Abstract
In the present paper, we introduce the notion of A-cone metric spaces over Banach algebra as a generalization of A-metric spaces and cone metric spaces over Banach algebra. We also defined generalized Lipschitz and expansive maps in such maps and establish some fixed point theorems for such maps in the setting of the new space. As an application, we prove a theorem for integral equation. We provide illustrative example to verify our results. Our results generalize and unify some well-known results in the literature.
Highlights
Gahler (1963) introduced the concept of 2-metric spaces as a generalization of an ordinary metric space
We introduce the notion of A-cone metric space and focus on fixed point theorem of generalized maps in such space
We introduce a new generalization of metric spaces, called as A-cone metric spaces over Banach algebra, study some fixed point theorems and an application to integral equations
Summary
Gahler (1963) introduced the concept of 2-metric spaces as a generalization of an ordinary metric space. On the other hand, Huang and Zhang (2007) generalized the notion of a metric space by replacing the real numbers by ordered Banach spaces and defined cone metric spaces and proved some fixed point theorems of contractive maps in such space using the normality condition. We introduce a new generalization of metric spaces, called as A-cone metric spaces over Banach algebra, study some fixed point theorems and an application to integral equations. For other definitions and related results on cone metric space over Banach algebra, we refer to Liu and Xu (2013). The pair (X, d) is called an A-cone metric space over Banach algebra. Definition 3.5 Let (X, d) is an A-cone metric space over Banach algebra A.
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