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The purpose of this paper is to give some properties for the so-called ε-pseudo weakly demicompact linear operators acting on Banach spaces. Some sufficient conditions on the entries of an unbounded 2 × 2 block operator matrix $$$\mathcal{L}_{0}$$$ ensuring its ε-pseudo weak demicompactness are provided. In addition, we develop, in the bounded case, the class of ε-pseudo Fredholm perturbation to investigate the essential pseudo-spectra of $$$\mathcal{L}_{0}$$$. The results are formulated in terms of some denseness conditions on the topological dual space.
In this paper, we conduct a mathematical analysis of a tumor growth model with treatments. The model consists of a system that describes the evolution of metastatic tumors and the number of cells present in the primary tumor. The former evolution is described by a transport equation, and the latter by an ordinary differential equation of Gompertzian type. The two dynamics are coupled through a nonlocal boundary condition that takes into account the tumor colonization rate. We prove an existence result where the main difficulty is to handle the coupling and to take into account the time discontinuities generated by treatment terms. The proof is based on a Banach fixed point theorem in a suitable functional space. We also develop a computational code based on the method of characteristics and present numerical tests that highlight the effects of different therapies.
The second fundamental form arising from an oriented minimal immersion of a closed surface in a space form satisfies several constraints. One of them is provided by the Gauss-Codazzi equation that can be rephrased as a semilinear problem on the surface. We discuss some results for these type of nonlinear problems and analyze the behaviors of the solutions when the hyperbolic norm of the second fundamental form is small.
2024
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