1. The aim of this paper is to study the nonlinear differential equationwhere AT is a nonlinear operator in a real Hubert space S, and £ is a linear differential operator in S with preassigned linear homogeneous boundary conditions. The idea is to reduce the problem to a finite dimensional setting and this technique has been used by several authors. We use here a method due to Cesari [4]. This method has been extensively developed in the existence analysis of differential equations by Cesari, Hale, Locker, Mawhin and others. For a detailed bibliography one is referred to Cesari [5]. In this paper, by applying results from the theory of monotone operators, we show that, under suitable monotonicity hypotheses on N, the equation Ex = Nx can be solved. In the present short presentation we restrict ourselves to the simplest hypotheses on E 9 N and S, even though the results obtained here hold under more general conditions. 2. Let S be the direct sum of the subspaces S 0 and S t and let P:S -> S 0 be a projection operator with null space S l9 and H:S l -* S x a linear operator such that (hj H(I -P)Ex = (I -P)x, x belonging to the domain oî E.If y is a solution of (1), then Ey = Ny implies H(I -P)Ey = H(I -P)Ny. Hence, (/ -P)y = H(I -P)Ny; and finally (2) y = Py + H(I -P)Ny.Thus, any solution of (1) is a solution of (2). If we also have that (h 2 ) EPx = PEx and (h 3 ) EH{I -P)Nx = {I -P)Nx, then from (2) we derive Ey = EPy + EH(I -P)Ny = PEy + (I -P)Ny.Hence, Ey -Ny = P(Ey -Ny). Thus, any solution y of (2) is a solution of (1) if and only if y satisfies (3) P(Ey -Ny) = 0. (1970). Primary 47H15.
AMS (MOS) subject classifications
We study the convergence of Gauss-Seidel and nonlinear successive overrelaxation methods for finding the minimum of a strictly convex functional defined on R".
By Schauder's fixed point theorem and alternative method (bifurcation theory) an abstract existence theorem at resonance for operational equations is proved which contains as particular cases rather different existence theorems for ordinary and partial differential equations as those of Lazer and Leach and of Landesman and Lazer.
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