Existence results for fractional evolution equations with superlinear growth nonlinear terms

We investigate the initial value problem to a class of fractional evo-lution equations with superlinear growth nonlinear functions in Banach spaces. When the linear operator generates an extendable compact C0-semigroup of contractions and the nonlinearity f : [0, T] x X -> Y is Caratheodory contin-uous, with X and Y two real Banach spaces such that X subset of Y, the existence of mild solutions to such problems are established without assuming that the nonlinear term satisfies the transversality condition by combining an approx-imation technique with Leray-Schauder continuation principle. The obtained results allow to treat, as an application, fractional parabolic equations with continuous superlinear growth nonlinear term.