In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 of

for any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

Using techniques from probability theory, it has been established that if μ is a probability measure on a separable, locally compact group, then the space of μ-harmonic functions on the group can be identified with C(X) for some compact, Hausdorff space X. The space X is known as the Poisson space of μ. We generalise this result in the context of a measure μ on a locally compact semigroup S, in particular establishing the existence of a Poisson space for non-separable groups. The proof is non-probabilistic, and depends on properties of projections on C(K)(K compact Hausdorff). We then show that if S is compact and the support of μ generates S, then the Poisson space associated with μ, is X, where X×G×Y is the Rees product representing the kernel of S.

In this paper the adjoint operator is derived for a multi-point differential operator with a varying structure in a suitably chosen Hilbert space. The formal differential operator is given by different differential expressions in the adjoining intervals. This form of adjoint operator is used to characterize self-adjointness.

We continue with the work of earlier papers concerning the use of partial dilferential equations to prove the uniform convergence of the eigenfunction expansion associated witha left definite two-parameter system of ordinary differential equations of the second order.

The paper considers a linear non-homogeneous boundary value problem for a class of neutral type functional differential equations. A necessary and sufficient condition for the existence of a unique solution of that problem is obtained.

This paper treats the global qualitative behaviour of all bifurcating configurations of whirling nonlinearly elastic strings with ends fixed on the axis of rotation.

If von Kármán's substitution is made in the Navier-Stokes equations, and boundary conditions corresponding to a flow in all of space with constant angular velocities at infinity are imposed, a boundary value problem analgous to those for flow above a rotating disk and between rotating disks is obtained. It is shown here that this problem has no solution.

In this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

Let A and B be commutative Noetherian local rings, such that B is a finitely generated free A-module. It is shown that if M is a balanced big Cohen–Macaulay A-module (that is, every system of parameters for A is an M-sequence), then M⊗AB is a balanced big Cohen-Macaulay B-module.

The following system of conservation laws is considered:

where σ: ℝ→ℝ is a smooth function monotonically increasing except in an interval. Two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other. This contrasts with the corresponding situation for strictly hyperbolic systems (σ'(u)>0 for all u), for which the two criteria are equivalent.

A previous attempt to systematize various conditions from multiparameter spectral theory is extended to weaker forms of definiteness. The latter not only occur in the established literature but are also under active investigation at present. Several algebraic and geometrical formulations exist, and questions concerning their equivalence are approached in a unified fashion where possible.

Let X be a set with infinite cardinality m and let Qm be the semigroupof balanced elements in ℐ(X), as described by Howie. If I is the ideal{αεQm:|Xα|<m} then the Rees factor Pm = Qm/I is O-bisimple and idempotent-generated. Its minimum non-trivial homomorphic image has both these properties and is congruence-free. Moreover, has depth 4, in the sense that [E()]4 = , [E()]3≠

It is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.

In this note we prove the following

Theorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.

The class of M-nilpotent rings is defined as a generalisation of the class of T-nilpotent rings. Certain results for radicals of T-nilpotent rings are shown to hold also in this larger class of rings.

In this paper we discuss the existence and multiplicity of solutions to some perturbed bifurcation problems. By using sub and supersolution techniques along with an anti-maximum principle, simple proofs of some “well known” local results of perturbed bifurcation theory are obtained. The existence of global continua of solutions is proved by using degree theory arguments.

Integral transformations T of the formal form

where G is a Meijer-G function are studied on the spaces ℐμ,r of functions f such that

Results are obtained on the boundedness, representation and range of the transformations, and on when they have an inverse of the same formal form.

By suitably coupling convexity and weight function methods, we prove uniqueness and continuous dependence theorems in linear elastodynamics in unbounded domains without definiteness conditions on the elasticities. The class of solutions considered allows the “growth” at large spatial distances.

Let S be an inverse semigroup and F a field. It is shown that if F has characteristic 0 and is not algebraic over its prime subfield then the algebra of S over F is semiprimitive (i.e. Jacobson semisimple). This generalises a well-known theorem on group algebras due to Amitsur. Similar results for the case in which F has prime characteristic are obtained under the additional hypotheses that S is completely semisimple or that S is E-unitary with a totally ordered semilattice.

The semilinear parabolic system ut + A(x, D)u = g(u) in (0, ∞) × Ω, Ω⊂ℝn bounded, u ∈ ℝN, with homogeneous boundary conditions B(x, D)u=0 on (0, ∞)×∂Ω is considered. The non-linearity g is assumed to be locally Lipschitz-continuous. It is shown that the orbit of a bounded regular solution u is relatively compact in .

The operators Δhf ≡ f(x) on function spaces and Δxn ≡ xn+1–xn on sequence spaces replace derivatives to yield analogues of the Kolmogorov inequality. Estimates for best constants are given for many spaces and for a few the best constants are actually given.