Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms

Abstract

We consider the bifurcation problem −u ′′ (t) = λ (u(t) + g(u(t))) , u(t) > 0, t ∈ I := (−1, 1), u(±1) = 0, where g(u) ∈ C 1 (R) is a periodic function with period 2π and λ > 0 is a bifurcation parameter. It is known that, under the appropriate conditions on g, λ is parameterized by the maximum norm α = ∥u λ ∥∞ of the solution u λ associated with λ and is written as λ = λ(α). If g(u) is periodic, then it is natural to expect that λ(α) is also oscillatory for α ≫ 1. We give a simple condition of g(u), by which we can easily check whether λ(α) is oscillatory and intersects the line λ = π 2 /4 infinitely many times for α ≫ 1 or not.

Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term

“…Introduction. Many problems in the fields of physics [1,2], mathematical biology [3,4,5,6,7,8,9] and control theory can be attributed to study of the nonlinear differential equations, especially it is almost periodicity because there is almost no phenomenon that is purely periodic [10,11,12]. Consequently, the qualitative theory of differential equations involving almost periodicity has been the new world-wide focus.…”

Asymptotic behavior of bifurcation curves of ODEs with oscillatory nonlinear diffusion

“…Added to these, there are several papers studying the asymptotic behavior of oscillatory bifurcation curves. We refer to [4,5,6,7,[14][15][16][17][18] and the references therein.…”

Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

“…On the other hand, there are several papers which treat the asymptotic behavior of oscillatory bifurcation curves. We refer to [7,[12][13][14][15][16][17][18][19] and the references therein. Our equation (1) contains both nonlinear diffusion term and oscillatory nonlinear terms.…”