A new method for the existence of solutions of nonlinear differential equations

Abstract

Many problems is analysis reduce to solving an equation Au=f, where A is an operator on a space into another space, u is the unknown, and f is a given element. The existence of a local solution near a trivial solution u0 of Au, = 0, where 0 is near f, is usually solved by the classical implicit function theorem. As a matter of fact, the Picard iteration procedure is used. Unfortunately, this procedure cannot be applied in some problems of partial differential equations. These cases occur if the loss of derivatives appears [6, 7, 31. To get around this difficulty a more sophisticated iteration procedure is needed. Moser [ 1 ] has found such a method which enables one to handle a loss of derivatives by using a rapidly converging iteration scheme. The Nash-Moser technique is nicely described in the papers [S, lo]. We prove in this paper several theorems on the existence of solutions, with the application to several types of specific problems. The main interest is that in many cases these results can be used in situations which up to now have required the Nash-Moser implicit function theorem, yet they have classical proofs. We also apply abstract results to show the existence of local invariant manifolds for singularly perturbed ordinary differential equations and we give a simple proof of a theorem of J. Moser for invariant tori [l]. The plan of our paper is as follows. In Section 2 we present proofs of several existence theorems in an abstract setting in a scale of Hilbert spaces. The hypotheses are those commonly found in problems which have up to now required the application of the Nash-Moser technique, however, the results have classical proofs based on the degree of mappings [S]. We shall give in the proof of Theorem 2.8 a new iteration procedure which retains supergeometric convergence properties. We require obligations only for the linearization of our equation. This is the main advance of this theorem. Section 3 of the paper presents applications of the theorems of Section 2. There are demonstrated proofs of the above mentioned theorems of