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Connecting orbits for nonlinear differential equations at resonance. (English) Zbl 1302.34098
The author examinates the problem of the existence of orbits connecting stationary points of the following equation \[ u'(t)=-Au(t)+\lambda u(t)+F(u(t))\quad\text{for }\;t>0, \] where \(\lambda\) is an eigenvalue of a sectorial operator \(A: X\supset D(A) \to X\) defined on a real Banach space \(X\) and \(F: X^{\alpha} \to X\) is a continuous mapping. \(X^{\alpha}\) for \(\alpha \in (0,1)\) denotes here a fractional space defined by the equality \(X^{\alpha}=D((A+\delta I)^{\alpha})\), where \(\delta >0\) is chosen in such a way that \(A+\delta I\) is a positive definite operator. It is assumed that the above equation is at resonance at infinity, that is ker\((\lambda I-A)\neq \{0\}\) and \(F\) is a bounded mapping.
For that purpose, imposing special geometrical conditions on the nonlinearity, the author proves the index formula for bounded orbits which is the tool to determine the Conley index for the maximal invariant set contained in a sufficiently large ball.
It is established that those geometrical conditions encompass the well-known Landesman- Lazer conditions as well as the strong resonance ones.
As an example the author considers the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37B30 Index theory for dynamical systems, Morse-Conley indices
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