On the constructive solution of nonlinear functional equations

Abstract

In the recent development of the theory of nonlinear operators in Banach spaces, a number of general existence theorems have been established for various classes of mappings by the use of compactness, convexity, or topological arguments that do not give a constructive procedure for the generation of the solutions thus proved to exist. Even in cases when the solutions are unique and one obtains them as limits of a precisely determined sequence of approximants, they often fail to satisfy an important and useful principle of constructivity in that the procedures have no effective control provided for the error at each state of the approximation. This is particularly the case for the Galerkin approximations used in the theory of operators of monotone type and for various fixed-point methods used in the theory of nonlinear accretive operators. About a decade ago, the writer developed existence results which satisfied these principles of constructivity for the case of a broad class of continuous monotone mappings in Hilbert spaces, and more generally, for continuous accretive mappings in Banach spaces X whose conjugate spaces X* are uniformly convex. These results were stated in the rather inaccessible paper [l] and developed in detail in the middle of a lengthy treatment of accretive operators in the writer’s paper [2]. It is our object here to develop these results explicitly and in detail in connection with the problem of constructivity. Our renewed interest in this question was stimulated by the recent paper by Bruck [3], who has developed an iteration procedure to obtain the solution of the equation (I + T)(u) = 0 for a continuous, bounded monotone operator T on a Hilbert space H with explicit control of the error. Bruck’s result has the curious feature that it depends on the assumption of the prior existence of the solution. Although the procedure we give is not an iteration procedure it corresponds to a simpler intuitive picture of the situation in the general context of accretive operators and the error control does not depend upon the assumption of the existence of a solution.