A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space that is complete with $X\ne\varnothing$, and let $f:X\to\mathbb{R}$ be upper semicontinuous on $X$ and bounded above. Let $G\ge0$ be real and let $g:X\times X\to\mathbb{R}$ satisfy $g(x,x)=0$ and $0\le g(x,y)\le G$ for all $x,y\in X$, be such that $g(\cdot,y):X\to\mathbb{R}$ is lower semicontinuous on $X$ for every $y\in X$, and be such that for every real $\eta>0$ there is a real $\beta>0$ with $d(x,y)<\eta$ for all $x,y\in X$ with $g(x,y)\le\beta$. Let $(c_{k})_{k\in\mathbb{N}}$ be a sequence of positive reals whose series converges, let $\varepsilon>0$ be real, and let $x_{1}\in X$ satisfy $f(x_{1})\ge\sup_{x\in X}f(x)-\varepsilon$.

Then there are $\bar{x}\in X$ and a sequence $(x_{k})_{k\in\mathbb{N}}$ in $X$ whose first term is the given point $x_{1}$ such that, for every $x\in X$, the series $\sum_{k=1}^{\infty}c_{k}\,g(x,x_{k})$ converges, and, writing

$$\Phi(x)=f(x)-\sum_{k=1}^{\infty}c_{k}\,g(x,x_{k})\qquad(x\in X),$$

the following hold.

1. (Centres)¶ The sequence $(x_{k})_{k\in\mathbb{N}}$ converges to $\bar{x}$ in $(X,d)$.

2. (Value)¶ $\Phi(\bar{x})\ge f(x_{1})$.

3. (Strict maximum)¶ $\Phi(x)<\Phi(\bar{x})$ for every $x\in X$ with $x\ne\bar{x}$.

4. (Localisation)¶ $c_{1}\,g(\bar{x},x_{1})\le\varepsilon$.