In the paper a topological degree is constructed for the class of maps of the form −A + F where M is a closed neighborhood retract in a Banach space E, A : D(A) E is a m-accretive map such that −A generates a compact semigroup and F : M → E is a locally Lipschitz map.The obtained degree is applied to studying the existence and branching of periodic points of differential inclusions of the type (2000): 47H10, 47H11, 47J15, 47J35, 37L05. Key words: evolution equation, accretive operator, topological degree, fixed point index, structure of solutions, branching, periodic solution, t has no fixed points on the boundary ∂ M U . Then as the topological degree of −A + F with respect to U ⊂ M we take the fixed point index of t . It has all the expected properties and it can be effectively applied to differential inclusions of the typeu ∈ −Au + F (t, u). It is noteworthy that the idea of the approach is consistent with the Krasnoselskii formula for the finite dimensional equationu = f (u), stating that the Brouwer degree of −f is equal to the fixed point index of the translation along trajectories operator.
Mathematics Subject ClassificationThe paper is organized as follows. The aim of the rest of Section 1 is to recall basic definitions and facts concerning semiproducts in Banach spaces, accretive operators as well as fixed point index. Section 2 is devoted to compactness and continuity properties of evolution problems, i.e. the properties of the operator (A, x, w) → A (x, w) := u, defined for a m-accretive A, x ∈ D(A) and w ∈ L 1 ([a, b], E), where u is the solution Vol. 7, 2007 Degree theory for perturbations of m-accretive operators 3Recall that a set-valued map A : D(A) E, where E is a Banach space and D(A) ⊂ E, is called an accretive operator if, for any x, y ∈ D(A), u ∈ Ax, v ∈ Ay and λ > 0,x − y ≤ x − y + λ(u − v) . Vol. 7, 2007 Degree theory for perturbations of m-accretive operators 5