Numer. Math. 18, 367-372 (t972) 9 by Springer-Verlag t972 A Generalization of the Potential Method for Conditional Maxima on the Banach, Reflexive Spaces TOMASZ PIETRZYKOWSKI Received September 9, 1970 Summary. The paper discusses conditions of the convergence of a potential method. The method consists of approximating a constrained maximum by un- constrained maxima of a potential function. A proof of the convergence is given when dealing with the local nonsingular maximum of a locally concave functional on a locally convex set in a Banach reflexive space. Some examples of applications are discussed. Finally there is presented (as an open problem) a suggestion for further weakening of the conditions of convergence. 0. Introduction The potential method, as described in [2], is automatically applicable for a very wide class of problems. However, the convergence of this method is ques- tionable even for the case of the l * space if the maximized function is not concave. On the other hand, the assumption of the local compactness, which is basic for the results of [2], excludes many important applications. These facts provide a motivation to investigate the convergence of this method under different assumptions, and it is proved here that the following conditions are satisfactory: the space is normed, linear and reflexive, the maximized function and the penalty function are continuous and locally concave in a maximum point which is non- singular (see Def. f). This presents a considerable progress in comparison with the results of [2], since such important spaces as Hilbert spaces, and lp and L p (t < p < + oo) spaces are reflexive but not locally compact. The applicability of the method is thus extended to some constrained variational problems. However, in the author's opinion, a possibility of some further generalization should be investigated. 1. Preliminaries Let X be a normed linear space, and x, y, z denote vectors of X. Let B (x, Q) and S (x, e) denote respectively a closed ball and a sphere in X with the center x and the radius Q (e ~ 0). neg is a real function of a real variable such that neg (t) = t for t negative and neg (t)= 0 otherwise. A functional / on X is called a/fine iff there exists a linear functional g on X such that for all x g(x)=](x)--](0). A space X is reflexive iff it is isomorphic with the space (X*)* where Y* denotes the normed space of linear continuous functionals over a linear normed space Y. The following are examples of reflexive spaces (see [3] P. 374): any Hilbert space, the space lp of all sequences absolutely summable in the power p (1 < p < + cr