  We introduce new contraction properties called the generalized
$p$-contraction property for $p$-energy forms as generalizations of many
well-known inequalities, such as Clarkson's inequalities, the strong
subadditivity and the ``Markov property'' in the theory of nonlinear Dirichlet
forms, and show that any $p$-energy form satisfying Clarkson's inequalities is
Fr\'{e}chet differentiable. We also verify the generalized $p$-contraction
property for $p$-energy forms constructed by Kigami [Mem. Eur. Math. Soc. 5
(2023)] and by Cao--Gu--Qiu [Adv. Math. 405 (2022), no. 108517]. As a general
framework of $p$-energy forms taking into consideration the generalized
$p$-contraction property, we introduce the notion of $p$-resistance form and
investigate fundamental properties for $p$-harmonic functions with respect to
$p$-resistance forms. In particular, some new estimates on scaling factors of
$p$-energy forms are obtained by establishing H\"{o}lder regularity estimates
for harmonic functions, and the $p$-walk dimensions of the generalized
Sierpi\'{n}ski carpets and $D$-dimensional level-$l$ Sierpi\'{n}ski gasket are
shown to be strictly greater than $p$.
