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Steepest descent and differential equations. (English) Zbl 0576.65053
Let H, K, S be real Hilbert spaces and $$F=H\to K$$, B:H$$\to S$$ operators with local Lipschitz derivatives. For the solution of (1) $$F(u)=0$$, $$B(u)=0$$ the steepest descent process (2) $$z'(t)=-P(x)(\nabla \phi)(z(t))$$, $$t\geq 0$$, $$z(0)=x^ 0$$, is considered where $$\phi (x)=\| F(x)\|^ 2$$ and P(x) is the orthogonal projection of H onto the nullspace of B’(x). In generalization of known results to this constrained case, it is shown that when (3) $$\| P(x)(\nabla \phi)(x)\| \geq c\| F(x)\|$$, $$x\in \Omega \subset H$$, $$c>0$$, and (4) rge(z)$$\in \Omega$$ then $$u=\lim_{\epsilon \to \infty}z(t)$$ exists and $$F(u)=0$$, $$B(u)=B(x^ 0)$$. The conditions (3) and (4) are discussed in some detail and examples involving several boundary value problems are given.
Reviewer: W.C.Rheinboldt

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
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