A Neumann problem for a system depending on the unknown boundary values of the solution

A semilinear system of second order ODEs under Neumann conditions is studied. The system has the particularity that its nonlinear term depends on the (unknown) Dirichlet values $y(0)$ and $y(1)$ of the solution. Asymptotic and non-asymptotic sufficient conditions of Landesman-Lazer type for existence of solutions are given. We generalize our previous results for a scalar equation, and a well known result by Nirenberg for a nonlinearity independent of $y(0)$ and $y(1)$

Sturm-Liouville boundary conditions for a second order ODE

We study the semilinear second order ODE u00 + g(t, u) = 0 under the following Sturm- Liouville boundary condition au(0) + bu0(0) = u0 and cu(T) + du0(T) = uT . We obtain solutions by topological methods. Moreover, we show that a solution may be constructed recursively, under appropriate conditions

Stability of Hahnfeldt Angiogenesis Models with Time Lags

Mathematical models of angiogenesis, pioneered by P. Hahnfeldt, are under study. To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M-matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies

On the solvability of the periodically forced relativistic pendulum equation on time scales

We study some properties of the range of the relativistic pendulum operator P, that is, the set of possible continuous T-periodic forcing terms p for which the equation Px = p admits a T-periodic solution over a T-periodic time scale T. Writing p(t) = p0(t) + p, we prove the existence of a nonempty compact interval I(p0), depending continuously on p0, such that the problem has a solution if and only if p ∈ I(p0) and at least two different solutions when p is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if T is small then I(p0) is a neighbourhood of 0 for arbitrary p0. The results in the present paper improve the smallness condition obtained in previous works for the continuous case T = R

On two-point boundary value problems in multi-ion electrodiffusion

AbstractThe solvability is established of certain two-point boundary value problems for nonlinear equations that arise in multi-ion electrodiffusion. Topological methods are adduced to prove the existence of solutions under appropriate conditions on the physical parameters

Superstring-Inspired E_6 Unification, Shadow Theta-Particles and Cosmology

We construct a new cosmological model considering the superstring-inspired E_6 unification in the 4-dimensional space at the early stage of the Universe. We develop a concept of parallel existence in Nature of the ordinary and shadow worlds with different cosmological evolutions.Comment: 7 page

On Visible Homelessness and the Micro-Aesthetics of Public Space

In this article, we investigate the circumstances that have produced the current municipal regulatory approach to homelessness in the City of Melbourne, Victoria, and the ways in which visibly homeless people are policed through a micro-aesthetics of their presence in public space, which involves the monitoring of their bodily demeanour and their physical possessions. Our study contributes to and draws from a range of debates, including studies of the governmental conjunction of poverty and crime, analysis of the co-implication of law and spatiality, research on the criminalisation of homelessness and homeless people, and the burgeoning criminological interest in the significance of the visual field for our understandings of crime and criminality. This article recounts how homelessness, public space and questions of aesthetics have recently coalesced in debates about the regulation of homelessness in the public space of Melbourne’s city centre. It approaches the issues through comparative consideration of genres of municipal management frameworks in other jurisdictions, detailed textual consideration of the Protocol on Homelessness in the City of Melbourne and an empirical study of visible homelessness in the public places of central Melbourne