Let
$\mathcal{A}$
be an index set, and
$C=\{C_{\alpha}\}_{\alpha\in \mathcal{A}}\in{}[1;\infty)^{\mathcal{A}}$
. Fuzzy quasi-triangular space is defined to be
$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$
, where *X* is a nonempty set, a fuzzy family
$\mathcal{M}_{C;\mathcal{A}}=\{M_{\alpha }:X\times X\times(0;\infty)\rightarrow(0;1],\alpha\in\mathcal{A}\}$
satisfies
$\forall_{\alpha\in\mathcal{A}}\forall_{x,y,z\in X}\forall _{t,s\in(0;\infty)}\{M_{\alpha}(x,y,t)\ast M_{\alpha}(y,z,s)\leq M_{\alpha}(x,z,C_{\alpha}(t+s))\}$
, and ∗ is the continuous *t*-norm
$\ast:[0;1]\times{}[0;1]\rightarrow{}[0;1]$
. In
$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$
, left (right)
$\mathcal{G}$
-families and
$\mathcal{W}$
-families
$\mathcal{K}_{C;\mathcal{A}}$
generated by
$\mathcal{M}_{C;\mathcal{A}}$
(
$\mathcal{K}_{C;\mathcal{A}}$
generalize
$\mathcal {M}_{C;\mathcal{A}}$
) are defined and described. Using families
$\mathcal {K}_{C;\mathcal{A}}$
, three kinds of left (right) fuzzy sets of Pompeiu-Hausdorff type on
$2^{X}\times2^{X}\times(0;\infty)$
are introduced. Using these fuzzy sets, three kinds of left (right) set-valued fuzzy contractions
$T:X\rightarrow2^{X}$
are constructed, and for such fuzzy contractions, conditions guaranteeing the existence of periodic points and left (right)
$\mathcal {M}_{C;\mathcal{A}}$
-convergence to these periodic points of dynamic processes
$(w^{m}:m\in\{0\}\cup\mathbb{N})$
,
$w^{m}\in T(w^{m-1})$
for
$m\in \mathbb{N}$
, starting at
$w^{0}\in X$
, are established. Moreover, in
$(X,\mathcal {M}_{C;\mathcal{A}},\ast)$
, using left (right)
$\mathcal{G}$
-families and
$\mathcal{W}$
-families
$\mathcal{K}_{C;\mathcal{A}}$
generated by
$\mathcal{M}_{C;\mathcal{A}}$
, two kinds of left (right) single-valued fuzzy contractions
$T:X\rightarrow X$
are constructed, and for such fuzzy contractions, the convergence, existence, approximation, uniqueness, periodic point, and fixed point result is also obtained. Examples are provided.