*(English)*Zbl 1145.35076

The authors prove the existence of integral solutions to the nonlocal Cauchy problem

in a Banach space $X$, where $A:D\left(A\right)\subset X\to X$ is $m$-accretive and such that $A$ generates a compact semigroup, $F:[0,T]\times X\to {2}^{X}$ has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and $g:C([0,T];\overline{D\left(A\right)})\to \overline{D\left(A\right)}$. The case when $A$ 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, ${u}^{\text{'}}\left(t\right)\in -Au\left(t\right)+f\left(t\right)$, on the same interval $I=[0,T]$, with initial condition $u\left(0\right)={u}_{0}$. For the linear problem, the integral solution is defined by means of the integral inequality

for all $x\in D\left(A\right)$, $y\in Ax$ and $u\left(0\right)={u}_{0}$, continuous. An integral solution for the nonlinear equation is defined as satisfying the linear equation for some integrable $f\left(t\right)\in F(t,u(t\left)\right)$. Then, using the existence for the linear equation, one shows the existence in the nonlinear case. An application is provided for heat equation.