Existence of Solutions for Fourth Order Boundary Value Problems on a Time Scale

Abstract

Dedicated to Allan Peterson on the occasion of his 60th Birthday

Let 𝕋 be a closed subset of ℝ with inf 𝕋 = − ∞ and sup 𝕋 = + ∞. For the fourth order nonlinear dynamic equation, u ^Δ4 = f (t, u, u ^Δ, u ^Δ2 , u ^Δ3 ), t ∈ 𝕋, where f: 𝕋 × ℝ⁴ → ℝ is continuous, we assume solutions of initial value problems are unique and extend to 𝕋. We consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on 𝕋.

Keywords:

• Time scale
• Nonlinear
• Boundary value problem
• Monotone sequence

2000 AMS Subject Classification:

• 39A99
• 34B15