Multiple fixed points of weakly inward mapping. (Q1860750)

This article deals with some fixed point theorems for weakly inward mappings, i.e., mappings \(A:D\to E\) defined on a convex subset of a Hilbert space \(E\) and satisfying the condition \[ Ax\in \overline{\{(1-\lambda) x+\lambda y: \lambda\geq 0,\;y\in D\}}. \] The main results are theorems about the existence of two and three fixed points for a mapping \(A\) that is potential \((A=g')\), weakly inward, locally Lipschitzian, and such that the functional \(\frac12\| x\|^2- g(x)\) satisfies the Palais-Smale condition. The proofs are based on the study of the Cauchy problem \(\frac{dx}{dt}= Ax-x\), \(x(0)= x_0\).