Equivariant Conley index versus degree for equivariant gradient maps

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August 2013, 6(4): 985-997. doi: 10.3934/dcdss.2013.6.985

Equivariant Conley index versus degree for equivariant gradient maps

Anna Go??biewska ^1,2,3, and S?awomir Rybicki ^1,2,3,

1.	Faculty of Mathematics and Computer Science
2.	Nicolaus Copernicus University
3.	ul. Chopina 12/18, PL-87-100 Toru?

Received October 2011 Revised April 2012 Published December 2012

Abstract
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In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.

Keywords: Equivariant Conley index, equivariant gradient degree, compact Lie group, strongly indefinite operator..

Mathematics Subject Classification: Primary: 37G40; Secondary: 58E0.

Citation: Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985

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[35]	Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.2307/1998452.
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[45]	CMBS Regional Conf. Ser. in Math., 5, AMS, Providence, R.I., 1970.

show all references

References:

[1]	J. Anal. Math., 76 (1998), 321-335. doi: 10.1007/BF02786940.
[2]	J. Fixed Point Theory App., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.
[3]	Israel Math. Conf. Proc., Conf. Nonlinear Analysis and Optimization, Haifa, Israel, (2008), AMS Contemporary Mathematics, 514 (2010), 45-84. doi: 10.1090/conm/514/10099.
[4]	Kluwer Academic Publishers, 2004.
[5]	J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.
[6]	Lect. Notes in Math., 1560, Springer-Verlag, Berlin, 1993.
[7]	Math. Z., 127 (1972), 105-126.
[8]	Academic Press, Inc. LTD, 1972.
[9]	Nonl. Anal. TMA, 12 (1988), 51-61. doi: 10.1016/0362-546X(88)90012-0.
[10]	CBMS Regional Conference Series in Mathematics, 38, AMS, Providence, R. I., 1978.
[11]	Ann. Inst. H.Poincaré, Ana. Non Lin., 2, (1985), 329-370.
[12]	Fund. Math., 185 (2005), 1-18. doi: 10.4064/fm185-1-1.
[13]	Springer-Verlag, Berlin, 1979.
[14]	Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.
[15]	Springer-Verlag, Berlin Heidelberg New York, 2000. doi: 10.1007/978-3-642-56936-4.
[16]	Nonl. Anal. TMA, 36 (1999), 101-118. doi: 10.1016/S0362-546X(98)00017-0.
[17]	Erg. Th. and Dynam. Sys., 7 (1987), 93-103. doi: 10.1017/S0143385700003825.
[18]	Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDLE 27, Birkhäuser, (1997), 247-272.
[19]	Studia Math., 134, (1999), 217-233.
[20]	Nonl. Anal TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.
[21]	Mem. AMS, 174, 1976.
[22]	Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDE, 15, Birkhäuser, (1995), 341-463.
[23]	J. Diff. Equat., 170 (2001), 22-50. doi: 10.1006/jdeq.2000.3818.
[24]	Nonl. Anal. TMA, 51 (2002), 33-66. doi: 10.1016/S0362-546X(01)00811-2.
[25]	Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.
[26]	Conf. Sem. Mat. Univ. Bari, 132 (1973).
[27]	Man. Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.
[28]	Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.
[29]	Courant Institute of Mathematical Sciences, New York University, 1974.
[30]	Comm. Pure Appl. Math., 23 (1970), 939-961.
[31]	J. Func. Anal., 7 (1971), 487-513.
[32]	Contributions to Nonlinear Functional Analysis, E Academic Press, New York, (1971), 11-36.
[33]	J. Diff. Equat., 202 (2004), 284-305. doi: 10.1016/j.jde.2004.03.037.
[34]	Nonl. Anal. TMA, 68 (2008), 1479-1516. doi: 10.1016/j.na.2006.12.039.
[35]	Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.2307/1998452.
[36]	Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.
[37]	Topol. Meth. Nonl. Anal., 9 (1997), 383-417.
[38]	Milan J. Math., 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.
[39]	Adv. Nonl. Stud., 11 (2011), 929-940.
[40]	TAMS, 291 (1985), 1-41. doi: 10.2307/1999893.
[41]	Bull. London Math. Soc., 22 (1990), 113-140. doi: 10.1112/blms/22.2.113.
[42]	Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.
[43]	Fundamental Principles of Mathematical Science, 258, Springer-Verlag, New York-Berlin, 1983.
[44]	Math. Z., 125 (1972), 359-364.
[45]	CMBS Regional Conf. Ser. in Math., 5, AMS, Providence, R.I., 1970.

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