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On the semilinear integro-differential nonlocal Cauchy problem

100%
Majcher P., Roszak M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2005
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tom 25
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nr 1
5-18
EN
In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x'(t) + Ax(t) = f(t,x(t),∫_{t₀}^{t} k(t,s,x(s))ds)$, x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T(t)}_{t > 0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.
2
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Mathematical modeling of the competition between acquired immunity and cancer

88%
Kolev M.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
289-296
EN
In this paper we propose and analyse a model of the competition between cancer and the acquired immune system. The model is a system of integro-differential bilinear equations. The role of the humoral response is analyzed. The simulations are related to the immunotherapy of tumors with antibodies.
3
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A nonlocal coagulation-fragmentation model

75%
Lachowicz M., Wrzosek D.
Applicationes Mathematicae
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2000
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tom 27
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nr 1
45-66
EN
A new nonlocal discrete model of cluster coagulation and fragmentation is proposed. In the model the spatial structure of the processes is taken into account: the clusters may coalesce at a distance between their centers and may diffuse in the physical space Ω. The model is expressed in terms of an infinite system of integro-differential bilinear equations. We prove that some results known in the spatially homogeneous case can be extended to the nonlocal model. In contrast to the corresponding local models the analysis can be carried out in the $L_1(Ω)$ setting. Our purpose is to study global (in time) existence, mass conservation and well-posedness of the model.
4
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Integro-differential equations on time scales with Henstock-Kurzweil delta integrals

63%
Sikorska-Nowak A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2011
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tom 31
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nr 1
71-90
EN
In this paper we prove existence theorems for integro - differential equations $x^Δ (t) = f(t,x(t),∫₀^t k(t,s,x(s))Δs)$, t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale.
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