Steepest descent and differential equations
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.
scientific article; zbMATH DE number 3921848