Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms

We investigate the existence of solutions for a class of fractional evolution equations with nonlocal initial conditions and superlinear growth nonlinear functions in Banach spaces. By using the compactness of semigroup generated by the linear operator, we neither assume any Lipschitz property of the nonlinear term nor the compactness of the nonlocal initial conditions. Moreover, the approximation technique coupled with the Hartmann-type inequality argument allows the treatment of nonlinear terms with superlinear growth. Then combining with the Leray-Schauder continuation principle, we prove the existence results. Finally, the results obtained are applied to fractional parabolic equations with continuous superlinearly growth nonlinearities and nonlocal initial conditions including periodic or antiperiodic conditions, multipoint conditions and integral-type conditions.