Topological degree methods for perturbations of operators generating compact C0C0 semigroups

The main aim of this paper is to construct a topological degree for maps -A+F:M∩D(A)→E-A+F:M∩D(A)→E where a densely defined closed operator A:D(A)→EA:D(A)→E of a Banach space E   is such that -A-A is the generator of a compact C0C0 semigroup, and F:M→EF:M→E is a locally Lipschitz map defined on a neighborhood retract M⊂EM⊂E. If M is a closed convex cone, then a degree formula allowing an effective computation of the degree is proved. This formula provides an infinite-dimensional counterpart of the well-known Krasnosel'skii theorem. By the use of the introduced topological degree and an abstract result concerning branching of fixed points, the bifurcation of periodic points of the parameterized boundary value problemu˙=-λAu+λF(t,u),λ>0,u(t)∈M,u(0)=u(T)is studied. Examples of applications to partial differential equations are discussed.