## Topological Degree Theory and ApplicationsSince the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, and it has become a valuable tool in nonlinear analysis. Presenting a survey of advances made in generalizations of degree theory during the past decade, this book focuses on topological degree theory in normed spaces and its applications. The authors begin by introducing the Brouwer degree theory in Rn, then consider the Leray-Schauder degree for compact mappings in normed spaces. Next, they explore the degree theory for condensing mappings, including applications to ODEs in Banach spaces. This is followed by a study of degree theory for A-proper mappings and its applications to semilinear operator equations with Fredholm mappings and periodic boundary value problems. The focus then turns to construction of Mawhin's coincidence degree for L-compact mappings, followed by a presentation of a degree theory for mappings of class (S+) and its perturbations with other monotone-type mappings. The final chapter studies the fixed point index theory in a cone of a Banach space and presents a notable new fixed point index for countably condensing maps. Examples and exercises complement each chapter. With its blend of old and new techniques, Topological Degree Theory and Applications forms an outstanding text for self-study or special topics courses and a valuable reference for anyone working in differential equations, analysis, or topology. |

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### Contents

BROUWER DEGREE THEORY | 1 |

LERAY SCHAUDER DEGREE THEORY | 25 |

DEGREE THEORY FOR SET CONTRACTIVE MAPS | 55 |

GENERALIZED DEGREE THEORY FOR APROPER MAPS | 75 |

COINCIDENCE DEGREE THEORY | 105 |

DEGREE THEORY FOR MONOTONETYPE MAPS | 127 |

FIXED POINT INDEX THEORY | 169 |

REFERENCES | 195 |

217 | |

### Common terms and phrases

a e OQ A-proper mappings A-proper with respect Amer assume assumption Brouwer degree class S+ coincidence degree completes the proof cone continuous compact mapping continuous function continuous mapping contradiction convergent countably condensing mapping Definition deg(f deg(I deg(L deg(T degree theory depend on t e exists finite dimensional subspace fixed point index fixed point theorem Fredholm mapping Hilbert space homeomorphism homotopy of mappings ind(T index theory index Zero type integer Ker(L L-compact Lemma Leray Schauder degree Let Q C R linear lower semicontinuous mapping of class mapping of index Math metric space multi-valued mapping Nonlinear Anal normed space O D(L open bounded subset open subsets operator point in Q projection scheme Proposition Q C E Q C X Q O D(T Q O P real Banach space Section separable Banach spaces Suppose topological degree upper semicontinuous