Global injectivity of C-1 maps of the real plane, inseparable leaves and the Palais-Smale condition

We study two sufficient conditions that imply global injectivity for a C-1 Map X : R-2 -> R-2 such that its Jacobian at any point of R-2 is not zero. One is based on the notion of half-Reeb component and the other on the Palais-Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais-Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais-Smale condition it is not a necessary condition for the global injectivity of the map X.50337738