### Abstract

We consider the nonlinear eigenvalue problem −u^{′′}(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f_{1}(u)=u^{3}+sin(u^{3})∕u, f(u)=f_{2}(u)=u+u^{p}sin(u^{q}) (0≤p<1, 1<q≤p+2) and λ>0 is a bifurcation parameter. It is known that, in this case, 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). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f_{1}(u) and f_{2}(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f_{1}(u) seems to be new.

Original language | English |
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Article number | 111594 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 190 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

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### Keywords

- Global structure
- Nonlinear eigenvalue problems
- Oscillatory bifurcation

### Cite this

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**Simple proof of stationary phase method and application to oscillatory bifurcation problems.** / Kato, Keiichi; Shibata, Tetsutaro.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Simple proof of stationary phase method and application to oscillatory bifurcation problems

AU - Kato, Keiichi

AU - Shibata, Tetsutaro

PY - 2020/1/1

Y1 - 2020/1/1

N2 - We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 10 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ‖∞ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.

AB - We consider the nonlinear eigenvalue problem −u′′(t)=λf(u(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0, where f(u)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (0≤p<1, 10 is a bifurcation parameter. It is known that, in this case, λ is parameterized by the maximum norm α=‖uλ‖∞ of the solution uλ associated with λ and is written as λ=λ(α). We simplify the argument of the stationary phase method and show the asymptotic formulas for λ(α) for f1(u) and f2(u) as α→∞ and α→0. In particular, the shape of bifurcation diagram of λ(α) for f1(u) seems to be new.

KW - Global structure

KW - Nonlinear eigenvalue problems

KW - Oscillatory bifurcation

UR - http://www.scopus.com/inward/record.url?scp=85070927243&partnerID=8YFLogxK

U2 - 10.1016/j.na.2019.111594

DO - 10.1016/j.na.2019.111594

M3 - Article

AN - SCOPUS:85070927243

VL - 190

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

M1 - 111594

ER -