We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces and l-infinity (S)) even the best possible bound 1 is attained for certain measures of noncompactness.
We consider a reaction-diffusion system of activator-inhibitor type which is subject to Turing's diffusion-driven instability. It is shown that unilateral obstacles of various type for the inhibitor, modeled by variational inequalities, lead to instability of the trivial solution in a parameter domain where it would be stable otherwise. The result is based on a previous joint work with I.-S. Kim, but a refinement of the underlying theoretical tool is developed. Moreover, a different regime of parameters...
Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...
We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential...
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