Abstract
In preceding articles ([3] and [5]), we began an examination of the structure of the solution set to the two-parameter system (Formula presented.) In this article, we treat the case left uncovered in our previous analysis; namely, we assume f (s,0) = 0 and g(0,t) = 0 for all s, t ∈ℝ. In this situation, solutions to (*) of the form (λ, μ, u, 0) or (λ, μ,0,v lie in linear subspaces of ℝ.2dx (C01 [a, b] 2. As such, they are neither locally expressable as functions of (λ, μ) nor are à priori bounded in terms of (λ, μ), as was crucial to the analysis in [3] and [5]. Nevertheless, we demonstrate that solutions to (*) of the form (λ, μ, u, v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b), where n and m are positive integers, arise as global secondary bifurcations from solutions of the form (λ,μ,u,0) with u having n − 1 simple zeros in a, b and from solutions of the form (λ, μ, o, v) with v having m − 1 simple zeros in (a, b). Moreover, we establish that solutions to (*) of the form (λ,μ,u,v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b) serve as a link between solutions of the form (λ, μ, u, 0) with u having n − 1 simple zeros in (a,b) and solutions of the form (λ, μ, 0, v) with v having m − 1 simple zeros in (a, b). The analysis in this article when combined with that in [3] and [5] provides a fairly comprehensive examination of the structure of the solution set to (*).
| Original language | English (US) |
|---|---|
| Pages (from-to) | 470-488 |
| Number of pages | 19 |
| Journal | Results in Mathematics |
| Volume | 22 |
| Issue number | 1 |
| DOIs | |
| State | Published - 1992 |
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- Applied Mathematics
Cite this
On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure. / Cantrell, Robert.
In: Results in Mathematics, Vol. 22, No. 1, 1992, p. 470-488.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure
AU - Cantrell, Robert
PY - 1992
Y1 - 1992
N2 - In preceding articles ([3] and [5]), we began an examination of the structure of the solution set to the two-parameter system (Formula presented.) In this article, we treat the case left uncovered in our previous analysis; namely, we assume f (s,0) = 0 and g(0,t) = 0 for all s, t ∈ℝ. In this situation, solutions to (*) of the form (λ, μ, u, 0) or (λ, μ,0,v lie in linear subspaces of ℝ.2dx (C01 [a, b] 2. As such, they are neither locally expressable as functions of (λ, μ) nor are à priori bounded in terms of (λ, μ), as was crucial to the analysis in [3] and [5]. Nevertheless, we demonstrate that solutions to (*) of the form (λ, μ, u, v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b), where n and m are positive integers, arise as global secondary bifurcations from solutions of the form (λ,μ,u,0) with u having n − 1 simple zeros in a, b and from solutions of the form (λ, μ, o, v) with v having m − 1 simple zeros in (a, b). Moreover, we establish that solutions to (*) of the form (λ,μ,u,v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b) serve as a link between solutions of the form (λ, μ, u, 0) with u having n − 1 simple zeros in (a,b) and solutions of the form (λ, μ, 0, v) with v having m − 1 simple zeros in (a, b). The analysis in this article when combined with that in [3] and [5] provides a fairly comprehensive examination of the structure of the solution set to (*).
AB - In preceding articles ([3] and [5]), we began an examination of the structure of the solution set to the two-parameter system (Formula presented.) In this article, we treat the case left uncovered in our previous analysis; namely, we assume f (s,0) = 0 and g(0,t) = 0 for all s, t ∈ℝ. In this situation, solutions to (*) of the form (λ, μ, u, 0) or (λ, μ,0,v lie in linear subspaces of ℝ.2dx (C01 [a, b] 2. As such, they are neither locally expressable as functions of (λ, μ) nor are à priori bounded in terms of (λ, μ), as was crucial to the analysis in [3] and [5]. Nevertheless, we demonstrate that solutions to (*) of the form (λ, μ, u, v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b), where n and m are positive integers, arise as global secondary bifurcations from solutions of the form (λ,μ,u,0) with u having n − 1 simple zeros in a, b and from solutions of the form (λ, μ, o, v) with v having m − 1 simple zeros in (a, b). Moreover, we establish that solutions to (*) of the form (λ,μ,u,v) with u having n − 1 simple zeros in (a, b) and v having m − 1 simple zeros in (a, b) serve as a link between solutions of the form (λ, μ, u, 0) with u having n − 1 simple zeros in (a,b) and solutions of the form (λ, μ, 0, v) with v having m − 1 simple zeros in (a, b). The analysis in this article when combined with that in [3] and [5] provides a fairly comprehensive examination of the structure of the solution set to (*).
UR - http://www.scopus.com/inward/record.url?scp=84871815261&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84871815261&partnerID=8YFLogxK
U2 - 10.1007/BF03323101
DO - 10.1007/BF03323101
M3 - Article
AN - SCOPUS:84871815261
VL - 22
SP - 470
EP - 488
JO - Results in Mathematics
JF - Results in Mathematics
SN - 0378-6218
IS - 1
ER -