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

Keiichi Kato, Tetsutaro Shibata

Research output: Contribution to journalArticle

### 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)=f1(u)=u3+sin(u3)∕u, f(u)=f2(u)=u+upsin(uq) (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=‖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.

Original language English 111594 Nonlinear Analysis, Theory, Methods and Applications 190 https://doi.org/10.1016/j.na.2019.111594 Published - 1 Jan 2020

### Fingerprint

Stationary Phase
Maximum Norm
Nonlinear Eigenvalue Problem
Bifurcation Diagram
Asymptotic Formula
Simplify
Bifurcation

### Keywords

• Global structure
• Nonlinear eigenvalue problems
• Oscillatory bifurcation

### Cite this

title = "Simple proof of stationary phase method and application to oscillatory bifurcation problems",
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)=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.",
keywords = "Global structure, Nonlinear eigenvalue problems, Oscillatory bifurcation",
author = "Keiichi Kato and Tetsutaro Shibata",
year = "2020",
month = "1",
day = "1",
doi = "10.1016/j.na.2019.111594",
language = "English",
volume = "190",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier Limited",

}

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 190, 111594, 01.01.2020.

Research output: Contribution to journalArticle

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 -