Abstract
In the present paper; we introduce and study an appropriate definition of proximal normal cones and proximal subdifferentials in L-p-spaces, 1 < p <= 2. The density theorem and various important properties for the proximal subdifferential are proved. Applications to calculus of variations is also given. Our results extend various results in [18, 281 from L-2 spaces to L-p spaces (1 < p <= 2).