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Boundary value problems with compatible boundary conditions

George L. Karakostas, P. K. Palamides (2005)

Czechoslovak Mathematical Journal

If Y is a subset of the space ℝ n × ℝ n , we call a pair of continuous functions U , V Y -compatible, if they map the space ℝ n into itself and satisfy U x · V y ≥ 0 , for all ( x , y ) ∈ Y with x · y ≥ 0 . (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential n -dimensional system is investigated, provided the boundary conditions are given via...

Cauchy Problem for Differential Equation with Caputo Derivative

Kilbas, Anatoly, Marzan, Sergei (2004)

Fractional Calculus and Applied Analysis

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution...

Chemotaxis models with a threshold cell density

Dariusz Wrzosek (2008)

Banach Center Publications

We consider a quasilinear parabolic system which has the structure of Patlak-Keller-Segel model of chemotaxis and contains a class of models with degenerate diffusion. A cell population is described in terms of volume fraction or density. In the latter case, it is assumed that there is a threshold value which the density of cells cannot exceed. Existence and uniqueness of solutions to the corresponding initial-boundary value problem and existence of space inhomogeneous stationary solutions are discussed....

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