On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure

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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 languageEnglish (US)
Pages (from-to)470-488
Number of pages19
JournalResults in Mathematics
Volume22
Issue number1
DOIs
StatePublished - 1992

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Sturm-Liouville
Boundary value problems
Boundary Value Problem
Zero
Solution Set
Form
Two Parameters
Bifurcation
Subspace
Integer

ASJC Scopus subject areas

  • Mathematics (miscellaneous)
  • Applied Mathematics

Cite this

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title = "On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure",
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 {\`a} 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 (*).",
author = "Robert Cantrell",
year = "1992",
doi = "10.1007/BF03323101",
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T1 - On solutions to coupled multiparameter nonlinear Sturm- Liouville boundary value problems whose state components have a specified nodal structure

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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 (*).

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