On the existence of solutions to boundary value problems on time scales

This work formulates existence theorems for solutions to two-point boundary value problems on time scales. The methods used include maximum principles, a priori bounds and topological degree theory.

Existence of solutions to some equilibrium problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.

Existence of solutions for p(x)-Laplacian problems on a bounded domain

In this paper we study the following p(x)-Laplacian problem: -div(a(x)&VERBAR;&DEL; u&VERBAR;(p(x)-2)&DEL; u)+b(x)&VERBAR; u&VERBAR;(p(x)-2)u = f(x, u), x ε &UOmega;, u = 0, on &PARTIAL; &UOmega;, where 1< p(1) &LE; p(x) &LE; p(2) < n, &UOmega; &SUB; R-n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W-0(1,p(x)) for the p(x)-Laplacian problems in the superlinear and sublinear cases. © 2004 Elsevier Inc. All rights reserved.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

Existence of complementary pairs of proportionally balanced designs

Proportionally balanced designs (pi BDs) were introduced by Gray and Matters in response to a need for the allocation of markers of the Queensland Core Skills Test to have a certain property. Subsequent papers extended the theoretical results relating to such designs and provided further instances and general constructions. This work focused on designs comprising blocks of precisely two sizes, and when each variety occurs with one of precisely two possible frequencies. Two designs based on the set V of varieties are complementary if, whenever B is a block of one, then its complement with regard to the set V is a block of the other. Here we present necessary conditions for the existence of complementary pairs of such pi BDs and provide lists of some restricted parameter sets satisfying these necessary conditions. The lists are arranged according to the number of blocks. We demonstrate that not all of these parameter sets give rise to designs. However we establish by construction of the sets of blocks that, for every feasible number of blocks less than or equal to 100, with the possible exception of 63, there exists at least one pair of complementary pi BDs. We also investigate the conditions under which the complementary design can be isomorphic to the original design, and again provide a list of feasible parameters for pairs of such designs with at most 400 blocks.

Moreton Bay, Queensland, Australia: An example of the co-existence of significant marine mammal populations and large-scale coastal development

Recent analyses assert that large marine vertebrates such as marine mammals are now 'functionally or entirely extinct in most coastal ecosystems'. Moreton Bay is a large diverse marine ecosystem bordering the fastest growing area in Australia. The human population is over 1.6 million and increasing yearly by between 10% and 13% with resultant impacts upon the adjoining marine environment. Nonetheless, significant populations of three species of marine mammals are resident within Moreton Bay and a further 14 species are seasonal or occasional visitors. This paper reviews the current and historical distributions and abundance of these species in the context of the current management regime and suggests initiatives to increase the resilience of marine mammal populations to the changes wrought by the burgeoning human population in coastal environments. (C) 2004 Elsevier Ltd. All rights reserved.

Confirmation of the existence of alloparasitoids in nature - host relationships of an Australian Coccophagus species that parasitizes mealy bugs

Heteronomous hyperparasitoids are parasitic wasps with sex-related host relationships that are unique to a group of genera in the chalcidoid family Aphelinidae. Females are primary parasitoids of various sedentary bugs (mainly, scale insects, mealy bugs, and whiteflies). Males, in contrast, are hyperparasitic, and they frequently develop at the expense of female conspecifics. Alloparasitoids constitute a special category of heteronomous hyperparasitoids, for their males never develop through female conspecifics. The existence of alloparasitic host relationships and the utility of the category 'alloparasitoid' have both been questioned. Here, we present results that confirm the existence of the alloparasitic way of life among heteronomous aphelinids. We investigated an undescribed species of Coccophagus (Hymenoptera: Aphelinidae), an Australian parasitoid that attacks the introduced lantana mealy bug, Phenacoccus parvus Morrison (Homoptera: Pseudococcidae), in Queensland. A year-long field survey regularly returned large numbers of female Coccophagus spec. near gurneyi individuals from P. parvus (total n = 4212), but only few males (n = 11). Males emerged from samples only when the encyrtid parasitoid Anagyrus diversicornis (Howard) (Hymenoptera: Encyrtidae) was present in samples in relatively high numbers. Laboratory oviposition tests confirmed that A. diversicornis is a male host and showed that males do not develop at the expense of conspecific females. Other studies show that males are attracted in numbers to virgin females held in cages above mealy bug-infested Lantana montevidensis (Spreng.) Briq. (Verbenaceae) in the field, demonstrating that they are common in the population as a whole. This confirms that the males need hosts other than conspecific females and that their usual hosts are present outside of the lantana/P. parvus system. The implications of these results for developing a realistic classification of heteronomous host relationships are discussed.

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

in this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary a deltaOhm and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1). (C) 2003 Published by Elsevier Inc.

Multiplicity results for second-order two-point boundary value problems with nonlinearities across several eigenvalues

We consider the boundary value problems for nonlinear second-order differential equations of the form u '' + a(t)f (u) = 0, 0 < t < 1, u(0) = u (1) = 0. We give conditions on the ratio f (s)/s at infinity and zero that guarantee the existence of solutions with prescribed nodal properties. Then we establish existence and multiplicity results for nodal solutions to the problem. The proofs of our main results are based upon bifurcation techniques. (c) 2004 Elsevier Ltd. All rights reserved.

Sharp Sobolev inequality involving a critical nonlinearity on a boundary

We consider the solvability of the Neumann problem for the equation -Delta u + lambda u = 0, partial derivative u/partial derivative v = Q(x)vertical bar u vertical bar(q-2)u on partial derivative Omega, where Q is a positive and continuous coefficient on partial derivative Omega, lambda is a parameter and q = 2(N - 1)/(N - 2) is a critical Sobolev exponent for the trace embedding of H-1(Omega) into L-q(partial derivative Omega). We investigate the joint effect of the mean curvature of partial derivative Omega and the shape of the graph of Q on the existence of solutions. As a by product we establish a sharp Sobolev inequality for the trace embedding. In Section 6 we establish the existence of solutions when a parameter lambda interferes with the spectrum of -Delta with the Neumann boundary conditions. We apply a min-max principle based on the topological linking.

Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

On multiple solutions of the exterior neumann problem involving critical sobolev exponent

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.