Stability in semilinear problems
The author derives results concerning the continuous dependence of solutions on the right-hand side for a semilinear operator equation \(Lu=\nabla g(u)\), by assuming that \(L: D(L)\subset H\to H\) (\(H\) -- a Hilbert space) is selfadjoint, which a closed range, and \(g: H\to \mathbb{R}\) is continuous convex on \(H\) and Gâteaux differentiable on \(D(L)\). Using these results, he obtains theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order \(\ddot u+ au= D_uF(t, u,\omega)\), \(t\in [0,\pi]\) a.e., with the Dirichlet boundary conditions \(y(0)= u(\pi)= 0\), where \(a\geq 1\), and \(\omega\) is a functional parameter.
scientific article; zbMATH DE number 1437339
scientific article