Cone $$b_2$$ b 2 -metric spaces over Banach algebra with applications

Abstract

In the present paper, we first introduce the concept of Cone b 2-metric space over Banach algebras which generalizes the notions of b 2-metric space and cone metric spaces over Banach algebra. Next, we define generalized Lipschitz and expansive maps in the new structure and establish the existence and uniqueness of fixed points for such mappings in Cone b 2-metric space over Banach algebra. The results presented here generalize and extend some recent results of Singh et al. (Comment Math 52(2):143-151, 2012) and Wang et al. (Math Japonica 29:631-636, 1984). Also, we illustrate the result by an appropriate example. Finally, an application to integral equations is given to demonstrate the effectiveness of our acquired results.