Elhadj Dahia
École Normale Supérieure de Bousaâda, Bousaâda, Algeria

Let $(X,d_{X})$ and $(Y,d_{Y})$ are pointed metric spaces and let $E$ be a Banach space, we say that a map $T:X\times Y\longrightarrow E$ is a 2-Lipschitz operator if there is a constant $C>0$ such that for each $x,x^{\prime }\in X$ and $y,y^{\prime }\in Y$, $$ \left\Vert T\left( x,y\right) -T\left( x,y^{\prime }\right) -T\left( x^{\prime },y\right) +T\left( x^{\prime },y^{\prime }\right) \right\Vert \leq C.d_{X}\left( x,x^{\prime }\right) d_{Y}\left( y,y^{\prime }\right) . $$ In this work, we introduce and investigate the compactness concept for the 2-Lipschitz operators, we show the basics of this new class of non-linear mappings. We see that the nature of this extension allow us to transfer some properties of the bilinear compact operators (and also the linear compact operators) to the 2-Lipschitz case.