Semilinear integrodifferential equations with nonlocal initial conditions

Of concern is the Cauchy problem for semilinear integrodifferential equations with nonlocal initial conditions. Under general and natural hypotheses, we establish some new theorems about the existence and uniqueness of solutions for the Cauchy problem. As a consequence, we unify and extend the corresponding theorems given previously for the Cauchy problem for differential equations or integrodifferential equations with nonlocal initial conditions. Moreover, we present two examples, one of which comes from heat conduction in materials with memory, to show that the existing results are not applicable to them, in contrast with ours.