Gradient method for non-uniformly convex functionals in Hilbert space (Q2755027)

The main result concerns the minimalization of twice Gâteaux differentiable convex real valued functionals \(\phi\) on Hilbert spaces with the local uniformity property of curvature \(m(\|u\|) \|h\|^2 \leq \langle \phi''(u) h,h \rangle \leq M(\|u\|) \|h\|^2\) for suitable positive functions \(m,M\). Namely, if \(M\) is increasing and \(m\) is decreasing with \(\lim_{t\to\infty} tm(t)=\infty\), then \(\phi\) admits a unique point of minimum \(u^*\), and given any vector \(u_0\), there can be constructed a constant \(\alpha(u_0)>0\) such that the iteration \(u_{n+1} := u_n-\alpha(u_0) \phi'(u_n)\) converges in linear order to \(u^*\). This result is then applied to give iterative solutions converging in linear order for the equation \(A(x)=b\) with a non-linear Hilbert space operator having self-adjoint Gâteaux derivative and satisfying some local uniform ellipticity properties. Finally, hence an iterative solution of \(A(x)=b\) is also developed for some interesting cases where \(A\) is a non-differentiable Hilbert space operator which can be transformed to a differentiable one in the energy space of a suitable linear operator.