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Let A$\mathcal{A}$ be an index set, and C={Cα}α∈A∈[1;∞)A$C=\{C_{\alpha}\}_{\alpha\in \mathcal{A}}\in{}[1;\infty)^{\mathcal{A}}$. Fuzzy quasi-triangular space is defined to be (X,MC;A,∗)$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$, where X is a nonempty set, a fuzzy family MC;A={Mα:X×X×(0;∞)→(0;1],α∈A}$\mathcal{M}_{C;\mathcal{A}}=\{M_{\alpha }:X\times X\times(0;\infty)\rightarrow(0;1],\alpha\in\mathcal{A}\}$ satisfies ∀α∈A∀x,y,z∈X∀t,s∈(0;∞){Mα(x,y,t)∗Mα(y,z,s)≤Mα(x,z,Cα(t+s))}$\forall_{\alpha\in\mathcal{A}}\forall_{x,y,z\in X}\forall _{t,s\in(0;\infty)}\{M_{\alpha}(x,y,t)\ast M_{\alpha}(y,z,s)\leq M_{\alpha}(x,z,C_{\alpha}(t+s))\}$, and ∗ is the continuous t-norm ∗:[0;1]×[0;1]→[0;1]$\ast:[0;1]\times{}[0;1]\rightarrow{}[0;1]$. In (X,MC;A,∗)$(X,\mathcal{M}_{C;\mathcal{A}},\ast)$, left (right) G$\mathcal{G}$-families and W$\mathcal{W}$-families KC;A$\mathcal{K}_{C;\mathcal{A}}$ generated by MC;A$\mathcal{M}_{C;\mathcal{A}}$ (KC;A$\mathcal{K}_{C;\mathcal{A}}$ generalize MC;A$\mathcal {M}_{C;\mathcal{A}}$) are defined and described. Using families KC;A$\mathcal {K}_{C;\mathcal{A}}$, three kinds of left (right) fuzzy sets of Pompeiu-Hausdorff type on 2X×2X×(0;∞)$2^{X}\times2^{X}\times(0;\infty)$ are introduced. Using these fuzzy sets, three kinds of left (right) set-valued fuzzy contractions T:X→2X$T:X\rightarrow2^{X}$ are constructed, and for such fuzzy contractions, conditions guaranteeing the existence of periodic points and left (right) MC;A$\mathcal {M}_{C;\mathcal{A}}$-convergence to these periodic points of dynamic processes (wm:m∈{0}∪N)$(w^{m}:m\in\{0\}\cup\mathbb{N})$, wm∈T(wm−1)$w^{m}\in T(w^{m-1})$ for m∈N$m\in \mathbb{N}$, starting at w0∈X$w^{0}\in X$, are established. Moreover, in (X,MC;A,∗)$(X,\mathcal {M}_{C;\mathcal{A}},\ast)$, using left (right) G$\mathcal{G}$-families and W$\mathcal{W}$-families KC;A$\mathcal{K}_{C;\mathcal{A}}$ generated by MC;A$\mathcal{M}_{C;\mathcal{A}}$, two kinds of left (right) single-valued fuzzy contractions T:X→X$T:X\rightarrow X$ are constructed, and for such fuzzy contractions, the convergence, existence, approximation, uniqueness, periodic point, and fixed point result is also obtained. Examples are provided.

Fuzzy quasi-triangular spaces, fuzzy sets of Pompeiu-Hausdorff type, and another extensions of Banach and Nadler theorems