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Analytic solutions of a second-order iterative functional differential equation near resonance

Houyu Zhao, Jianguo Si (2009)

Annales Polonici Mathematici

We study existence of analytic solutions of a second-order iterative functional differential equation x ' ' ( z ) = ∑ j = 0 k ∑ t = 1 ∞ C t , j ( z ) ( x [ j ] ( z ) ) t + G ( z ) in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form α ² y ' ' ( α z ) y ' ( z ) = α y ' ( α z ) y ' ' ( z ) + [ y ' ( z ) ] ³ [ ∑ j = 0 k ∑ t = 1 ∞ C t , j ( y ( z ) ) ( y ( α j z ) ) t + G ( y ( z ) ) ] invertible analytic solutions of the form y ( α y - 1 ( z ) ) for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...

Angiogenesis process with vessel impairment for Gompertzian and logistic type of tumour growth

Jan Poleszczuk, Urszula Foryś (2009)

Applicationes Mathematicae

We propose two models of vessel impairment in the process of tumour angiogenesis and we consider three types of treatment: standard chemotherapy, antiangiogenic treatment and a combined treatment. The models are based on the idea of Hahnfeldt et al. that the carrying capacity for any solid tumour depends on its vessel density. In the models proposed the carrying capacity also depends on the process of vessel impairment. In the first model a logistic type equation is used to describe the neoplastic...

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