A global bifurcation index for set-valued perturbations of Fredholm operators

The Krasnosel'skii formula for parabolic differential inclusions with state constraints

“…Then all above results remain true, in particular: the solution map Σ : − → E representing it, such that x / ∈ q p −1 (x) for x ∈ ∂V , the fixed point index Ind ((p, q) , V ) is well-defined (cf. [13,Th. 4.5]).…”

“…Properties of a c-admissible ϕ strongly depend on a decomposition ϕ = f • ψ or a pair (p, q) representing it. When studying c-admissible maps one has to take into account representing pairs (for a detailed discussion of c-admissible maps, related topics and some references -see [13]). In particular: if ϕ : X ⊸ Y is cell-like, then the canonical pair (p ϕ , q ϕ ), where the graph Gr(ϕ) := {(x, y) ∈ X × Y | y ∈ ϕ(x)}, p ϕ : Gr(ϕ) → X and q ϕ : Gr(ϕ) → Y are projections, is c-admissible and represents ϕ.…”

“…4.5]). This index has the usual properties such as: the existence, the localization, the additivity and the homotopy invariance (see [13]).…”

On the global bifurcation of periodic solutions of differential inclusions in Hilbert spaces

“…The bifurcation problem for differential and operator inclusions has been studied by many researchers (see, e.g., [1][2][3][4][5][6][7]). Commonly, this problem is treated by topological methods, the base of which consists in the evaluation of the bifurcation index, being the topological invariant which represents the difference of the topological degrees of the corresponding multimap when the parameter passes through the critical value.…”