Document Type : Original Article
Authors
1
Department of Mathematics, Sagar Institute of Engineering, Technology and Research, Ratibad Bhopal (M.P.), India
2
Department of Mathematics, Desh Bhagat University, Mandi Gobindgarh, Punjab, India
3
Desh Bhagat university
Abstract
Let \(C=\{C_{\alpha}\}_{\alpha\in\mathcal{A}}\in[1;\infty)^{\mathcal{A}}\) with index set \(\mathcal{A}\). A quasi-triangular space \((X,\mathcal{P}_{C;\mathcal{A}})\) is a set X with family \(\mathcal{P}_{C;\mathcal{A}}=\{p_{\alpha}:X^{2}\rightarrow[0,\infty),\alpha \in \mathcal{A}\}\) satisfying \(\forall_{\alpha\in\mathcal{A}}\forall _{u,v,w\in X}\{p_{\alpha}(u,w)\leq C_{\alpha}[p_{\alpha }(u,v)+p_{\alpha }(v,w)]\}\). In \((X,\mathcal{P}_{C;\mathcal{A}})\), using the left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal {P}_{C;\mathcal{A}}\) (\(\mathcal{P}_{C;\mathcal{A}}\) is a particular case of \(\mathcal {J}_{C;\mathcal{A}}\)), we establish theorems concerning left (right) \(\mathcal {P}_{C;\mathcal{A}}\)-convergence, existence, periodic point, fixed point, and (when\((X,\mathcal{P}_{C;\mathcal{A}})\) is separable) uniqueness for \(\mathcal{J}_{C;\mathcal{A}}\)-contractions and weak \(\mathcal {J}_{C;\mathcal{A}}\)-contractions \(T:X\rightarrow X\) satisfying
\begin{eqnarray*}\begin{aligned}
& \forall_{x,y\in X}\forall _{\alpha\in\mathcal{A}}\forall_{\varepsilon>0}\exists_{\eta >0}\exists _{r\in\mathbb{N}}\forall_{s,l\in\mathbb{N}} \{J_{\alpha }(T^{[s]}(x),T^{[s+r]}(x)) + J_{\alpha }(T^{[l]}(y),T^{[l+r]}(y)) < \eta+\varepsilon \\ & \Rightarrow C_{\alpha }J_{\alpha}(T^{[s+r]}(x),T^{[l+r]}(y))<\varepsilon\}
\end{aligned}\end{eqnarray*}
and
\begin{eqnarray*}\begin{aligned}
& \exists _{w^{0}\in X}\forall_{\alpha\in\mathcal{A}}\forall_{\varepsilon >0}\exists_{\eta>0}\exists_{r\in\mathbb{N}} \forall_{s,l\in \mathbb{N}} \{J_{\alpha}(T^{[s+r]}(w^{0}), T^{[s]}(w^{0}))+J_{\alpha}(T^{[l]}(w^{0}), T^{[l+r]}(w^{0}))<\eta+\varepsilon \\ & \Rightarrow C_{\alpha}J_{\alpha }(T^{[s+r]}(w^{0}),T^{[l+r]}(w^{0}))<\varepsilon\},
\end{aligned} \end{eqnarray*}
respectively. The spaces \((X,\mathcal{P}_{C;\mathcal{A}})\), in particular, generalize metric, ultrametric, quasi-metric, ultra-quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra-quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra-quasi-gauge, and partial quasi-gauge spaces. Results are new in all these spaces. Examples are provided.
Keywords
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