This paper is a study of the oscillatory and asymptotic behaviour of solutions of the second order nonhomogeneous linear differential equation y"+P(x)y=f(x), and the associated homogeneous equation. Conditions are determined, under which the nonhomogeneous equation is oscillatory if and only if the homogeneous equation is oscillatory.

Minimal s-Urysohn and minimal s-regular spaces are studied. An s-Urysohn (respectively, s-regular) space (X, $\mathfrak{T}$) is said to be minimal s-Urysohn (respectively, minimal s-regular) if for no topology $\mathfrak{T}^{\prime}$ on X which is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) is s-Urysohn (respectively s-regular). Several characterizations and other related properties of these classes of spaces have been obtained. The present paper is a study of minimal P-spaces where P refers to the property of being an s-Urysohn space or an s-regular space. A P-space (X, $\mathfrak{T}$) is said to be minimal P if for no topology $\mathfrak{T}^{\prime}$ on X such that $\mathfrak{T}^{\prime}$ is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) has the property P. A space X is said to be s-Urysohn [2] if for any two distinct points x and y of X there exist semi-open set U and V containing x and y respectively such that $clU{\bigcap}clV={\phi}$, where clU denotes the closure of U. A space X is said to be s-regular [6] if for any point x and a closed set F not containing x there exist disjoint semi-open sets U and V such that $x{\in}U$ and $F{\subseteq}V$. Throughout the paper the spaces are assumed to be Hausdorff.

Consider the system (i) x'=Ax. Let ${\Phi}$ be its fundamental matrix solution. If there is w>0 such that A(t+w)-A(t) commutes with ${\Phi}$ for all t, then we call this system a "generalized" Floquet system or a "G. F. system". We show that $A(t+w)-A(t)=B_1$=constant if and only if $A(t)=C+B_1t/w+Q(t)$, Q is periodic of period w>0. For this A(t) We prove that if all eigenvalues of $B_1$ have negative real parts, then the origin is asumptotically stable. We find a growth condition for a continuous D(t) which guarantees that all solutions of z'=[A(t)+D(t)]z are bounded if all solutions of the G. F. system (i) are bounded. Combining the foregoing results yield a class of perturbed G. F. operators all of whose solutions are bounded.

In this paper we consider the semilinear hyperbolic symmetric system of the first-order. The existence and uniqueness of the solution are proved, under certain conditions, some properties of the solution are investigated.

A linear system $\dot{x}= A(t)x$, with A(t+w)=A(t) is considered. A step function approximation of a periodic matrix is constructed. The stability criteria is discussed.