NEW QUALITATIVE PROPERTIES OF SOLUTIONS TO NONLINEAR NONLOCAL CAUCHY PROBLEMS

We introduce new concepts of asymptotically anti-periodic function and semi-Lipschitz continuity. The former is a natural generalization of the well-known anti-periodic function. Then, sufficient conditions, ensuring the existence of asymptotically anti-periodic mild solutions to a Cauchy problem of nonlinear evolution equation with nonlocal initial condition, are established. It is mentioned that one of our main results is proved in the absence of the compactness and Lipschitz continuity of nonlocal item and of the Lipschitz continuity of nonlinearity. Finally, an example is presented as an application.