Topological Degree Theory and Applications
Since 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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A-proper mappings A-proper with respect assume assumption Brouwer degree class S+ coincidence degree completes the proof cone constant continuous compact mapping continuous function continuous mapping contradiction convergent countably condensing mapping define a mapping Definition deg(I deg(L degree theory diﬀerential equations exists finite dimensional subspace fixed point index Fredholm mapping G D(T G dQ PI G Tx Hilbert space homeomorphism homotopy of mappings Im(L implies ind(T index theory index zero type integer Ker(L KPQ)T L-compact Lemma Leray Schauder degree lower semicontinuous mapping of class mapping of index Math maximal monotone mapping multi-valued mapping nonempty nonlinear normed space O’Regan open bounded subset open subset PI D(T projection scheme Proposition Q PI quasinormal real Banach space relatively compact Schauder basis Section separable Banach spaces sequence Suppose topological degree upper semicontinuous xo G Y. Q. Chen