Fixed point theorems without continuity condition in non-solid extended abstract metric spaces with applications

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.