Existence results for fractional evolution equations with superlinear growth nonlinear terms

Abstract

We investigate the initial value problem to a class of fractional evolution equations with superlinear growth nonlinear functions in Banach spaces. When the linear operator generates an extendable compact $ C_0 $-semigroup of contractions and the nonlinearity $ f:[0,T]\times X\to Y $ is Carathéodory continuous, with $ X $ and $ Y $ two real Banach spaces such that $ X\subseteq Y $, the existence of mild solutions to such problems are established without assuming that the nonlinear term satisfies the transversality condition by combining an approximation technique with Leray-Schauder continuation principle. The obtained results allow to treat, as an application, fractional parabolic equations with continuous superlinear growth nonlinear term.