Julio G. Dix

Multiple positive periodic solutions for a nonlinear first order functional difference equation

We obtain sufficient conditions for the existence of at least three non-negative periodic solutions for the first order functional difference equation Our main tool is the Leggett–Williams fixed point theorem, and our main application is a hematopoiesis model in population dynamics.

INFINITE FREQUENCY SOLUTIONS TO DISCONTINUOUS CONTROL SYSTEMS WITH VARIABLE DELAY

In this article, we study the frequency of zeros in the solution tou′(t)=f(t, u(t−τ(t))), whereτ is a delay that depends on time, andf is a discontinuous function. First we show examples of solutions that have infinite frequency for autonomous systems with variable delay, and for non-autonomous systems with constant delay. Then we prove that infinite frequency solutions cannot come from finite frequency data. Also we prove that under certain conditions on the delay, the zero solution is the only solution that has infinitely many zeros in each interval of a fixed length.

Existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4

Abstract This article shows the existence of positive periodic solutions for retarded, advanced, neutral, and ordinary differential equations whose order is a multiple of 4. First, we find a positive Green’s function explicitly. Then assuming that the coefficient of the unknown in the linear part of the equation is bounded above and below by positive constants, we find one and then three solutions by applying the Krasnoselskii and Legget–William fixed point theorems.

Oscillation of solutions to a higher-order neutral PDE with distributed deviating arguments

This article presents conditions for the oscillation of solutions to neutral partial differential equations. The order of these equations can be even or odd, and the deviating arguments can be distributed over an interval. We also extend our results to a nonlinear equation and to a system of equations.