Dynamical Systems Method for solving nonlinear operator equations

Consider an operator equation (*) B(u) + u = 0 in a real Hilbert space, where > 0 is a small constant. The DSM (Dynamical Systems Method) for solving equation (*) consists of finding and solving a Cauchy problem: u̇ = Φ(t, u), u(0) = u0, t ≥ 0, which has the following properties: 1) it has a global solution u(t), 2) this solution tends to a limit as time tends to infinity, i.e., u(∞) exists, 3) this limit solves the equation B(u) = 0, i.e., B(u(∞)) = 0. Existence of the unique solution is proved by the DSM for equation (*) with operators B defined on all of H and satisfying a spectral assumption: ||[B′(u) + I]−1|| ≤ c/ for any u ∈ H, where c > 0 is a constant independent of u and ∈ (0, 0). If = 0 and equation (**) B(u) = 0 is solvable, the DSM yields a solution to (**). The case when B is a monotone, hemicontinuous, defined on all of H operator is also studied, and DSM is justified for this case, that is, above properties 1),2), and 3) are proved. A sufficient condition for surjectivity of a nonlinear map is given. Meyer’s generalization of the Hadamard theorem about global homeomorphisms is proved by the DSM. The DSM method is justified for non-differentiable, hemicontinuous, monotone, defined on all of H operators. 2000 AMS Subject Classification:34R30, 35R25, 35R30, 37C35, 37L05, 37N30, 47A52, 47J06, 65M30, 65N21.