In this paper, several fixed point results for different contractions in extended abstract metric spaces with non-solid cones are established. The definitions of Cauchy sequences, convergent sequences, and completeness with respect to normality in the non-solid extended abstract metric spaces are reintroduced. We prove that the fixed point exists and is unique in these spaces by removing the orbital continuity assumption. Our results generalize and extend several significant theorems in classical abstract metric spaces and standard metric spaces. A consequence, we present several examples demonstrating that our main theorems serve as powerful tools for solving equations in different space frameworks, for both solid and non-solid cones.

The purpose of this paper was to introduce several $b$-generalized contractive mappings in the framework of cone $b$-metric spaces over Banach algebras. The obtained contractions generalized and extended the counterparts in metric spaces, cone metric spaces, and $b$-metric spaces. Moreover, via weakening the completeness of the spaces, we gave some fixed point theorems for asymptotically regular mappings without considering the orbital continuity and $k$-continuity of the mappings. Those who need a specification is our results do not rely on the continuity of $b$-metric and the normality of cones. In addition, some nontrivial examples were presented to illustrate the superiority of our fixed point theorems.