Codifferentiable functions in Banach spaces: methods and applications to problems of variation calculus

For the study of special classes of nonsmooth functions, specific tools and methods are usually employed. Thus, for the class of qusidifferentiable functions, such a tool is Quasidifferential Calculus. The notion of codifferential allows one to construct continuous approximations of nonsmooth functions. This approach is investigated in detail for the finite-dimensional case. In the present paper, the notion of codifferential is generalized to the case of abstract spaces. Calculus of codifferentials is consrtucted, necessary conditions for an extremum of a codifferentiable function defined on a normed space are formulated, a numerical method for finding stationary points of the functional (the method of codifferential descent) is derived, a convergence theorem is proved. The efficiency of the theory described is demonstrated on some problems of Calculus of Variations. By means of the notion of codifferential, all known optimality conditions for classical variational problems were almost automatically obtained as well as necessary conditions for a minmax variational problem. Bibliogr. 17.