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

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• 作者：Kazimierz Włodarczyk
• 关键词：fuzzy quasi ; triangular space ; Pompeiu ; Hausdorff fuzzy set ; set ; valued fuzzy contraction ; single ; valued fuzzy contraction ; periodic point ; fixed point ; convergence of dynamic process
• 刊名：Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：2,199 KB
• 刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
• 出版者：Springer International Publishing
• ISSN：1687-1812

文摘

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.