Some periodic and fixed point theorems on quasi-b-gauge spaces

Abstract

Abstract The notions of a quasi-b-gauge space ( U , Q s ; Ω ) $(U,\textsl{Q}_{s ; \Omega })$ and a left (right) J s ; Ω $\mathcal{J}_{s ; \Omega }$ -family of generalized quasi-pseudo-b-distances generated by ( U , Q s ; Ω ) $(U,\textsl{Q}_{s ; \Omega })$ are introduced. Moreover, by using this left (right) J s ; Ω $\mathcal{J}_{s ; \Omega }$ -family, we define the left (right) J s ; Ω $\mathcal{J}_{s ; \Omega }$ -sequential completeness, and we initiate the Nadler type contractions for set-valued mappings T : U → C l J s ; Ω ( U ) $T:U\rightarrow Cl^{\mathcal{J}_{s ; \Omega }}(U)$ and the Banach type contractions for single-valued mappings T : U → U $T: U \rightarrow U$ , which are not necessarily continuous. Furthermore, we develop novel periodic and fixed point results for these mappings in the new setting, which generalize and improve the existing fixed point results in the literature. Examples validating our obtained results are also given.

Keywords

• Generalized quasi-pseudo-b-distance
• Banach contraction
• Nadler contraction
• Quasi-b-gauge space
• Fixed point
• Periodic point