Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

In this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

Maximal solutions for the ∞-eigenvalue problem

In this article we prove that the first eigenvalue of the {\infty}-Laplacian\left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v,|\nabla v|-\lambda% _{1,\infty}(\Omega)v\}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as {\ell\nearrow 1} of concave problems of the form\left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v_{\ell},|\nabla v_{% \ell}|-\lambda_{1,\infty}(\Omega)v_{\ell}^{\ell}\}&\displaystyle=0&&% \displaystyle\text{in }\Omega,\\ \displaystyle v_{\ell}&\displaystyle=0&&\displaystyle\text{on }\partial\Omega.% \end{aligned}\right.In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed {1<p<\infty}.

The Maximal Solution and Comparison Theorems for the Periodic Discrete-Time Riccati Equation

Properties and comparison theorems for the maximal solution of the periodic discrete-time Riccati equation are supplemented by an extension of some earlier results and analysis, for the discrete-time Riccati equation to the periodic case.