In the paper “A nonlinear complementarity problem for monotone functions”, Bull. Austral. Math. Soc. 20 (1979), 227–231, Nanda and Patel proved that for a monotone function that fixes the origin, the complementarity problem admits a solution. In this note we give a short proof of the same result under weaker assumptions.

Using degree theory we establish the inverse mapping theorem and a Sard-Smale theorem for analytic contractive vector fields in Fréchet spaces. Applying these theorems we obtain Cronin's theorem relating topological degree to the number of zeros of analytic contractive vector fields.

The role of convexity in the development of mathematical programming is reviewed. Many recent generalizations of convexity and their applications to optimization theory are discussed.

A well-known inequality relating the areas and squares of the sides of two triangles is generalised to higher-dimensional euclidean spaces. Extension of the results to non-euclidean spaces is also considered.

We conjecture that a complete hypersurface of non-positive Ricci curvature in the Euclidean space must be unbounded. We prove this under the additional assumption that all sectional curvatures of the hypersurface are bounded away from negative infinity.

Let R be a semiprime ring in which for each x in R there exists a positive integer n = n(x) < 1 such that (xy)n = xnyn for all y in R. Then R is commutative.

Planar graphs have recently been characterised as those which have no strict elegant odd ring of circuits. Here we generalise that result by showing that its dual yields a theorem that is valid for all graphs.

We construct a semigroup S with an epimorphically embedded proper subband U. The band U furnishes an example of a regular semigroup which is not saturated, thus answering a question posed by Hall, Semigroup Forum (to appear).

It is shown that a transformation group (X, T, Π) is of strong characteristic 0 if and only if it is of P-strong characteristic 0 for some replete semigroup P in the phase group, provided that all orbit closures are compact. It is shown also that, under certain conditions, (X, T, Π) is of P-strong characteristic 0 if and only if (X × X, T, Π × Π) is Liapunov stable.

Let X, Y be rearrangement invariant spaces and let M = M(Y, X) be the space of all multipliers of Y into X. It is shown that if X = YM and some technical conditions are satisfied, then the K-functional K(t, f, X, Y) is equivalent to the expression

where ψ is the inverse of the fundamental function ϕM of M, defined by

The adjoint of a composition operator CT on the L2-space of an atomic measure is computed and a characterization for an operator to be a composition operator is given in this short note. The dimensions of kernel and co-kernel of CT are calculated in order to characterise Fredholm composition operators. Finally, essentially normal composition operators are studied on l2.

The notion of accretiveness for multi-valued nonlinear maps is defined in locally convex spaces and it is used to obtain a locally convex space version of a result of M.G. Crandall and J.A. Nohel.

The order structure of the space of all continuous linear operators on an ordered Banach space is studied. The main topic is the Robinson property, that is, the norm of a positive linear operator is attained on the positive unit cone.

A new result is obtained for the oscillation of second order linear ordinary differential equations with alternating coefficients. This oscillation result extends a recent oscillation criterion due to Kamenev [.Mat. Zametki 23 (1978), 249–251].