Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study the existence of classical and generalized solutions for nonlinear differential inclusions $x'(t) \in F(t,x(t))$ in Hilbert spaces in which the multifunction $F$ on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condensing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper.

Keywords: differential inclusions, set-valued integral, semimonotone and hemicontinuous multifunctions, condensing multifunctions

MSC numbers: Primary 34A60; Secondary 49K24, 47H05