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Topological degree and boundary value problems for nonlinear differential equations. (English) Zbl 0798.34025
Furi, Massimo (ed.) et al., Topological methods for ordinary differential equations. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 24- July 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1537, 74-142 (1993).
This paper surveys recent results in degree theory and its applications to boundary value problems for nonlinear ordinary differential equations. Section 2 contains preliminary material on degree theory of Leray- Schauder type and on existence theorems based upon the homotopy invariance of degree, Zbl 0798.34024 (the so-called Leray-Schauder continuation method; cf. also the preceding review). The degree is constructed for an important in applications class of mappings of the form $L+N$, where $L$ is a linear Fredholm operator of index zero and $N$ is nonlinear $L$-compact (this is essentially the author’s coincidence degree). In Section 3 it is shown that the degree associated to the $T$- periodic solutions of the autonomous system of $n$ equations ${x}^{\text{'}}=g\left(x\right)$ equals the Brouwer degree of $g$ on a bounded subset of ${ℝ}^{n}$. This result and the continuation method are used in Section 4 in order to derive existence results for periodic solutions to systems of the form ${x}^{\text{'}}=g\left(x\right)+e\left(t,x\right)$, where $e$ is a $T$-periodic (in $t\right)$ perturbation of the autonomous term $g$. Section 5 deals with superlinear second order equations ${u}^{\text{'}\text{'}}+g\left(u\right)=p\left(t,u,{u}^{\text{'}}\right)$ and related planar Hamiltonian systems. Existence of a solution (in the case of periodic and Dirichlet boundary conditions) is shown by a continuation principle recently developed by Capietto-Mawhin-Zanolin. The new feature of this principle is that it can be successfully applied to situations (like here) where there are no global bounds for all solutions. However, for a suitably restricted class of solutions such bounds are shown to exist. In Section 6 some second order equations with restoring force having a singularity are considered, and Section 7 describes recent results by Ortega showing how degree theory may be used in order to study stability of periodic solutions to second order equations and first order planar systems. The paper contains a large number of bibliographical remarks and a bibliography of 141 titles.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 47H11 Degree theory (nonlinear operators) 47J25 Iterative procedures (nonlinear operator equations) 34C25 Periodic solutions of ODE