A generalization of the Moreau–Yosida regularization

In many applications, one deals with nonsmooth functions, e.g., in dynamical systems, mechanics, or optimization. order to establish theoretical results, it is often beneficial regularize the functions an intermediate step. this work, we investigate properties of a generalization Moreau–Yosida regularization on normed space where replace quadratic kernel infimal convolution more general function. More precisely, for function f:X→(−∞,+∞] defined (X,‖⋅‖) and given parameters p>1 ε>0, generalized byfε(u)=infv∈X⁡{1pε‖u−v‖p+f(v)},u∈X. We show that satisfies same as classical case p=2, provided X not Hilbert space. further convergence result sense Mosco-convergence parameter ε tends zero.