TY - JOUR
A2 - Zheng, Sining
AU - Shibata, Tetsutaro
PY - 2012
DA - 2012/12/30
TI - Asymptotic Behavior of Bifurcation Curve for Sine-Gordon-Type Differential Equation
SP - 753857
VL - 2012
AB - 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).
SN - 1085-3375
UR - https://doi.org/10.1155/2012/753857
DO - 10.1155/2012/753857
JF - Abstract and Applied Analysis
PB - Hindawi Publishing Corporation
KW -
ER -