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

Abstract

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.

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

“…Moreover, one immediately obtains that, for any t ≥ 0, u ∈ H 1 (R N ), µ, ν ∈ [0, 1], G(t, u, µ) − G(t, u, ν) L 2 ≤ |ρ(µ) − ρ(ν)|(1 + u H 1 ) for some ρ ∈ C([0, 1]). Therefore G satisfies (18), (19) and (20). Hence it follows that Θ (ǫ) T is well defined and we can apply Proposition 3.4 to infer that Θ (ǫ) T is an ultimately compact operator (for any ǫ ∈ [0, 1]).…”

“…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.…”

Connecting orbits for nonlinear differential equations at resonance

“…Furthermore, by Theorem 2.1 from [9] we have the following Theorem 3.2. Under the assumptions (A1), (A2) and (A3), if λ = λ k for some k ≥ 1, is an eigenvalue of A, then there exists a a decomposition X = X + ⊕X − ⊕X 0 on closed subspaces, such that…”

“…In view of Theorems 3.4 and 3.5 from [9], the semiflow Φ is continuous and admissible with respect to any bounded set N ⊂ X α .…”

“…where u( · ; s, x) : [0, +∞) → X α is a weak solution of (3.9) starting at x. Theorems 3.4 and 3.5 from [9] say that the family {Ψ s } s∈[0,1] is continuous and any bonded set is admissible. Hence we have a homotopy between Ψ 1 = Φ and the semiflow Ψ 0 associated withu (t) = −Au(t) + λu(t) + P F (P u(t)), t > 0.…”