Resumen: We give an original representation integral formula for the resolvent families associated to the fractional Cauchy equation with memory effects (Formula presented.) where (Formula presented.) and A is a sectorial operator on a Banach space X. Moreover, we get spatial bounds for the resolvent families in order to study global or blow up mild solutions when the nonlinearity f satisfies a locally Lipschitz condition. The case of critical nonlinearities is also treated. Idioma: Inglés DOI: 10.1002/mana.201800342 Año: 2020 Publicado en: Mathematische Nachrichten 293 (2020), 1846-1872 ISSN: 0025-584X Factor impacto JCR: 1.228 (2020) Categ. JCR: MATHEMATICS rank: 111 / 330 = 0.336 (2020) - Q2 - T2 Factor impacto SCIMAGO: 0.913 - Mathematics (miscellaneous) (Q1)