofs of both results, thereby integrating the contributions of Wold (1943) and Debreu (1954). Through a series of examples we show that our representation results are robust; they cannot be improved upon by dropping any of our conditions. An example is also presented to show that existence of degenerate indifference classes is compatible with the representation of monotone preferences. Our study thereby indicates that while the presence of substitution possibilities can be useful in representing preferences, they are not necessary for such results to hold.

Lp (1 < p ⩽ 2) solutions of one-dimensional BSDEs whose generator is weakly monotonic in y and non-Lipschitz in z

ABSTRACTIn this paper, we start with establishing the existence of a minimal (maximal) Lp (1 < p ⩽ 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of Lp (1 < p ⩽ 2) solutions to one-dimensional BSDEs with q-order (1 ⩽ q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of Lp (1 < p ⩽ 2) solutions is also given for BSDEs whose generator g is q-order (1 ⩽ q < p) weakly monotonic with a general growth in y and uniformly continuous in z.

A representation theorem for generators of BSDEs with general growth generators in [formula omitted] and its applications

In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in y and a Lipschitz condition in z. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in y. Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.

Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term

We study the Schrödinger equations − Δ u + V ( x ) u = f ( x , u ) -\Delta u + V(x)u = f(x,u) in R N \mathbb {R}^N and − Δ u − λ u = f ( x , u ) -\Delta u - \lambda u = f(x,u) in a bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N . We assume that f f is superlinear but of subcritical growth and u ↦ f ( x , u ) / | u | u\mapsto f(x,u)/|u| is nondecreasing. In R N \mathbb {R}^N we also assume that V V and f f are periodic in x 1 , … , x N x_1,\ldots ,x_N . We show that these equations have a ground state and that there exist infinitely many solutions if f f is odd in u u . Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802–3822], where u ↦ f ( x , u ) / | u | u\mapsto f(x,u)/|u| was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.

General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general g-supermartingales

The main purpose of this paper is to prove an existence and uniqueness result for solutions of a multidimensional backward stochastic differential equation (BSDE) with a general time interval (including the deterministic and stochastic cases), where the generator g of the BSDE is weakly monotonic and of general growth in y, and Lipschitz continuous in z, both non-uniformly with respect to t. And, the corresponding comparison theorem for the solutions of one-dimensional BSDEs is provided. As applications, we establish a nonlinear Doob-Meyer’s decomposition theorem for general continuous g-supermartingales under an additional assumption of the generator g. Some new problems in our setting arise naturally and are well overcome. These results generalize and improve some known works.

Wellposedness of Mean Field Games with Common Noise under a Weak Monotonicity Condition

In this paper, we consider Mean Field Games in the presence of common noise relaxing the usual independence assumption of individual random noise. We assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property. Our main result is showing existence and uniqueness of a Mean Field Game solution using the Stochastic Maximum Principle. The uniqueness is a result of a monotonicity property similar to that of Lasry and Lions. We use the Banach Fixed Point Theorem to establish an existence over small time duration and show that it can be extended over an arbitrary finite time duration.

Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

We study the existence of solutions of the following nonlinear Schrödinger equation−Δu+(V(x)−μ|x|2)u=f(x,u) for x∈RN∖{0}, where V:RN→R and f:RN×R→R are periodic in x∈RN. We assume that 0 does not lie in the spectrum of −Δ+V and μ<(N−2)24, N≥3. The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small μ≥0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If μ<0 and 0 lies below the spectrum of −Δ+V, then ground state solutions do not exist.

Existence of solutions to one-dimensional BSDEs with semi-linear growth and general growth generators

This paper is devoted to solving a one-dimensional backward stochastic differential equation (BSDE in short) when the generator g has a semi-linear growth and a general growth in (y,z). This condition is not only strictly weaker than the linear growth condition of g in (y,z), but also the (weak) monotonicity and general growth condition of g in y together with the linear growth condition of g in z. We establish, in this setting, three existence results on a solution and the minimal (maximal) solution to the BSDE, where the generator g may be discontinuous in y. These results virtually unify and improve some existing results.

Formula omitted] solutions of multidimensional BSDEs with weak monotonicity and general growth generators

In this paper, we first establish the existence and uniqueness of Lp(p>1) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g. Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of Lp(p>1) solutions for this kind of BSDEs.

On ergodic algorithms for equilibrium problems

In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159---164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371---386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation (BSDE) whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonicity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonicity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.

Existence of Equilibria in Auctions with Interdependent Values: Two Symmetic Bidders

For two symmetric bidders, weak monotonicity conditions are shown to imply existence of an equilibrium in mixed behavioral strategies for a sealed-bid first-price auction of an item for which each bidder's value depends on every bidder's observed signal. Such an equilibrium has atomless distributions of bids and thus is unaffected by how tied bids are resolved.