We are interested here in proving the existence of solutions to the (generalised) boundary value problem

where A is a continuous n×n matrix on R+ = [0, ∞), F is a continuous n vector on R+ × S (S = a suitable subset of Rn), T is a bounded linear operator defined on (or on a subspace of) C[R+, Rn], the space of all bounded and continuous Rn-valued functions on R+, and r is a fixed vector in Rn. There is an abundance of papers dealing with the problem ((I), (II)) on finite intervals, either in its full generality (cf., for example, (1), (2), (3), (4), (6)), or for special cases of the operator T. The reader is especially referred to the work of Shreve (7), (8) for such problems on infinite intervals for scalar equations. A series representation of the solutions is given by Kravchenko and Yablonskii (5). Most of our methods are extensions of the corresponding ones on finite intervals with some variations concerning the application of fixed-point theorems. Examples of interesting operators T are

where V(t), M, N are n×n matrices with V(t) integrable.

The paper examines blow-up phenomena for the inequality \renewcommand{\theequation}{*} \begin{equation} u_t - Lu - \vert u \vert^{q-1} u \leqslant v_t - Lv - \vert v \vert^{q-1}v \end{equation} in the half-space ${\bb R}^1_+ \times {\bb R}^n,\; n \geqslant 1$ , where $L$ is a linear second-order partial differential operator in divergence form.

The paper studies weak solutions of (*) that belong only locally to the corresponding Sobolev spaces in the half-space ${\bb R}^1_+ \times {\bb R}^n$ . It also requires no conditions for the behavior of solutions of (*) on the hyperplane $t = 0$ .

The existence of critical blow-up exponents is obtained for solutions of (*) as a special case of a comparison principle for the corresponding solutions of (*). For example, the well-known Fujita result is a consequence of the comparison principle.

The approach developed in the paper is directly applicable to the study of analogous problems involving nonlinear differential operators. Its elliptic analogue has been recently developed by the authors.

Properties of solutions of , are studied in the cause uH(t, u)>0 for u≠0. It is shown that two inequalities may always be associated with (I) in such a way that if one of these inequalities has a small positive solution and the other inequality has a small negative solution, then (I) is oscillatory. Further asymptotic properties of (I) are studied under assumptions involving intermediate antiderivatives P(i)(t), with P(n) = Q. Several results of this type ensure the non-existence of positive solutions of (I).

In this paper, we study the oscillation of nth-order differential equations. Recently, Atkinson and the present authors studied (separately) the comparison properties of differential inequalities. Kartsatos treated the nth-order ordinary case and proposed several open problems.

The purpose of this paper is to answer one of them in the affirmative concerning more general functional differential equations. The result is that if under several conditions, the equation

is oscillatory for n even or a solution x(t) of (1) is oscillatory or for n odd, then this is also the case for the equation

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

K. W. Chang generalizing a result of Lazer [3], proved in [4] the following THEOREM 1. Suppose that f:I → R+ = (0, + ∞), I = ([t0, + ∞), t0 0, is a non-decreasing function whose derivatives of orders ≧ 3 exist are continuous on ([t0, + ∞). Moreover, limt→+∞f(t) = +∞ and for some α, 1≧ α ≧ 2, and F =f-1/α then every solution x(t) of the equation tends to zero as t - + ∞.

In this paper we are dealing with differential equations of the forms:

Ei

where the functions pi are positive.

By a solution of an equation of the above forms, we mean a function x(t) ∈ C2 [c, + ∞) where c is a non-negative constant, which satisfies the corresponding equation on the whole interval [c, +∞). By an oscillatory solution of (Ei), we mean a solution with arbitrarily large zeros.