Alpár Richárd Mészáros

Ulam-Hyers stability of dynamic equations on time scales via Picard operators

In this paper we study the Ulam-Hyers stability of some linear and nonlinear dynamic equations and integral equations on time scales. We use both direct and operatorial methods and we propose a unified approach to Ulam-Hyers stability based on the theory of Picard operators (see [28,33]). Our results extend some recent results from [24,25,8,14,13,5,6] to dynamic equations and are more general than the results from [1]. The operatorial point of view, based on the theory of Picard operators, allows to discuss the Ulam-Hyers stability of many types of differential- and integral equations on time scales and also to obtain simple and structured proofs to the existing results, but as we point out at our final remarks there are also a few disadvantages.

First Order Mean Field Games with Density Constraints: Pressure Equals Price

In this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler equations a la Brenier. By this observation we manage to obtain a minimal regularity, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model.

Ulam-Hyers stability of elliptic partial differential equations in Sobolev spaces

In the present paper we study the Ulam-Hyers stability of some elliptic partial differential equations on bounded domains with Lipschitz boundary. We use direct techniques and also some abstract methods of Picard operators. The novelty of our approach consists in the fact that we are working in Sobolev spaces and we do not need to know the explicit solutions of the problems or the Green functions of the elliptic operators. We show that in some cases the Ulam-Hyers stability of linear elliptic problems mainly follows from standard estimations for elliptic PDEs, Cauchy-Schwartz and Poincare type inequalities or Lax-Milgram type theorems. We obtain powerful results in the sense that working in Sobolev spaces, we can control also the derivatives of the solutions, instead of the known point-wise estimations. Moreover our results for the nonlinear problems generalize in some sense some recent results from the literature (see for example Lazar (2012) [8]).

Uniqueness issues for evolution equations with density constraints

In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first order systems modeling crowd motion with hard congestion effects, introduced recently by \emph{Maury et al.} The monotonicity of the velocity field implies that the

On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field

2-

On nonlinear cross-diffusion systems: an optimal transport approach

Wasserstein distance along two solutions is

A variational approach to second order mean field games with density constraints: the stationary case

\lambda

BV Estimates in Optimal Transportation and Applications

-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an

Advection-diffusion equations with density constraints

L^1-

Wendroff type inequalities on time scales via Picard operators

contraction property. In this case, by the regularization effect of the non-degenerate diffusion, the result follows even if the given velocity field is only