Krasnosel'skii type formula and translation along trajectories method for evolution equations

Abstract

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.

Periodic solutions for nonlinear evolution equations at resonance

“…The principle for the equations on any Banach space has been recently considered in [5] in the case when −A generates a compact C 0 semigroup and in [6] for A being an m-accretive operator. In [8], a similar results were obtained when −A generates a semigroup of contractions and F is condensing. For the results when the operator A is replaced by a time-dependent family {A(t)} t≥0 see [9].…”

The Krasnosel’skiĭ-Quittner formula and instability of a reaction-diffusion system with unilateral obstacles

“…In order to formulate the Krasnosel'skiȋ-Quittner formula for (4) or (5), we have to introduce the metric projection P K : V → K which is defined such that for v ∈ V the image u = P K v is the unique point from K which has closest distance (with respect to • ) to v. Since V is a Hilbert space and K ⊆ V is nonempty, closed and convex, it is well-known that u = P K v is uniquely defined by that requirement and characterized by (i.e. the unique solution of) the variational inequality (6) u ∈ K, u − v, ϕ − u ≥ 0 for all ϕ ∈ K.…”

“…This is not completely true, since recently the Krasnoselskiȋ formula has been obtained also for (Lipschitz or compact) perturbations for linear generators of semigroups. We remark that just to see that the terms in the Krasnoselskiȋ formula are defined, some compactness requirements are necessary: Either compactness of the semigroup [3,5] or at least on the set where the nonlinearity assumes its values [4], or compactness of the nonlinearity [6] are required.…”

Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces

“…See also [2] for results in the case when A is a general single valued m-accretive operator. In [6] the averaging principle were studied in the case when −A generates a C 0 semigroup of contractions and F is a condensing map with respect to the Hausdorff measure of noncompactness. Obtained results were used to study the periodic solutions of the first order hyperbolic equations.…”