We derive interior second derivative estimates for solutions of equations of Monge-Ampère type which extend those of Pogorelov for the case of affine boundary values. A key ingredient in our method is the existence of a strong solution of the homogeneous Monge-Ampère equation.

Let r be a given prime. Then a monoid M is the endomorphism monoid of a field of characteristic r if and only if either M is a finite cyclic group or M is a right cancellative monoid and M has an element of infinite order in its centre. The main lemma is the technical base of the present and other papers.

In a Banach lattice or the hermitian part of a C*-algebra, every element a admits a decomposition a = a+ − a− such that and N(−a) = ‖a−‖, where N is the canonical half-norm of the positive cones. In general ordered Banach spaces, this property is related to the order structure of the duality map and the metric projectability of the positive cones, and it turns out to be equivalent to an “orthogonal” decomposability.

It is well-known that every sufficiently large positive integer is the sum of seven cubes. Both proofs of this result, due to Linnik and Watson, are ineffective. Here we show that Watson's proof may be made effective.

Sublinear elliptic differential inequalities with variable coefficients are studied. Sufficient conditions are given that all solutions are oscillatory in exterior domains. Riccati inequalities are used to establish sublinear oscillation criteria.

Let B be the class of functions ω(z) regular in |z| < 1 and satisfying ω(0) = 0, |ω(z)|<1 in |z|<1. We denote by P(A, B), −1 ≤ B < A ≤1, the class of functions p(z) = l+p1z+… regular in |z| < 1 and such that p(z) = [1+Aω(z)]/[1+Bω(z)] for some ω(z) ∈ Β. This paper establishes sharp lower and upper bounds on |z| = r<1 for the functional

where p(z) varies in P(A, B). The results are then used to study certain geometric properties of the corresponding class of meromorphic starlike univalent functions

Let SΓ be a vector space graph. A graphic subspace of SΓ need not be a direct summand with a graphic complement. A necessary and sufficient condition for the existence of a graphic complement is given. Also, it is shown that every graphic subspace possesses an o-special basis which extends to an o-special basis of SΓ.

A Radon–Nikodým theorem for operator-valued measures is applied to deduce the existence and uniqueness of conditional expectations in certain cases.

We show that a certain local pinching condition on a compact minimal submanifold in a sphere leads to a condition on the codimension of the immersion.

In this paper we give necessary and sufficient conditions for exponential dichotomy of a general class of strongly continuous semigroups of operators defined on a Banach space. As a particular case we obtain a Datko theorem for exponential stability of a strongly continuous semigroup of class C0 defined on a Banach space.

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

The Schwarz-Pick lemma,

for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:

where Φ(z, r) is a quantity determined by the non-Euclidean area of the image of

and ψ(z, r) is that determined by the non-Euclidean length of the image of the boundary of D(z, r). The multiplicities in both images by f are not counted.

A result by the author on the elimination of randomization (or relaxation) for variational problems is partially extended and then used to obtain a very general result on the denseness of the set of original control functions in the set of relaxed control functions. Also, a slight extension of Aumann's theorem on the integrals of multifunctions is shown to follow directly from the elimination result.