By Eberhard Zeidler (auth.)

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Extra resources for Nonlinear Functional Analysis and its Applications: II/B: Nonlinear Monotone Operators

Example text

Conversely, iff: [0, 1]-+ IRis monotone increasing, then for u, vEX, (A(u + v)- Au, v) = f(1)- f(O) ~ 0. 4. The Main Theorem on Strongly Monotone Operators, and the Projection-Iteration Method We consider the operator equation Au= b, UEX, (47) and make the following assumptions. , there are numbers c > 0 and L > 0 such that, for all u, v E X, (Au- Av,u- and IIAu- v) ~ cllu- vll 2 Avll :::;; Lllu- vii. 504 25. Lipschitz Continuous, Strongly Monotone Operators (H2) Let dim X = oo, and let (Xn) be a Galer kin scheme in the separable H-space X, where Xn = span {w1n, ...

The function

0. (b) Convergence of the iterative method - Aun = - Aun- 1 Un = - t(Lun- 1 0 on oG, - f) on G, n = 1, 2, ...

32c) X •. : X--+ X. is where u 0 = 0, and X 1 <:; X 2 <:; .. ·<:;X with X= the orthogonal projection operator from X onto the finite-dimensional linear subspace x. of X. , UEX, together with the Euler equation f'(u) = 0, UEX, where X is a real reflexive B-space. In this connection, we shall prove the following fundamental results: the main the main the main the main theorem theorem theorem theorem on minimum problems; on convex minimum problems; on monotone potential operators; on pseudomonotone potential operators.