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Cone \(b_2\)-metric spaces over Banach algebra with applications. (English) Zbl 1433.46013
Summary: 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 B. Singh et al. [Commentat. Math. 52, No. 2, 143–151 (2012; Zbl 1304.54095)] and S. Wang et al. [Math. Japon. 29, 631–636 (1984; Zbl 0554.54023)]. 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.

MSC:
46B20 Geometry and structure of normed linear spaces
46B40 Ordered normed spaces
46J10 Banach algebras of continuous functions, function algebras
47H10 Fixed-point theorems
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