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Degree Theory and Fixed Point Theory
2016Degree theory deals with equations of the form \(\varphi (u) = h\) on a space X (finite of infinite dimensional). It addresses the questions of existence, uniqueness, or multiplicity of solutions and their distribution in the space. Moreover, it examines how sensitive are these properties to variations of \(\varphi\) and h.
Leszek Gasiński +1 more
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1995
Throughout this section, we take Ω to be a bounded open subset of ℝ n and take cl Ω ∋ × → f(x) ∈ ℝ n to be continuous. We wish to estimate the number of solutions x in cl Ω of the equation $$f(x) = 0.$$ (1.1) We shall often be content with demonstrating that there is (at least) one solution.
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Throughout this section, we take Ω to be a bounded open subset of ℝ n and take cl Ω ∋ × → f(x) ∈ ℝ n to be continuous. We wish to estimate the number of solutions x in cl Ω of the equation $$f(x) = 0.$$ (1.1) We shall often be content with demonstrating that there is (at least) one solution.
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2002
For almost a century, degree theory has been a very important tool in analysis of existence and multiplicity results for solutions to nonlinear equations in euclidean spaces and manifolds. For example, the study of ordinary and partial differential equations has been considerably improved by degree theory.
Antonio Villanacci +3 more
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For almost a century, degree theory has been a very important tool in analysis of existence and multiplicity results for solutions to nonlinear equations in euclidean spaces and manifolds. For example, the study of ordinary and partial differential equations has been considerably improved by degree theory.
Antonio Villanacci +3 more
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2015
In the preceding chapters we saw several ways to show that two open subsets of \(\mathbf{R}^{n}\), and more generally two manifolds, are not diffeomorphic.
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In the preceding chapters we saw several ways to show that two open subsets of \(\mathbf{R}^{n}\), and more generally two manifolds, are not diffeomorphic.
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