# Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces]{Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces

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.

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