the moore-penrose inverse: a fixed point method

‎in this paper‎, ‎we establish several fixed-point theorems in the setting of orthogonal spaces (o-spaces)‎. ‎our primary contribution is to extend the fixed-point theorem of cabrera‎, ‎harjani‎, ‎and sadarangani $[text{ann‎. ‎univ‎. ‎ferrara} (2013)$ 59:251--258] to o-sets‎, ‎broadening its applicability‎. ‎additionally‎, ‎we present novel results concerning the moore--penrose inverse‎, ‎exploring its properties and potential applications in functional analysis‎. ‎these findings contribute to both fixed-point theory and generalized inverses in partially structured spaces‎. ‎also‎, ‎we show that the moore--penrose inverse of $a$‎, ‎denoted by $a^dagger$‎, ‎can be computed as the fixed point of a contractive‎, ‎orthogonal‎, ‎and $perp$-preserving mapping $t$‎.