Periodic solutions for nonlinear evolution equations at resonance

Abstract

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

A Landesman–Lazer type result for periodic parabolic problems on RN at resonance

“…Assumptions (5) and (6) will be reffered to as Landesman-Lazer type conditions, which had been widely used in the literature in the context of evolutionary partial differential equations -see e.g. [12], [4], as well as recent papers [7], [18] and [19]. The novelty of this paper may be viewed in the fact that we study the problem on an unbounded domain, which entails a few issues concerning compactness.…”

“…In order to find T -periodic solutions of (1) we shall look for fixed points of Φ T by use of local fixed point index theory. Motivated by [4], [7] and [18], we prove a resonant version of averaging principle. Roughly speaking, it states that the topological properties of our equation can be described in terms of the average functionF : N → N of F, restricted to the kernel of operator A, given byF…”

The averaging principle and periodic solutions for nonlinear evolution equations at resonance

“…The resonant version of averaging principle was proved in [18] in the case when A is a generator of a compact C 0 semigroup (not necessary sectorial) and F : [0, +∞)× X → X is a continuous map. Obtained result were used to prove the criteria on the existence of T -periodic solution for (1.2) under the assumption that f satisfies Landesman-Lazer conditions.…”

“…Then the resonant averaging principle says that, for small ε > 0, the fixed point index of Φ T (ε, · ) is equal to the Brouwer degree of −g. In Section 5 we formulate geometrical conditions (G1) and (G2) (see page 18) and apply the resonant averaging principle to prove the second result, the index formula for periodic solutions, which express the fixed point index of the translation along trajectories operator Φ T on sufficiently large ball, in terms of conditions (G1) and (G2). Finally, in Section 6 we provide applications for particular partial differential equations.…”

Invariant sets and connecting orbits for nonlinear evolution equations at resonance

“…(see[11], Theorem 3.5,[6], Proposition 4.3.3). If λ + µ / ∈ σ(A), then h(Φ, {0}) = h(Φ, {0}) = Σ b l , where b l := 0 if λ + µ < λ 1 and b l := l i=1…”