On Uniqueness of Fixed Points and Their Regularity

In this paper, we study the problem of uniqueness of fixed points for operators acting from a Banach space X into a subspace Y with a stronger norm. Our main objective is to preserve the expected regularity of fixed points, as determined by the norm of Y, while analyzing their uniqueness without imposing the classical or generalized contraction condition on Y. The results presented here provide generalized uniqueness theorems that extend existing fixed-point theorems to a broader class of operators and function spaces. The results are used to study fractional initial value problems in generalized Hölder spaces.