Uniqueness of unimodal maximizers for the improvement dynamics

Let $b$ be a kernel satisfying Assumption MainKernel, let the nonlinearity satisfy Assumption MainNonl, and let $\mathcal{C}_K$ be the admissible class on which the functional $\mathcal{P}$ and the improvement dynamics are defined. Uniqueness of unimodal maximizers. There exists a unique maximizer of $\mathcal{P}$ in $\mathcal{C}_K$, and this maximizer is a global attractor for the improvement dynamics

$$U_j=\mathcal{T}(U_{j-1}),\qquad j\in\mathbb{N}.$$

The numerical scheme is reported to have robust convergence properties, while the preceding analysis establishes monotonicity of $\mathcal{P}$ and shows that accumulation points solve the nonlinear eigenvalue problem. Uniqueness and independence of the initial datum remain unproved in the supplied text.