Applications of global bifurcation to existence theorems for Sturm-Liouville problems

• Autorzy: Jacek Gulgowski
• Języki publikacji: EN

Abstrakty

EN

We prove an existence theorem for Sturm–Liouville problems
⎧u''(t) + φ(t,u(t),u'(t)) = 0 for a.e. t ∈ (a,b), (∗)
⎨
⎩l(u) = 0,
where $φ: [a,b] × ℝ^{k} × ℝ^{k} → ℝ^[k}$ is a Carathéodory map. We assume that
φ(t,x,y) = m₁φ₀(t,x,y) + o(|x|+|y|) as |x|+|y| → 0 and
φ(t,x,y) = m₂φ₀(t,x,y) + o(|x|+|y|) as |x|+|y| → ∞, where m₁,m₂ are positive constants and φ₀ belongs to a class of nonlinear maps. The proof bases on global bifurcation results. We define a map $f: (0,∞) × C¹([a,b],ℝ^{k}) → C¹([a,b],ℝ^{k}) such that if f(1,u) = 0, then u is a solution of (∗). Then we show that there exists a connected set C of nontrivial zeroes of f such that there exist (λ₁,u₁),(λ₂,u₂)∈C with λ₁ < 1 < λ₂. In the last section we give examples of maps φ₀ leading to specific existence theorems.

Kategorie tematyczne

• 34B24: Sturm-Liouville theory
• 34C23: Bifurcation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2004
• Tom: 83
• Numer: 3
• Strony: 221-229

Daty

wydano

2004

Twórcy

autor

Jacek Gulgowski

• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Identyfikatory

DOI

10.4064/ap83-3-4