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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....

Computation of topological degree in ordered Banach spaces with lattice structure and applications

Yujun Cui (2013)

Applications of Mathematics

Using the cone theory and the lattice structure, we establish some methods of computation of the topological degree for the nonlinear operators which are not assumed to be cone mappings. As applications, existence results of nontrivial solutions for singular Sturm-Liouville problems are given. The nonlinearity in the equations can take negative values and may be unbounded from below.

Continuous dependence on parameters for second order discrete BVP’s

Marek Galewski, Szymon Głąb (2012)

Open Mathematics

Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.

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