We consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.

Let Mn be the semigroup, under composition, of endomorphisms of an n-dimensional vector space. Let E be the set of idempotents of Mn. It has been shown that each singular element of Mn() may be expressed as a composition of elements of E. In this paper the minimum number of idempotents needed in this composition is determined. This is given in terms of n and one parameter dependent on the element. Further, it is shown that En–1⊂<E>, while En=<E>.

This paper studies the equation

where the differential operator Ln is defined by

and a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case where

where N is an integer with l≦N≦n–1.

Some sufficient conditions are obtained on the coefficient g and the initial values Φ and ψfor the solution ot the non-linear hyperbolic equation

to change sign in the first quadrant. An example is given to show that is not sufficient in the linear case.

Given and similar , modelled on radial Laplace-Beltrami operators (ρp = , in this paper we begin the study of transmutations which leads to elliptic equations Working with and transmutations Qm → −D2 for m > −½ and −D2 → for m < −½, we obtain a transmutation formulation and derivation of many results of generalized axially symmetric potential theory in the first case and in both cases generalized Hilbert transforms (different). Canonical generalizations are then automatic using general transmutation theory.

A variational formulation of the Hele—Shaw flow model of the point injection of fluid into a laminar cell is introduced. The analysis concerning the existence, uniqueness and regularity of a solution to the variational problem is presented.

We study the boundary-value problem, for (λ/k,ψ),

Here ∆ denotes the Laplacian, H is the Heaviside step function and one of A or k is a given positive constant. We define

and usually omit the subscript. Throughout we are interested in solutions with ψ>0 in Ω and hence with λ/=0.

In the special case Ω = B(0, R), denoting the explicit exact solutions by ℑe, the following statements are true, (a) The set Aψ, issimply-connected, (b) Along ℑe, the diameter of Aψ tendsto zero when the area of Aψ, tends to zero.

For doubly-symmetrised solutions in domains Ω such as rectangles, it is shown that the statements (a) and (b) above remain true.

We establish upper and lower bounds for various norms of solutions and their gradients for the equation ut = div (|∇u|m−1 ∇u) in ℝN in terms of the norms of the initial data. Based on the L∞ estimate of ∇u, we conclude that u(x, t) is Lipschitz continuous in space-time, for all t>0, whenever u(x,0) is in L1(ℝN).

A recently developed asymptotic theory of higher-order differential equations is applied to problems of right-definite type to determine the numbers M+, M− of linearly independent solutions with a convergent Dirichlet integral, M+ and M− referring to the usual upper and lower λ.-half-planes. Particular attention is given to the phenomenon noted by Karlsson in which one of M+ and M− is maximal but not the other. Conditions are given under which M+ (say) is maximal and M− is the same, one less, and two less.