The existence of nonnegative nontrivial solutions to the problemy″+N−1xy′+λq(x)f(y)=0y′(0)=y(1)=0is discussed when the functionq(x) is allowed to be zero on a set of positive measure andf:[0,∞)→[0,∞) is continuous, strictly increasing, and allowing for the casef(0)=0. A description of the set of λ's for which there is a solution is obtained based on the properties off.

We find a solution to the radial Laplacian equation y + N − 1 x y ′ + λ a ( x ) f ( y ) = 0 , y ′ ( 0 ) = y ( 1 ) = 0 y + \frac {{N - 1}}{x}y’ + \lambda a(x)f(y) = 0,y’ (0) = y(1) = 0 when a may change sign and is "sufficiently positive". The function f is qualitatively like e y {e^y} , and we conclude solutions for 0 ≤ λ ≤ λ 0 0 \leq \lambda \leq {\lambda _0} .

The existence of nonnegative solutions of boundary value problems of the form x″(t) + λa(t) ƒ(x(t), y(t)) = 0y″(t) + λb(t) g(x(t), y(t)) = 0x(0) = x(1) = y(0) = y(1) = 0 is investigated where ƒ and g are functions from Rp × Rq into Rp and Rq, respectively, and a, b are functions defined on [0, 1] with values in the spaces of diagonal matrices of order p and q, respectively. All functions are assumed to be continuous and nonnegativity as well as monotonicity and growth conditions are imposed on ƒ and g; the existence results hold for a bounded interval of values of λ if ƒ and g are assumed to grow fast and have only nonzero values while if ƒ(0, 0) = 0, g(0, 0) = 0 it is shown that under reasonable conditions there exists a nontrivial nonnegative solution for every λ > 0.

The boundary value problems \[ ({\text{P}}_1 )\begin{array}{*{20}c} {y'' + \frac{{n - 1}}{x}y' + f(x,y) = 0,} \\ {y'(0) = y(1) = 0,} \\ \end{array} \] and \[ \begin{gathered} y'' + f(x,y) = 0, \hfill \\ ({\text{P}}_2 )\quad \alpha y(0) - \beta y'(0) = 0, \hfill \\ \gamma y(1) + \delta y'(1) = 0 \hfill \\ \end{gathered} \] with the function $f(x,y)$ satisfying conditions that allow for singularities to be present are studied with the view of obtaining general existence, uniqueness, and approximation of positive solutions. Furthermore, the behavior of solutions near $x = 1$ is described for the special case $f(x,y) = a(x)y^{ - p} $.

Questions 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.

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 such that if 0 < τ < τ∗ there are no positive almost periodic solutions while for τ > τ0 they do exist. A priori bounds on the set of positive solutions and uniqueness results are also obtained.

The boundary value problem is studied with a view to obtaining the existence of positive solutions in C 1 ([0, 1])∩ C 2 ((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f ( x, y ) = a ( x ) y − p , p >0. This boundary value problem arises in the search of positive radially symmetric solutions to where Ω is the open unit ball in ℝ N , centered at the origin, Γ is its boundary and | x | is the Euclidean norm of x .

The boundary value problem y″ + f(x, y) = 0αy(0) − βy′(0) = 0γy(1) + δy′(1) = 0 is studied under conditions which are modeled on the problem y″ + a(x)y−p = 0y(0) = y(1) = 0 or y′(0) = y(1) = 0, which occurs in a number of applications. The technical arguments involve properties of solutions (i.e., concavity) and the use of iterative techniques. A new fixed point theorem for cones, for decreasing mappings, is proved and used. It is of interest in its own right since most such theorems are for monotone increasing mappings.

Click to increase image sizeClick to decrease image sizeKeywords: Functional differentialAMS(MOS):: 34K2034K30

This chapter illustrates the recent work on equations with state dependent delays, with emphasis on particular models and on the emerging theory from the dynamical systems point of view. Several new results are presented. The chapter describes examples of differential equations with state dependent delays which arise in physics, automatic control, neural networks, infectious diseases, population growth, and cell production. Some of these models differ considerably from others, and most of them do not look simple. Typically the delay is not given explicitly as a function of what seems to be the natural state variable; the delay may be defined implicitly by a functional, integral or differential equation and should often be considered as part of the state variables.

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.

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.

A threshold model is proposed for humoral immune response to replicating antigen. The model incorporates two trigger functions, the first of which causes lymphocyte growth, and the second of which causes antibody growth. Both high and low zone tolerance as well as loss of immunity are incorporated into the model. Various specific models are analyzed to illustrate different possible responses.