Let $f : N \to P$ be a smooth map between $n$-dimensional oriented manifolds which has only fold singularities. Such a map is called a fold-map. For a connected closed oriented manifold $P$, we shall define a fold-cobordism class of a fold-map into $P$ of degree m under a certain cobordism equivalence. Let $\Omega _{fold,m}(P)$ denote the set of all foldcobordism classes of fold-maps into $P$ of degree $m$. Let $F^{m}$ denote the space $\lim _{k\to \infty}F_{k}^{m}$, where $F_{k}^{m}$ denotes the space of all base point preserving maps of degree $m$ of $S^{k-1}$. In this paper we shall prove that there exists a surjection of $\Omega _{fold,m}(P)$ to the set of homotopy classes $[P,F^{m}]$, which induces many fold-cobordism invariants.

Several criteria for the asymptotic exponential stability of a class of Hilbert space-valued, non-autonomous stochastic evolution equations with variable delays are presented. This formulation is particularly suitable for the treatment of mild solutions of general stochastic delay partial differential equations. The principal technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to obtain the required exponential stability. As a consequence, stability results from A. Ichikawa [8] are generalized to cover a class of non-autonomous stochastic delay evolution equations. In particular, we improve the recent results of T. Taniguchi [14] to remove the time delay interval restriction there.