Localisation at a Sequentially Strict Maximum of a Quadratically Penalised Difference

In the setting of Real Hilbert Spaces: Standing Notation and Background and Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, let $E$ be a real inner product space, with its norm $|\cdot|$ and distance $d$ as fixed there, and let $E\times E$ be the product of $E$ with itself, a real inner product space by Properties of the Product of Two Real Inner Product Spaces (i), whose distance is written $d_{\times}$. Let $|s|$ be the absolute value of a real number $s$.

Let $A\subseteq E$ be nonempty and let $u,v:A\to\mathbb{R}$; let $-v:A\to\mathbb{R}$ be the function whose value at $y\in A$ is the additive inverse of $v(y)$. Assume that $u$ and $-v$ have closed superlevel sets in $E$. Let $\alpha\in\mathbb{R}$, let $\tfrac{\alpha}{2}$ be the quotient of $\alpha$ by $2=1+1$, and let $\Phi:A\times A\to\mathbb{R}$ be given by

$$\Phi(x,y)=u(x)-v(y)-\tfrac{\alpha}{2}\,|x-y|^{2},$$

where $|x-y|^{2}=|x-y|\,|x-y|$. Suppose $\Phi$ attains a sequentially strict maximum on $A\times A$ at a point $(\bar{x},\bar{y})$, the ambient metric space being $(E\times E,d_{\times})$.

(i) Then for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\eta\in\mathbb{R}$ such that every $(x,y)\in A\times A$ satisfying

$$\Phi(\bar{x},\bar{y})-\eta<\Phi(x,y)$$

satisfies also

$$|x-\bar{x}|<\varepsilon,\qquad |y-\bar{y}|<\varepsilon,\qquad |u(x)-u(\bar{x})|<\varepsilon,\qquad |v(y)-v(\bar{y})|<\varepsilon .$$