A Perturbed Maximum Principle of Borwein-Preiss Type in a Real Hilbert Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, let $A$ be a nonempty subset of $H$, and let $\Phi:A\to\mathbb{R}$ be bounded above and have closed superlevel sets in $H$. Let $\mu,\lambda\in\mathbb{R}$ be positive and let $x_{0}\in A$ satisfy

$$\sup_{x\in A}\Phi(x)\le\Phi(x_{0})+\mu\lambda^{2}.$$

Then there exist $\bar{y}\in H$ and $\bar{x}\in A$ such that the following hold, where $\Psi:A\to\mathbb{R}$ denotes the function given by

$$\Psi(x)=\Phi(x)-\mu\,|x-\bar{y}|^{2}.$$

1. (Localisation)(i) $|\bar{y}-x_{0}|\le 4\lambda$, $|\bar{x}-x_{0}|\le 4\lambda$ and $|\bar{x}-\bar{y}|\le 8\lambda$.

2. (Sequentially strict maximum)(ii) The function $\Psi$ attains a sequentially strict maximum on $A$ at $\bar{x}$.

3. (Near-optimality)(iii) $\displaystyle\sup_{x\in A}\Phi(x)\le\Phi(\bar{x})+2\mu\lambda^{2}$.