Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations
| dc.contributor.author | Shibata, Tetsutaro | |
| dc.date.accessioned | 2021-11-05T20:05:01Z | |
| dc.date.available | 2021-11-05T20:05:01Z | |
| dc.date.issued | 2019-05-07 | |
| dc.description.abstract | We study the nonlinear eigenvalue problem -u″(t) = λ(u(t)p + g(u(t))), u(t) > 0, t ∈ (-1, 1), u(±1) = 0, where g(u) = h(u) sin(ur), p, r are given constants satisfying p ≥ 0, 0 < r ≤ 1 and λ > 0 is a parameter. It is known that under suitable conditions on h, λ is parameterized by the maximum norm α = ∥uα∥∞ of the solution uλ associated with λ and λ = λ(α) is a continuous function for α > 0. When p = 1, h(u) ≡ 1 and r = 1/2, this equation has been introduced by Chen [4] as a model equation such that there exist infinitely many solutions near λ = π2/4. We prove that λ(α). It is found that the shapes of bifurcation curves depend on the condition p > 1 or p < 1. The main tools of the proof are time-map argument and stationary phase method. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 11 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Shibata, T. (2019). Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations. <i>Electronic Journal of Differential Equations, 2019</i>(62), pp. 1-11. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14795 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2019, San Marcos, Texas: Texas State University and University of North Texas. | |
| dc.subject | Oscillatory bifurcation | |
| dc.subject | Time-map argument | |
| dc.subject | Stationary phase method | |
| dc.title | Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations | |
| dc.type | Article |