The continuation of the Conley index for singular perturbations and the Conley indices in gradient-like systems. I (Q5955605)

In this long paper two topics of the Conley index theory are considered: the singular perturbation of the index and the existence of isolated invariant sets for gradient-like systems. For a real \(\varepsilon\), let \(X^\varepsilon\) and \(Y\) be metric spaces, let \(\psi^\varepsilon=(\phi^\varepsilon,\xi^\varepsilon)\) be a semiflow on \(X^\varepsilon\times Y\), let \(\pi\) be a semiflow on \(Y\), and let \(0_\varepsilon\) be a given point in \(X^\varepsilon\). Assume that if \((\varepsilon_n,t_n,y_n)\) is a sequence in \(\mathbb R\times [0,\infty)\times Y\) which tends to \((0,t,y)\) then \(\phi^{\varepsilon_n}(t_n,x_n,y_n)\to 0_{\varepsilon_n}\) in \(X^{\varepsilon_n}\), \(\text{ dist}(\phi^{\varepsilon_n}(\cdot,x_n,y_n),0_{\varepsilon_n})\to 0\) in \(L^p[0,t]\), and \(\xi^{\varepsilon_n}(t_n,x_n,y_n)\to \pi(t,y)\) in \(Y\) provided \(x_n\to 0_{\varepsilon_n}\) in \(X^{\varepsilon_n}\) as \(n\to \infty\). Furthermore, assume that the semiflows satisfy some compactness hypotheses. For \(r>0\) let \(A^\varepsilon\) be a ball in \(X^\varepsilon\) of radius \(r\) centered at \(0_\varepsilon\). Let \(I_\pi\) be an isolated invariant set of \(\pi\) and let \(B\) be its isolating neighborhood. NEWLINENEWLINENEWLINETheorem A, the main result on singular perturbation, asserts that if \(X^\varepsilon\) satisfies some uniform contractibility hypothesis then there exists an \(\varepsilon_0>0\) such that for \(|\varepsilon|\leq \varepsilon_0\) the maximal invariant set \(I^\varepsilon\) of \(\psi^\varepsilon\) in \(A^\varepsilon\times B\) is isolated and the Conley indices \(h(I^\varepsilon)\) and \(h(I_\pi)\) are equal each to the other. NEWLINENEWLINENEWLINEAs an application of Theorem A, a result on a boundary value problem on a thin domain is proved. NEWLINENEWLINENEWLINEAssume now that \(\pi\) is a gradient like semiflow with respect to a Lyapunov function \(V\) on a Banach space \(X\), \(E\) is the set of equilibria of \(\pi\), and \(\Gamma\) is a compact component of \(E\). Under some compactness assumptions on \(\pi\) and Sard theorem type assumptions on \(V\), Theorem B, the main result on gradient-like systems, asserts the existence of an isolated invariant set \(I\) which contains \(\Gamma\) and calculates \(h(I)\) in the case \(\Gamma\) is of local minimum type or of mountain-pass type.