Summable Orbits and the Minimal Caristi Potential

Abstract

Let $(M,d)$ be a complete metric space and $f:M\to M$. Associated with $f$ is the \emph{orbit potential} \[ \varphi_f(x)=\sum_{n\ge0} d\bigl(f^n(x),f^{n+1}(x)\bigr)\in[0,+\infty], \] whose finiteness at a single point expresses the summability of the corresponding forward orbit. We show that, whenever $\varphi_f$ is \lsc, the map $f$ has a fixed point if and only if some orbit is summable. The lower semicontinuity of each individual gap $x\mapsto d(f^n(x),f^{n+1}(x))$ is a convenient sufficient condition for this hypothesis, and we exhibit an example in which it fails while $\varphi_f$ remains \lsc, so that the criterion applies strictly beyond that condition. Under the same hypothesis we observe that the existence of a summable orbit is equivalent to $f$ being a Caristi map, and that $\varphi_f$ is then the \emph{minimal} Caristi potential, in the sense that $\varphi_f\le\varphi-\inf\varphi$ for every admissible potential $\varphi$. Finally we delimit the reach of the criterion among generalized contractions: it recovers the Bianchini--Grandolfi contractions (those governed by a summable comparison function), and in particular the Banach contraction principle, but it does not subsume the Boyd--Wong or Matkowski classes, whose orbits need not be summable; the comparison function $\psi(t)=t/(1+t)$ marks this boundary explicitly.