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Random coincidence degree theory with applications to random differential inclusions

Enayet U, Tarafdar, P. Watson, George Xian-Zhi Yuan (1996)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to establish a random coincidence degree theory. This degree theory possesses all the usual properties of the deterministic degree theory such as existence of solutions, excision and Borsuk’s odd mapping theorem. Our degree theory provides a method for proving the existence of random solutions of the equation L x ∈ N ( ω , x ) where L : dom L ⊂ X → Z is a linear Fredholm mapping of index zero and N : Ω × G ¯ → 2 Z is a noncompact Carathéodory mapping. Applications to random differential inclusions are also considered.

Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions

Jan Eisner (2000)

Mathematica Bohemica

Sufficient conditions for destabilizing effects of certain unilateral boundary conditions and for the existence of bifurcation points for spatial patterns to reaction-diffusion systems of the activator-inhibitor type are proved. The conditions are related with the mollification method employed to overcome difficulties connected with empty interiors of appropriate convex cones.

Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions II, Examples

Jan Eisner (2001)

Mathematica Bohemica

The destabilizing effect of four different types of multivalued conditions describing the influence of semipermeable membranes or of unilateral inner sources to the reaction-diffusion system is investigated. The validity of the assumptions sufficient for the destabilization which were stated in the first part is verified for these cases. Thus the existence of points at which the spatial patterns bifurcate from trivial solutions is proved.

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