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April  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

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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
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show all references

References:
[1]

J. Anal. Math., 76 (1998), 321-335. doi: 10.1007/BF02786940.  Google Scholar

[2]

J. Fixed Point Theory App., 8 (2010), 1-74. doi: 10.1007/s11784-010-0033-9.  Google Scholar

[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.  Google Scholar

[4]

Kluwer Academic Publishers, 2004.  Google Scholar

[5]

J. Fixed P. Th. and Appl., 7 (2010), 145-160. doi: 10.1007/s11784-010-0013-0.  Google Scholar

[6]

Lect. Notes in Math., 1560, Springer-Verlag, Berlin, 1993.  Google Scholar

[7]

Math. Z., 127 (1972), 105-126.  Google Scholar

[8]

Academic Press, Inc. LTD, 1972.  Google Scholar

[9]

Nonl. Anal. TMA, 12 (1988), 51-61. doi: 10.1016/0362-546X(88)90012-0.  Google Scholar

[10]

CBMS Regional Conference Series in Mathematics, 38, AMS, Providence, R. I., 1978.  Google Scholar

[11]

Ann. Inst. H.Poincaré, Ana. Non Lin., 2, (1985), 329-370.  Google Scholar

[12]

Fund. Math., 185 (2005), 1-18. doi: 10.4064/fm185-1-1.  Google Scholar

[13]

Springer-Verlag, Berlin, 1979.  Google Scholar

[14]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[15]

Springer-Verlag, Berlin Heidelberg New York, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[16]

Nonl. Anal. TMA, 36 (1999), 101-118. doi: 10.1016/S0362-546X(98)00017-0.  Google Scholar

[17]

Erg. Th. and Dynam. Sys., 7 (1987), 93-103. doi: 10.1017/S0143385700003825.  Google Scholar

[18]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDLE 27, Birkhäuser, (1997), 247-272.  Google Scholar

[19]

Studia Math., 134, (1999), 217-233.  Google Scholar

[20]

Nonl. Anal TMA, 74 (2011), 1823-1834. doi: 10.1016/j.na.2010.10.055.  Google Scholar

[21]

Mem. AMS, 174, 1976.  Google Scholar

[22]

Topological Nonlinear Analysis, Degree, Singularity and Variations, PNDE, 15, Birkhäuser, (1995), 341-463.  Google Scholar

[23]

J. Diff. Equat., 170 (2001), 22-50. doi: 10.1006/jdeq.2000.3818.  Google Scholar

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Nonl. Anal. TMA, 51 (2002), 33-66. doi: 10.1016/S0362-546X(01)00811-2.  Google Scholar

[25]

Nonl. Anal. TMA, 73 (2010), 2779-2791. doi: 10.1016/j.na.2010.06.001.  Google Scholar

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Conf. Sem. Mat. Univ. Bari, 132 (1973).  Google Scholar

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Man. Math., 63 (1989), 99-114. doi: 10.1007/BF01173705.  Google Scholar

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Handbook of Dynamical Systems, 2, North-Holland, Amsterdam, (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.  Google Scholar

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Courant Institute of Mathematical Sciences, New York University, 1974.  Google Scholar

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[31]

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Nonl. Anal. TMA, 68 (2008), 1479-1516. doi: 10.1016/j.na.2006.12.039.  Google Scholar

[35]

Trans. Amer. Math. Soc., 269 (1982), 351-382. doi: 10.2307/1998452.  Google Scholar

[36]

Nonl. Anal. TMA, 23 (1994), 83-102. doi: 10.1016/0362-546X(94)90253-4.  Google Scholar

[37]

Topol. Meth. Nonl. Anal., 9 (1997), 383-417.  Google Scholar

[38]

Milan J. Math., 73 (2005), 103-144. doi: 10.1007/s00032-005-0040-2.  Google Scholar

[39]

Adv. Nonl. Stud., 11 (2011), 929-940.  Google Scholar

[40]

TAMS, 291 (1985), 1-41. doi: 10.2307/1999893.  Google Scholar

[41]

Bull. London Math. Soc., 22 (1990), 113-140. doi: 10.1112/blms/22.2.113.  Google Scholar

[42]

Invent. Math., 100 (1990), 63-95. doi: 10.1007/BF01231181.  Google Scholar

[43]

Fundamental Principles of Mathematical Science, 258, Springer-Verlag, New York-Berlin, 1983.  Google Scholar

[44]

Math. Z., 125 (1972), 359-364.  Google Scholar

[45]

CMBS Regional Conf. Ser. in Math., 5, AMS, Providence, R.I., 1970.  Google Scholar

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