We survey and announce some current results on the existence, the viability, and the topological structure of the viable solutions of differential equations and inclusion in Banach spaces under set constraints. Some new results concerning semilinear differential inclusions with state variables constrained to the so-called regular and strictly regular sets, together with their applications, are presented and discussed.

Suppose $X$ is a real $q$-uniformly smooth Banach space and $F,K:X\to X$ with $D\left(K\right)=F\left(X\right)=X$ are accretive maps. Under various continuity assumptions on $F$ and $K$ such that $0=u+KFu$ has a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is imposed on $K$ and the operators $K$ and $F$ need not be defined on compact subsets of $X$. Our method of proof is of independent interest.

We introduce the notion of local uniform linear convexity of bounded convex domains with respect to their Kobayashi distances.

Let $X$ be a Banach space whose characteristic of noncompact convexity is less than $1$ and satisfies the nonstrict Opial condition. Let $C$ be a bounded closed convex subset of $X$, $KC\left(C\right)$ the family of all compact convex subsets of $C$, and $T$ a nonexpansive mapping from $C$ into $KC\left(C\right)$. We prove that $T$ has a fixed point. The nonstrict Opial condition can be removed if, in addition, $T$ is a $1$-$\chi$-contractive mapping.