The Krasnosel'skii type degree formula for the equationu = −Au+F (u) where A : D(A) → E is a linear operator on a separable Banach space E such that −A is a generator of a C 0 semigroup of bounded linear operators of E and F : E → E is a locally Lipschitz k-set contraction, is provided. D(A)), then the topological degree of −A + F with respect to V is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajectories. It is applied to the non-autonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro-und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field to any spiral remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane.

Abstract. We study the existence of T -periodic solutions (T > 0) for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of well known Landesman-Lazer and strong resonance conditions. Obtained index formula is used to derive the criteria determining the existence of T -periodic solutions for the heat equation being at resonance at infinity.

We are concerned with periodic problems for nonlinear evolution equations at resonance of the formu(t) = −Au(t) + F (t, u(t)), where a densely defined linear operator A : D(A) → X on a Banach space X is such that −A generates a compact C 0 semigroup and F : [0, +∞) × X → X is a nonlinear perturbation. Imposing appropriate Landesman-Lazer type conditions on the nonlinear term F , we prove a formula expressing the fixed point index of the associated translation along trajectories operator, in the terms of a time averaging of F restricted to Ker A. By the formula, we show that the translation operator has a nonzero fixed point index and, in consequence, we conclude that the equation admits a periodic solution. T 0 P F (s, x) ds for x ∈ Nwhere P : X → X is a topological projection onto N with Ker P = M . First, we are concerned with an equatioṅ u(t) = −Au(t) + εF (t, u(t)), t > 0

We are interested in the differential equationü(t) = −Au(t) − cAu(t) + λu(t) + F (t, u(t)), where c > 0 is a damping factor, A is a sectorial operator and F is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that λ is an eigenvalue of A and F is a bounded map. We introduce new geometrical conditions for the nonlinearity F and use topological degree methods to find T -periodic solutions for this equation as fixed points of Poincaré operator.