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A notion of boundedness for hyperfunctions and Massera type theorems

Yasunori Okada (2012)

Banach Center Publications

For some classes of periodic linear ordinary differential equations and functional equations, it is known that the existence of a bounded solution in the future implies the existence of a periodic solution. In order to think on such phenomena for hyperfunction solutions to linear functional equations, we introduced a notion of bounded hyperfunctions, and translated the problems into the problems on analytic solutions to some equations in complex domains. In this article, after...

A rigidity phenomenon for the Hardy-Littlewood maximal function

Stefan Steinerberger (2015)

Studia Mathematica

The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let f C α ( , ) be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator ( A x f ) ( r ) = 1 / 2 r x - r x + r f ( z ) d z has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as those nonconstant...

Almost homoclinic solutions for a certain class of mixed type functional differential equations

Joanna Janczewska (2011)

Annales Polonici Mathematici

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: q ̈ ( t ) + V q ( t , q ( t ) ) + u ( t , q ( t ) , q ( t - T ) , q ( t + T ) ) = f ( t ) , where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable....

An epidemic model with a time delay in transmission

Q. J. A. Khan, E. V. Krishnan (2003)

Applications of Mathematics

We study a mathematical model which was originally suggested by Greenhalgh and Das and takes into account the delay in the recruitment of infected persons. The stability of the equilibria are also discussed. In addition, we show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise by Hopf bifurcation.

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