Pseudo-homotopies between maps on g-growth hyperspaces of continua

We introduce the concept of g-growth hyperspace: if $X$ is a continuum, then a non-empty subset $\mathcal H$ of $2^X$ is a g-growth hyperspace of $X$ provided that if $\mathcal A$ is a subcontinuum of $2^X$ and $\mathcal A \cap \mathcal H \neq \emptyset $, then $\bigcup \mathcal A \in \mathcal H$. We study pseudo-homotopies between maps of hyperspaces of continua. As a consequence, we show that pseudo-contractibility and contractibility are equivalent in g-growth hyperspaces.