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Leray-Schauder continuation theorems in the absence of a priori bounds. (English) Zbl 0912.47040

The Leray-Schauder continuation principle uses the following basic idea: Let $X$ be a Banach space, ${\Omega }\subset X×\left[0,1\right]$ a bounded open set, $F:\overline{{\Omega }}\to X$ a compact continuous operator, and ${\Sigma }$ the (possibly empty) set of solutions of the equation $x=F\left(x,\lambda \right)$ in $\overline{{\Omega }}$. If the a priori estimate ${\Sigma }\cap \partial {\Omega }=\varnothing$ holds and the map $I-F\left(·,0\right)$ has nonzero degree, then ${\Sigma }$ contains a continuum along which $\lambda$ assumes all values between 0 and 1. Loosely speaking, this principle is useful whenever it is difficult to find invariant balls for the operator $F\left(·,1\right)$, and thus classical fixed point principles do not apply.

In this interesting survey article, the author discusses several variants of continuation principles where a priori bounds are not required explicitly. This essentially enlarges the class of nonlinear problems for which existence may be established by means of topological arguments. Some of the existence theorems given here are taken from previous papers by A. Capietto [Boll. Unione Mat. Ital., VII. Ser. B 8, No. 1, 135-150 (1994; Zbl 0812.47062)]and A. Capietto, F. Zanolin and the author [J. Differ. Equations 88, No. 2, 347-395 (1990; Zbl 0718.34053)], M. Henrard [Differ. Equations Dyn. Syst. 3, No. 1, 43-60 (1995; Zbl 0874.34056)], R. Precup [this journal 5, No. 2, 385-396 (1995; Zbl 0847.34028)], A. Capietto, M. Henrard, F. Zanolin and the author [this journal 3, No. 1, 81-100 (1994; Zbl 0808.34028) and 6, No. 1, 175-188 (1995; Zbl 0849.34018)], and others. The abstract results are illustrated by various applications to ordinary and functional differential equations.

##### MSC:
 47J05 Equations involving nonlinear operators (general) 34B15 Nonlinear boundary value problems for ODE 34C25 Periodic solutions of ODE