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A dialectical approach on evolution of matter in the microcosm and macrocosmos
Series: De Gruyter STEM
Proceedings of the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, May 29–June 1, 2018
An Unhurried Introduction

Abstract

In this article, we study the generalized parabolic parametric Marcinkiewicz integral operators $ℳΩ,h,Φ,λ(r)$ related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to L p spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.

Abstract

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation

$u′′+c⁢u′+(λ⁢a+⁢(x)-μ⁢a-⁢(x))⁢g⁢(u)=0,$

where $λ,μ>0$ are parameters, $c∈ℝ$, $a⁢(x)$ is a locally integrable P-periodic sign-changing weight function, and $g:[0,1]→ℝ$ is a continuous function such that $g⁢(0)=g⁢(1)=0$, $g⁢(u)>0$ for all $u∈]0,1[$, with superlinear growth at zero. A typical example for $g⁢(u)$, that is of interest in population genetics, is the logistic-type nonlinearity $g⁢(u)=u2⁢(1-u)$. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of $a⁢(x)$. More precisely, when m is the number of intervals of positivity of $a⁢(x)$ in a P-periodicity interval, we prove the existence of $3m-1$ non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.