Abstract
In this papier, a homotopy index (Conley index) which can be applied to non-autonomous differential equations is defined. It is proved that the index is well defined, and several theorems concerning its basic properties are established. The second part of this paper is concerned with the application of this index to (non-autonomous) ordinary differential equations as well as (non-autonomous) semilinear parabolic equations. Finally, several existence results for bounded solutions of asymptotically linear non-autonomous equations are proved. We also consider the existence of recurrent or Poisson stable solutions.
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Notes
It is tacitly assumed that \(X\subset \mathfrak {U}\) as well as \(\mathbb {R}\times X\subset \mathfrak {U}\).
If an evolution operator (or process) is defined as the “solution operator” of, e.g., a differential equation, a mapping u is a solution of the evolution operator if and only if it is a solution of the differential equation. From that point of view, this is a generalized notion of a solution. If the evolution operator is in fact a semiflow, this definition coincides with the one in [12]. In [12], entire solutions of a process are defined analogously.
However, often, the evolution operator itself is considered to be “the” solution. One could thus be tempted to replace the term “solution” by “motion”, but at least in [11] or [1], this notion is only used for (two-sided) flows. Another possible term would be “continuation”, which has a completely different meaning in this context. It should also noted that, at least concerning partial differential equations, it is not uncommon that the term solution requires disambiguation.
Defined for all \(t\in \mathbb {R}^+\)
There are several variants of Conley indices, and consequently the exact meaning of the term index pair varies. To disambiguate between different kinds of index pairs, Rybakowski introduced the notion FM-index pair in [2].
The semiflow is continuous.
i.e., the only compact invariant subset is the empty set.
By Zorn’s lemma, there is a positively invariant minimal subset, which is then shown to be invariant.
The index as constructed here depends on the behavior of the evolution operator for large initial times. The restriction to positive initial times is not an artificial one but reflects this property.
As weak as possible, as strong as necessary.
Perhaps, the term isolating superset would be more appropriate.
Otherwise, there is nothing to prove.
In contrast to Lemma 3.9, H is obviously well defined.
A rather strong assumption but presumably necessary.
\((y,x)\pi \mathbb {R}^+\subset N\cup \{\Diamond \}\) implies \((y,x)\pi \mathbb {R}^+\subset N\) for all \((y,x)\in N\).
By an abuse of notation, we write \(d(y,\Sigma (y_0)) := \inf _{{\tilde{y}}\in \Sigma (y_0)} d(y,{\tilde{y}})\).
Of our omnipresent semiflow \(\pi \).
Addition and scalar multiplication in Y are assumed to be continuous.
Here, it is used that \(y_0\) and thus r are independent of n.
The definition of an isolating neighborhood depends on a skew-product semiflow, which is usually fixed. Here, this skew-product semiflow \(\tilde{\pi }_{h,\lambda }\) is considered to be variable and thus mentioned explicitly.
The proofs here are original, but they probably do not contain new ideas.
Or strongly admissible, which is equivalent in this case.
As defined in [12, Section 3.6].
In contrast to the previous section, it is required that \(\delta \) can be chosen uniformly in t, which is a slightly stronger assumption.
The embedding is defined as usual: \(X^\alpha \ni \left[ x\right] \mapsto x\in C^1({\bar{\Omega }})\).
The situation in the literature is difficult. There are proofs of this claim which appear to be wrong. Lemma 37.8 in [12] is useful, but the details are given only for the Sobolev case (which is similar).
With respect to inclusion.
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Jänig, A. A non-autonomous Conley index. J. Fixed Point Theory Appl. 19, 1825–1870 (2017). https://doi.org/10.1007/s11784-016-0307-y
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DOI: https://doi.org/10.1007/s11784-016-0307-y
Keywords
- Non-autonomous Conley index
- Semilinear parabolic equations
- Recurrent solutions
- Poisson stable solutions