Degree Theory

  • Dumitru Motreanu
  • Viorica Venera Motreanu
  • Nikolaos Papageorgiou


This chapter provides the fundamental elements of degree theory used later in the book for showing abstract results of critical point theory or bifurcation theory as well as for the study of the existence and multiplicity of solutions to nonlinear problems. The first section of the chapter introduces Brouwer’s degree and its important applications such as Brouwer’s fixed point theorem, Borsuk’s theorem, Borsuk–Ulam, and Lyusternik–Schnirelmann–Borsuk theorems. The second section sets forth the Leray–Schauder degree theory for compact perturbations of the identity. The third section amounts to a description of the degree for (S)+maps using Galerkin approximations and construction of the degree theory for multifunctions of the form f + A with f an (S)+-map and A a maximal monotone operator. Comments and historical notes are given in a remarks section.


Degree Theory Leray-Schauder Degree Compact Perturbation Galerkin Approximation Borsuk 
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Copyright information

© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  • Dumitru Motreanu
    • 1
  • Viorica Venera Motreanu
    • 2
  • Nikolaos Papageorgiou
    • 3
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Department of MathematicsNational Technical UniversityAthensGreece

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