Abstract
In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone b -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of X . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone b -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.
Highlights
It is well known that the fixed point theory is widely applied to almost all fields of quantitative sciences such as computer science, physics, and biology, especially since the famous Banach contraction principle was introduced in 1922 [1]
In 1968, Kannan [2] studied the following meaningful fixed point theorem, which is a generalization of Banach contraction principle
E mapping satisfying the contractive condition is known as Kannan-type contraction mapping, which is highly interesting since the contraction mapping does not need to be continuous
Summary
It is well known that the fixed point theory is widely applied to almost all fields of quantitative sciences such as computer science, physics, and biology, especially since the famous Banach contraction principle was introduced in 1922 [1]. We prove that the completeness of cone b-metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X. Let (X, d) be a T-orbitally compact cone b-metric space over Banach algebra A with a unit e and the coefficient s ≥ 1, where T: X ⟶ X is a generalized Reich–Ciric–Rus-type contraction mapping and orbitally continuous. Let (X, d) be a boundedly compact cone b-metric space over Banach algebra A with a unit e and the coefficient s ≥ 1. Let (X, d) be a T-orbitally compact cone b-metric space over Banach algebra A with a unit e and the coefficient s ≥ 1, where T: X ⟶ X is a generalized Kannantype contraction mapping and orbitally continuous. + a2d(y, Ty) + a3d(x, y), is clearly true. erefore, the mapping T has a unique fixed point in X by eorem 6
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