The fixed point index of the Poincare` translation operator associated to an ordinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclidean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincare` translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation operator associated with non-autonomous equations will be deduced as consequences.

The fixed point index of the Poincaré translation operator on differentiable manifolds / M. FURI; M. PERA; M. SPADINI. - STAMPA. - (2005), pp. 741-782.

The fixed point index of the Poincaré translation operator on differentiable manifolds

FURI, MASSIMO;PERA, MARIA PATRIZIA;SPADINI, MARCO
2005

Abstract

The fixed point index of the Poincare` translation operator associated to an ordinary differential equation is a very useful tool for establishing the existence of periodic solutions. In this chapter we focus on ODEs on differentiable manifolds embedded in Euclidean spaces. Our purpose is twofold: on the one hand we aim to provide a short and accessible introduction to some topological tools (such as the Topological Degree, the Degree of a tangent vector field and the Fixed Point Index) that are useful in Nonlinear Analysis; on the other hand we offer a unifying approach to several results about the fixed point index of the Poincare` translation operator that were previously scattered among a number of publications. Our main concern will be a formula for the computation of the fixed point index of the flow operator induced on a manifold by a first order autonomous ordinary differential equation. Other formulas for the fixed point index of the translation operator associated with non-autonomous equations will be deduced as consequences.
2005
9781402032219
Handbook of topological fixed point theory
741
782
M. FURI; M. PERA; M. SPADINI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/307421
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