On regularization in Banach spaces
The aim of the paper is to extend to Banach spaces previous results on regularization given in \textit{J. M. Lasry} and \textit{P. L. Lions} [Isr. J. Math. 55, 257-266 (1986; Zbl 0631.49018)]. The author proves that it is possible to associate to any bounded below lower semicontinuous (l.s.c.) proper function \(f\), defined in a Banach space, a family of \(C^1\) functions approximating \(f\) from below obtaining good properties for minimization. The method reduces, in case when \(X\) is a Hilbert space, to that one introduced by J. M. Lasry and P. L. Lions, which is based upon the Moreau-Yosida approximation. The main results are contained in Theorem 1, where under suitable assumptions on the function \(f\), which is defined in a Banach space whose norm and dual norm are both locally uniformly rotund, regularization properties for the two-parameter family of approximates \(f_{t,s}\) and for \(df_{t,s}\) are established. Moreover, an approximation from below, i.e., \(f_{t,s}\leq f\), is proved and it is shown that the method preserves the infimum of \(f\) and the associated set of minimizers. Furthermore, convergence properties for \(f_{t,s}\) to \(f\) are also obtained. Several corollaries show that some known results on approximation of uniformly continuous functions can be derived from Theorem 1. Also approximation by twice Gâteaux differentiable \(C^{1,1}\) functions in separable Banach spaces are studied. Finally, a convergence result for the derivatives of the approximates \(df_{t,s}\) to the Clarke subdifferential \(\partial f\) is obtained in case when \(f\) is defined in a Hilbert space.
scientific article; zbMATH DE number 944337