Abstract Differential Equations and \(L^{q,\alpha } \)-Hölder Functions

We introduce the class of \(L^{q,\alpha }\) $L^{q,\alpha }$ -Hölder functions and study the local and global existence and uniqueness of solution for abstract evolution differential equations assuming that the non-linear term is a \(L^{q,\alpha }\) $L^{q,\alpha }$ -Hölder function. In the last section, some examples with applications to partial differential equations are presented.