The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations $F_k\left[u\right]=0$, where $F_k\left[u\right]$ is the elementary symmetric function of order $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix $D^{2}u$. For example, $F_1\left[u\right]$ is the Laplacian $\Delta u$ and $F_n\left[u\right]$ is the real Monge-Ampère operator det $D^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give several...

We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J($\frac{1-\lambda }{\lambda }$x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation vn = Φ($\frac{1}{n}$, $v_{n-1}$) (resp. $v_{\lambda }$ = Φ(λ, $v_{\lambda }$)) where J is the Shapley operator of the game. We study the evolution equation u'(t) =...

A bifurcation problem for variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-B_{\lambda }U\left(t\right)-G(\lambda ,U\left(t\right)),Z-U\left(t\right))\ge 0\text{for}\text{all}Z\in K,\text{a.a.}t\ge 0$$ is studied, where $K$ is a closed convex cone in ${ℝ}^{\kappa }$, $\kappa \ge 3$, $B_{\lambda }$ is a $\kappa \times \kappa$ matrix, $G$ is a small perturbation, $\lambda$ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

We prove the existence of a positive solution to the BVP $${\left(\Phi \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=f(t,u\left(t\right)),u^{\text{'}}\left(0\right)=u\left(1\right)=0,$$ imposing some conditions on Φ and f. In particular, we assume $\Phi \left(t\right)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_{\infty }$ bound for the solution is provided by the $L_{\infty }$ norm of any test function with negative energy.