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The splitting lemmas for nonsmooth functionals on Hilbert spaces I
1. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China |
References:
[1] |
A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional,, Adv. Nonlinear Stud., 9 (2009), 597.
|
[2] |
T. Bartsch, Critical point theory on partially ordered hilbert spaces,, J. Funct. Anal., 186 (2001), 117.
doi: 10.1006/jfan.2001.3789. |
[3] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.
doi: 10.1007/s002090050492. |
[4] |
T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations,, in, (2008), 41.
doi: 10.1016/B978-044452833-9.50003-6. |
[5] |
P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Quebec, 22 (1998), 131.
|
[6] |
P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds,, Topol. Methods Nonlinear Anal., 16 (2000), 279.
|
[7] |
M. Berger, "Nonlinearity and Functional Analysis,", Acad. Press, (1977).
|
[8] |
H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465.
|
[9] |
F. E. Browder, Fixed point theory and nonlinear problem,, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1.
doi: 10.1090/S0273-0979-1983-15153-4. |
[10] |
K. C. Chang, "Infinite Dimensional Morse Theory and its Applications,", Univ. de Montreal, 97 (1985).
|
[11] |
K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem,", Birkhäuser, (1993).
|
[12] |
K. C. Chang, "Methods in Nonlinear Analysis,", Springer Monogaphs in Mathematics, (2005).
|
[13] |
K. C. Chang, $H^1$ versus $C^1$ isolated critical points,, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441.
|
[14] |
S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces,, Adv. Nonlinear Stud., 9 (2009), 679.
|
[15] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).
|
[16] |
J. B. Conway, "A Course in Functional Analysis,", Springer, (1990).
|
[17] |
J. N. Corvellec, Morse theory for continuous functionals,, J. Math. Anal. Appl., 196 (1995), 1050.
doi: 10.1006/jmaa.1995.1460. |
[18] |
J. Dieudonné, "Fondements de L'Analyse Moderne,", Gauthier-Villars, (1963).
|
[19] |
D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces,, Proc. Amer. Math. Soc., 135 (2007), 921.
doi: 10.1090/S0002-9939-06-08662-X. |
[20] |
D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals,, Topological Methods in Nonlinear Analysis, 29 (2007), 35.
|
[21] |
P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings,, J. Funct. Anal., 124 (1994), 1.
doi: 10.1006/jfan.1994.1096. |
[22] |
N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, A.I.H.P. Analyse Non linéaire, 6 (1989), 321.
|
[23] |
D. Gromoll and W. Meyer, On differentiable functions with isolated critical points,, Topology, 8 (1969), 361.
|
[24] |
H. Hofer, The topological degree at a critical point of mountain pass type,, in, 45 (1986), 501.
|
[25] |
D. Husemoller, "Fibre Bundle,", Springer-Verlag, (1975).
|
[26] |
A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions,, in, 204 (1997), 139.
doi: 10.1090/conm/204/02627. |
[27] |
M. Jiang, A generalization of Morse lemma and its applications,, Nonlinear Analysis, 36 (1999), 943.
doi: 10.1016/S0362-546X(97)00701-3. |
[28] |
E. Kreyszig, "Introduction Functional Analysis with Applications,", John wiley & Sons. Ins. 1978., (1978).
|
[29] |
S. Lang, "Differential Manifolds,", $2^{nd}$ edition, (1985).
doi: 10.1007/978-1-4684-0265-0. |
[30] |
C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems,, J. Funct. Anal., 221 (2005), 439.
doi: 10.1016/j.jfa.2004.09.010. |
[31] |
C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum,, J. Differential Equations, 244 (2008), 2498.
doi: 10.1016/j.jde.2008.02.021. |
[32] |
G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034],, J. Funct. Anal., 261 (2011), 542.
doi: 10.1016/j.jfa.2009.01.001. |
[33] |
G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, (). Google Scholar |
[34] |
G. Lu, Some critical point theorems and applications,, preprint, (). Google Scholar |
[35] |
G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, (). Google Scholar |
[36] |
Jean Mawhin and Michel Willem, On the generalized Morse Lemma,, Bull. Soc. Math., 37 (1985), 23.
|
[37] |
Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences 74, 74 (1989).
|
[38] |
A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability,, Abstract and Applied Analysis, 5 (2000), 113.
doi: 10.1155/S1085337500000245. |
[39] |
J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289.
doi: 10.1007/BF02786939. |
[40] |
K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators,", Mathematical Surveys and Monographs 161, 161 (2010).
|
[41] |
M. Schechter, "Principles of Functional Analysis,", Academic Press, (1971).
|
[42] |
W. Schirotzek, "Nonsmooth Analysis,", Springer, (2007).
doi: 10.1007/978-3-540-71333-3. |
[43] |
I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order,", [in Russian], (1973).
|
[44] |
I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems,", Teubner, (1986).
|
[45] |
A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem,, Math. Ann., 263 (1983), 303.
doi: 10.1007/BF01457133. |
[46] |
S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems,, J. Sov. Math., 67 (1993), 2713.
doi: 10.1007/BF01455151. |
show all references
References:
[1] |
A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional,, Adv. Nonlinear Stud., 9 (2009), 597.
|
[2] |
T. Bartsch, Critical point theory on partially ordered hilbert spaces,, J. Funct. Anal., 186 (2001), 117.
doi: 10.1006/jfan.2001.3789. |
[3] |
T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.
doi: 10.1007/s002090050492. |
[4] |
T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations,, in, (2008), 41.
doi: 10.1016/B978-044452833-9.50003-6. |
[5] |
P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Quebec, 22 (1998), 131.
|
[6] |
P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds,, Topol. Methods Nonlinear Anal., 16 (2000), 279.
|
[7] |
M. Berger, "Nonlinearity and Functional Analysis,", Acad. Press, (1977).
|
[8] |
H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465.
|
[9] |
F. E. Browder, Fixed point theory and nonlinear problem,, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1.
doi: 10.1090/S0273-0979-1983-15153-4. |
[10] |
K. C. Chang, "Infinite Dimensional Morse Theory and its Applications,", Univ. de Montreal, 97 (1985).
|
[11] |
K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem,", Birkhäuser, (1993).
|
[12] |
K. C. Chang, "Methods in Nonlinear Analysis,", Springer Monogaphs in Mathematics, (2005).
|
[13] |
K. C. Chang, $H^1$ versus $C^1$ isolated critical points,, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441.
|
[14] |
S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces,, Adv. Nonlinear Stud., 9 (2009), 679.
|
[15] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).
|
[16] |
J. B. Conway, "A Course in Functional Analysis,", Springer, (1990).
|
[17] |
J. N. Corvellec, Morse theory for continuous functionals,, J. Math. Anal. Appl., 196 (1995), 1050.
doi: 10.1006/jmaa.1995.1460. |
[18] |
J. Dieudonné, "Fondements de L'Analyse Moderne,", Gauthier-Villars, (1963).
|
[19] |
D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces,, Proc. Amer. Math. Soc., 135 (2007), 921.
doi: 10.1090/S0002-9939-06-08662-X. |
[20] |
D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals,, Topological Methods in Nonlinear Analysis, 29 (2007), 35.
|
[21] |
P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings,, J. Funct. Anal., 124 (1994), 1.
doi: 10.1006/jfan.1994.1096. |
[22] |
N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, A.I.H.P. Analyse Non linéaire, 6 (1989), 321.
|
[23] |
D. Gromoll and W. Meyer, On differentiable functions with isolated critical points,, Topology, 8 (1969), 361.
|
[24] |
H. Hofer, The topological degree at a critical point of mountain pass type,, in, 45 (1986), 501.
|
[25] |
D. Husemoller, "Fibre Bundle,", Springer-Verlag, (1975).
|
[26] |
A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions,, in, 204 (1997), 139.
doi: 10.1090/conm/204/02627. |
[27] |
M. Jiang, A generalization of Morse lemma and its applications,, Nonlinear Analysis, 36 (1999), 943.
doi: 10.1016/S0362-546X(97)00701-3. |
[28] |
E. Kreyszig, "Introduction Functional Analysis with Applications,", John wiley & Sons. Ins. 1978., (1978).
|
[29] |
S. Lang, "Differential Manifolds,", $2^{nd}$ edition, (1985).
doi: 10.1007/978-1-4684-0265-0. |
[30] |
C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems,, J. Funct. Anal., 221 (2005), 439.
doi: 10.1016/j.jfa.2004.09.010. |
[31] |
C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum,, J. Differential Equations, 244 (2008), 2498.
doi: 10.1016/j.jde.2008.02.021. |
[32] |
G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034],, J. Funct. Anal., 261 (2011), 542.
doi: 10.1016/j.jfa.2009.01.001. |
[33] |
G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, (). Google Scholar |
[34] |
G. Lu, Some critical point theorems and applications,, preprint, (). Google Scholar |
[35] |
G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, (). Google Scholar |
[36] |
Jean Mawhin and Michel Willem, On the generalized Morse Lemma,, Bull. Soc. Math., 37 (1985), 23.
|
[37] |
Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences 74, 74 (1989).
|
[38] |
A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability,, Abstract and Applied Analysis, 5 (2000), 113.
doi: 10.1155/S1085337500000245. |
[39] |
J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289.
doi: 10.1007/BF02786939. |
[40] |
K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators,", Mathematical Surveys and Monographs 161, 161 (2010).
|
[41] |
M. Schechter, "Principles of Functional Analysis,", Academic Press, (1971).
|
[42] |
W. Schirotzek, "Nonsmooth Analysis,", Springer, (2007).
doi: 10.1007/978-3-540-71333-3. |
[43] |
I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order,", [in Russian], (1973).
|
[44] |
I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems,", Teubner, (1986).
|
[45] |
A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem,, Math. Ann., 263 (1983), 303.
doi: 10.1007/BF01457133. |
[46] |
S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems,, J. Sov. Math., 67 (1993), 2713.
doi: 10.1007/BF01455151. |
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