{
  "id": "2105.12193",
  "version": "v1",
  "published": "2021-05-25T20:14:35.000Z",
  "updated": "2021-05-25T20:14:35.000Z",
  "title": "Bifurcation Theory for Fredholm Operators",
  "authors": [
    "Julián López-Gómez",
    "Juan Carlos Sampedro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper consists of four parts. It begins by using the authors's generalized Schauder formula, \\cite{JJ}, and the algebraic multiplicity, $\\chi$, of Esquinas and L\\'{o}pez-G\\'{o}mez \\cite{ELG,Es,LG01} to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick--Pejsachowicz--Rabier degree, \\cite{JJ2}. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of $\\mathfrak{F}(\\lambda,u)=0$ at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Further, we combine the theorem of structure of analytic manifolds with a brilliant idea of Buffoni and Toland \\cite{BT} to show that the solution sets of the most paradigmatic one-dimensional boundary value problems with analytic nonlinearities actually consist of global analytic arcs of curve. Finally, the unilateral theorems of \\cite{LG01,LG02}, as well as the refinement of Xi and Wang \\cite{XW}, are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-25T20:14:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34B15",
      "35J93",
      "47H11",
      "58C40"
    ],
    "keywords": [
      "bifurcation theory",
      "fredholm operators",
      "paradigmatic one-dimensional boundary value problems",
      "multidimensional quasilinear boundary value problem",
      "solution set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}