Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces

Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′(u) − F ′(v)‖ ≤ ω(‖u − v‖), where ω ∈ C([0,∞)), ω(0) = 0, ω(r) is strictly growing on [0,∞). Denote A(u) := F ′(u), where F ′(u) is the Fréchet derivative of F , and Aa := A + aI. Assume that (*) ‖A−1 a (u)‖ ≤ c1 |a|b , |a| > 0, b > 0, a ∈ L. Here a may be a complex number, and L is a smooth path on the complex a-plane, joining the origin and some point on the complex a−plane, 0 < |a| < 0, where 0 > 0 is a small fixed number, such that for any a ∈ L estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) u̇(t) = −A a(t)(u(t))[F (u(t)) + a(t)u(t)− f ], u(0) = u0, u̇ = du dt , converges to y as t → +∞, where a(t) ∈ L, F (y) = f , r(t) := |a(t)|, and r(t) = c4(t + c2) −c3 , where cj > 0 are some suitably chosen constants, j = 2, 3, 4. Existence of a solution y to the equation F (u) = f is assumed. It is also assumed that the equation F (wa) + awa − f = 0 is uniquely solvable for any f ∈ X, a ∈ L, and lim|a|→0,a∈L ‖wa − y‖ = 0. MSC 2000, 47J05, 47J06, 47J35