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Nonlinear functional analysis

This manuscript provides a brief introduction to nonlinear functional analysis. We start out with calculus in Banach spaces, review differentiation and integration, derive the implicit function theorem (using the uniform contraction principle) and apply the result to prove existence and uniqueness of solutions for ordinary differential equations in Banach spaces. Next we introduce the mapping degree in both finite (Brouwer degree) and infinite dimensional (Leray-Schauder degree) Banach spaces. Several applications to game theory, integral equations, and ordinary differential equations are discussed. As an application we consider partial differential equations and prove existence and uniqueness for solutions of the stationary Navier-Stokes equation. Finally, we give a brief discussion of monotone operators.