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Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances. (English) Zbl 1258.47075

Definition 1. Let \(X\) be a nonempty set.
(i) A quasi-pseudometric on \(X\) is a map \(p: X\times X\to [0,\infty)\) such that: \((P_1)\): \(\forall x\in X\{p(x,x)= 0\}\), and \((P_2)\): \(\forall x,y,z\in X\{p(x,z)\leq p(x,y)+ p(y,z)\}\). For given quasi-pseudometric \(p\) on \(X\), a pair \((X,p)\) is called a quasi-pseudometric space. A quasi-pseudometric space \((X,p)\) is called Hausdorff if \(\forall x,y\in X\{x\neq y\Rightarrow p(x,y)> 0 \vee p(y,x)> 0\}\).
(ii) Each family \({\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}\) of quasi-pseudometrics \(p_\alpha: X\times X\to [0,\infty)\), \(\alpha\in{\mathcal A}\), is called a quasi-gauge space.
(iii) Let the family \({\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}\) be a quasi-gauge on \(X\). The topology \(\tau({\mathcal P})\), having as a subbase the family \({\mathcal B}(D)= \{B(x,\varepsilon_\alpha) : x\in X\), \(\varepsilon_\alpha> 0\), \(\alpha\in{\mathcal A}\}\) of all balls \(B(x,\varepsilon_\alpha)= \{y\in X: p_\alpha(x,y)< \varepsilon_\alpha\), \(x\in X\), \(\alpha> 0\), \(\alpha\in{\mathcal A}\}\), is called the topology induced by \({\mathcal P}\) on \(X\).
(iv) A topological space \((X,\tau)\) such that there is a quasi-gauge \({\mathcal P}\) on \(X\) with \(\tau= \tau({\mathcal P})\) is called a quasi-gauge space and is denoted by \((X,{\mathcal P})\).
(v) A quasi-gauge space \((X,{\mathcal P})\) is called Hausdorff if the quasi-gauge \({\mathcal P}\) has the property: \(\forall x,y\in X\{x\neq y\Rightarrow\exists\alpha \in{\mathcal A}\{ p_\alpha(x,y)> 0 \vee\,p_\alpha(y,x)> 0\}\}\).
Recently, the concept of generalized pseudodistance was introduced by the authors [Appl. Math. Lett. 24, No. 3, 325-328 (2011; Zbl 1206.54068); Fixed Point Theory Appl. 2011, Article ID 712706 (2011; Zbl 1213.81161)].
In the present paper, the authors introduce the notions of generalized quasipseudodistances in quasi-gauge spaces.
Definition 2. Let \((X,{\mathcal P})\) be a quasi-gauge space. The family \({\mathcal I}= \{J_\alpha: \alpha\in{\mathcal A}\}\) of maps \(J_\alpha: X\times X\to [0,\infty)\), \(\alpha\in{\mathcal A}\), is said to be of left (right) \({\mathcal I}\)-family of generalized quasipseudodistances on \(X\) if the following conditions hold:
\(({\mathcal I}_1)\): \(\forall\alpha\in{\mathcal A}\), \(\forall x,y,z\in X\{J_\alpha(x,z)\leq J_\alpha(x,y)+ J_\alpha(y,z)\}\); and
\(({\mathcal I}_2)\) For any sequences \((u_m: m\in \mathbb {N})\) and \((v_m: m\in \mathbb {N})\) in \(X\) satisfying \[ \forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\;\sup_{n> m}\,J_\alpha(u_m,u_n)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\;\sup_{n> m}\, J_\alpha(u_n, u_m)= 0\Biggr\}\Biggr) \] and \[ \forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, (J_\alpha(v_m,u_m)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, J_\alpha(u_m, v_m)= 0\Biggr\}\Biggr), \] the following holds: \[ \forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, p_\alpha(v_m,u_m)= 0\Biggr\}\;\Biggl(\forall\alpha\in{\mathcal A}\Biggl\{\lim_{m\to\infty}\, p_\alpha(u_m, v_m)= 0\Biggr\}\Biggr). \] In this paper, the notion of generalized quasipseudodistances in quasi-gauge sapces is used in natural way to define contractions of leader type and to obtain for these contractions periodic and fixed point theorems without Hausdorff and sequentially complete assumptions about these spaces and without graph assumptions about these contractions.

MSC:

47H99 Nonlinear operators and their properties
54A05 Topological spaces and generalizations (closure spaces, etc.)
54E15 Uniform structures and generalizations
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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