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Degree Theory

In: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

Author

Listed:
  • Dumitru Motreanu

    (University of Perpignan, Department of Mathematics)

  • Viorica Venera Motreanu

    (Ben-Gurion University of the Negev, Department of Mathematics)

  • Nikolaos Papageorgiou

    (National Technical University, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Dumitru Motreanu & Viorica Venera Motreanu & Nikolaos Papageorgiou, 2014. "Degree Theory," Springer Books, in: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, edition 127, chapter 0, pages 61-96, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-9323-5_4
    DOI: 10.1007/978-1-4614-9323-5_4
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