We obtain a positive solution, a negative solution, and a sign-changing solution for a class of $p$-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.

We introduce the notion of $L^1$-completeness for a stochastic flow on manifold and prove a necessary and sufficient condition for a flow to be $L^1$-complete. $L^1$-completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort of $L^1$-functional space, natural for manifolds where no Riemannian metric is specified.

For a given closed and translation invariant subspace $Y$ of the bounded and uniformly continuous functions, we will give criteria for the existence of solutions $u\in Y$ to the equation $u^{\prime }\left(t\right)+A\left(u\left(t\right)\right)+\omega u\left(t\right)∍f\left(t\right),t\in ℝ$, or of solutions $u$ asymptotically close to $Y$ for the inhomogeneous differential equation $u^{\prime }\left(t\right)+A\left(u\left(t\right)\right)+\omega u\left(t\right)∍f\left(t\right),t>0,u\left(0\right)=u_0$, in general Banach spaces, where $A$ denotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space $Y$ are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior.