A Linear Perturbed Palais-Smale Condition For Lower Semicontinuous Functions On Banach Spaces

In this paper, we discuss linear perturbed Palais-Smale condition for real-valued functions on Banach spaces with Radon-Nikodym property (RNP) . We mainly show the following results:1. By an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition;2.In terms of strongly exposed points, we present a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions defined on a closed bounded set of a Banach space with the Radon-Nikodym property.