Juan A. Gatica

Singular nonlinear boundary value problems for second-order ordinary differential equations

A long-period storage accumulator for storing heat water is used as a storing medium. The water is enclosed in large thin-walled containers which are arranged under the ground. The water pressure forces acting on the container walls are carried by the surrounding of the container.

Positive almost periodic solutions of some delay integral equations

The delay integral equation x(t) = ∝tt − τ f(s, x(s)) ds which arises in models for the spread of epidemics, is studied with the aim of establishing the existence of positive almost periodic solutions for large values of τ when f(t, x) is uniformly almost periodic in t for x in compact subsets of R+. Under reasonable assumptions on f it is shown that there exist two positive numbers τ∗ τ0 they do exist. A priori bounds on the set of positive solutions and uniqueness results are also obtained.

Fixed point techniques in a cone with applications

Abstract This article is concerned with the existence of fixed points of compact operators mapping a cone in a Banach space into itself. Applications to two-point boundary value problems in ordinary differential equations and to an integral equation of K. E. Swick, modeling single species population growth, are given. A main feature of our results is that nonzero fixed points are obtained even though zero is known to be a fixed point.

Positive solutions to semilinear problems with coefficient that changes sign

where Ω will be assumed to be a bounded domain in R with smooth boundary. We will concentrate on the case when the coefficient a(x) is allowed to change sign; this brings about an interesting mathematical problem. This is not a new problem: due to its appearance in many mathematical models in physics, several special cases of it have been studied since the middle 1800’s. For example, it was used by Lord Kelvin [T,W] and J. Lane [L,J] to study the equilibrium configuration of mass in a spherical cloud gas; this model was further studied by R.H. Fowler [F,R], leaving the Emden Fowler equation which, in a generalized form is still studied today, as seen in, for example [AP]. For a review of the work done up to 1975 in the ordinary differential equations setting, see [W]; for a review of the work done for the existence of positive solutions of semilinear elliptic equations, see [L]. Positive solutions are usually the ones of interest and because of the difficulties that this brings in using the techniques developed for nonlinear functional analysis, most of the recent work assumes nonnegativity of a(x) and f(u) in order to generate positive operators using the Green’s function of −∆ on the region where the problem is posed. In nuclear physics, a form of (1.1) was proposed and used by E. Fermi [F,E] and by L. H. Thomas [T,L], and now it is widely known as the Thomas-Fermi equation. Both the Emden-Fowler equation as well as the Thomas Fermi equation continue to be subjects of considerable interest in theoretical physics to this day, as a search on the Wide World Web will quickly convince you. In combustion theory equation (1.1) has also been used as a model, for example in [ACP], [BK],[CL], [G],[L]. Equation (1.1) has also found applications to geometry: see [N] As mentioned above, most of the literature deals with the problem under the assumption that the coefficient function a(x) be positive in its domain; we are interested in the case when this function is allowed to change sign. In particular we are interested in determining sufficient conditions on a(x) to assure the existence of positive solutions for

Predator-prey models with almost periodic coefficients

This paper deals with the question of persistence of differential equations with almost periodic coefficients which model predator-prey systems. It is shown that persistence does occur in many models that incorporate almost periodic time dependence in the prey equation

A THRESHOLD MODEL OF ANTIGEN ANTIBODY DYNAMICS WITH FADING MEMORY

Publisher Summary This chapter describes a threshold model of antigen antibody dynamics with fading memory. Thresholds play an important role in most views of the immune response and are mathematically difficult to model. Waltman and Butz, borrowing from the mathematical theory of epidemics, proposed the use of an integral threshold to mark the onset of B-cell proliferation. Such a constraint leads to functional rather than ordinary differential equations. Gatica and Waltman enlarged the class of allowable threshold functions. The model described in the chapter continues the integral threshold approach but allows more complicated dynamics. Specifically, it allows for the reversible binding of antigen to B-cell receptors. The integral constraint can then be interpreted as an averaging of the effect of the number of bound surface receptors. The model allows this average to be weighted—perhaps in favor of more recent bindings and, hence, allows memory to fade. Both modifications make the model more realistic.

Iterative procedures for nonlinear second order boundary value problems

SummaryQuestions of the existence of positive solutions of second order nonlinear boundary value problems with separated boundary conditions are investigated. The nonlinearities are such that the linearization about the trivial solution does not exist or is trivial. The methods are thus applicable when shooting methods or ordinary bifurcation techniques cannot be applied. The conditions on the nonlinearity are quite modest and both super and sublinear problems can be included.

A threshold model of antigen-antibody dynamics

A model for the B-cell response to antigen challenge is proposed which makes use of an integral constraint to model the thresholds. This type of constraint leads to functional, rather than ordinary, differential equations. The model differs from previous models of this type in that it allows for the disassociation of the receptor-antigen complexes and allows for a weighting function in the integral constraint.