Resolubility of linear Cauchy problems on Fréchet spaces and a de- layed Kaldors model - Silva, Alex Pereira da (2019)
The long-run aim of this thesis is to solve delay differential equations with infinite delay of the type d dt u(t) = Au(t) + ∫t-∞ u(s)k(t - s)ds+ f (t, u(t)), on Fréchet spaces under an extended theory of groups of linear operators; where A is a linear operator, k(s) ≥ 0 satisfies ∫∞0 k(s)ds = 1 and f is a nonlinear map. In order to pursue such a goal we study a discrete delay model which explains the natural economic fluctuations considering how economic stability is affected by the role of the fiscal and monetary policies and a possible government inefficiency concerning its fiscal policy decision-making. On the other hand, we start to develop such an extended theory by considering linear Cauchy problems associated to a continuous linear operator on Fréchet spaces, for which we establish necessary and sufficient conditions for generation of a uniformly continuous group which provides the unique solution. Further consequences arises by considering pseudodifferential operators with constant coefficients defined on a particular Fréchet space of distributions, namely FL2loc, and special attention is given to the distributional solution of the heat equation on FL2loc for all time, which extends the standard solution on Hilbert spaces for positive time.