The authors prove the existence of integral solutions to the nonlocal Cauchy problem
in a Banach space , where is -accretive and such that generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when depends on time is also considered.
The proof of the basic result is constructed as follows: First, one considers the linear counterpart of the above equation, , on the same interval , with initial condition . For the linear problem, the integral solution is defined by means of the integral inequality
for all , and , continuous. An integral solution for the nonlinear equation is defined as satisfying the linear equation for some integrable . Then, using the existence for the linear equation, one shows the existence in the nonlinear case. An application is provided for heat equation.