Asymptotic length of bifurcation curves related to inverse bifurcation problems

We consider the nonlinear eigenvalue problemu″(t)+λf(u(t))=0,u(t)>0,t∈I=:(−1,1),u(1)=u(−1)=0, where f(u)=u+g(u)f(u)=u+g(u) and λ>0λ>0 is a parameter. Our interest is the asymptotic length L(g,α)L(g,α) of the bifurcation curve λ=λ(g,α)λ=λ(g,α) (α=‖uλ‖∞>0α=‖uλ‖∞>0). By the notion of L(g,α)L(g,α), we will propose a new framework of inverse bifurcation problems. Precisely, we consider whether it is possible to characterize the unknown nonlinear term g(u)g(u) by L(g,α)L(g,α). If g(u)=g1(u)=(u/2)sin⁡ug(u)=g1(u)=(u/2)sin⁡u or g2(u)=u+(1/2)sin⁡ug2(u)=u+(1/2)sin⁡u, then the bifurcation curves λ=λ(gi,α)λ=λ(gi,α) (i=1,2i=1,2) are continuous functions of α>0α>0 and λ(gi,α)→π2/4λ(gi,α)→π2/4 as α→∞α→∞. Furthermore, they intersect the line λ=π2/4λ=π2/4 infinitely many times as α→∞α→∞. In this paper, by establishing the precise asymptotic formula for λ(gi,α)λ(gi,α) for α≫1α≫1, we obtain the precise asymptotic length L(gi,α)L(gi,α) of λ(gi,α)λ(gi,α) for α≫1α≫1. Then we consider the set {g∈C2(R¯+):λ(g,α)→π2/4 as α→∞}, and study the existence and non-existence of g   whose asymptotic formula for L(g,α)L(g,α) is equal to those of L(gi,α)L(gi,α) up to the second term.