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Asymptotic Behavior of Bifurcation Curve for Sine‐Gordon‐Type Differential Equation

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  • Tetsutaro Shibata

Abstract

We consider the nonlinear eigenvalue problems for the equation −u″(t) + sin u(t) = λu(t), u(t) > 0, t ∈ I = :(0, 1), u(0) = u(1) = 0, where λ > 0 is a parameter. It is known that for a given ξ > 0, there exists a unique solution pair (uξ,λ(ξ))∈C2(I¯)×ℝ+ with ∥uξ∥∞=ξ. We establish the precise asymptotic formulas for bifurcation curve λ(ξ) as ξ → ∞ and ξ → 0 to see how the oscillation property of sin u has effect on the behavior of λ(ξ). We also establish the precise asymptotic formula for bifurcation curve λ(α) (α=∥uλ∥2) to show the difference between λ(ξ) and λ(α).

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Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:753857
DOI: 10.1155/2012/753857
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