November  2019, 12(7): 1807-1833. doi: 10.3934/dcdss.2019119

The work of Norman Dancer

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Prof. Norman Dancer's help during the preparation of this article is gratefully acknowledged. The author also thanks the editors for advices leading to an improved version of this article

Received  June 2018 Revised  July 2018 Published  December 2018

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions.

The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work. 

Citation: Yihong Du. The work of Norman Dancer. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1807-1833. doi: 10.3934/dcdss.2019119
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114. Google Scholar

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl.(4), 93 (1972), 231-246. doi: 10.1007/BF02412022. Google Scholar

[3]

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[5]

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[7]

B. BuffoniE. N. Dancer and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal., 152 (2000), 207-240. doi: 10.1007/s002050000086. Google Scholar

[8]

B. BuffoniE. N. Dancer and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal., 152 (2000), 241-271. doi: 10.1007/s002050000087. Google Scholar

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E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. Google Scholar

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fucik Spectrum for the p-Laplacian, J. Differ. Equ., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645. Google Scholar

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E. N. Dancer, Bifurcation theory in real Banach space, Proc. London Math. Soc., (3) 23 (1971), 699-734. doi: 10.1112/plms/s3-23.4.699. Google Scholar

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E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., (3) 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747. Google Scholar

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E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1977), 283-300. doi: 10.1017/S0308210500019648. Google Scholar

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E. N. Dancer, On the existence of solutions of certain asymptotically homogeneous problems, Math. Z., 177 (1981), 33-48. doi: 10.1007/BF01214337. Google Scholar

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E. N. Dancer, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal., 6 (1982), 667-686. doi: 10.1016/0362-546X(82)90037-2. Google Scholar

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E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

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E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4. Google Scholar

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E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math., 371 (1986), 46-66. doi: 10.1515/crll.1986.371.46. Google Scholar

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show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709. doi: 10.1137/1018114. Google Scholar

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl.(4), 93 (1972), 231-246. doi: 10.1007/BF02412022. Google Scholar

[3]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $\mathbb R^3$ and a conjecture of De Giorgi, J. Am. Math. Soc., 13 (2000), 725-739. doi: 10.1090/S0894-0347-00-00345-3. Google Scholar

[4]

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294. doi: 10.1002/cpa.3160410302. Google Scholar

[5]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Springfield, MO, 2006. Google Scholar

[6]

K. Borsuk, Drei Sätze über die $n$-dimensionale Sphäre, Fund. Math., 20 (1933), 236-243.Google Scholar

[7]

B. BuffoniE. N. Dancer and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal., 152 (2000), 207-240. doi: 10.1007/s002050000086. Google Scholar

[8]

B. BuffoniE. N. Dancer and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal., 152 (2000), 241-271. doi: 10.1007/s002050000087. Google Scholar

[9]

D. Clark, A variant of the Lyusternik-Schnirelmann theory, Indiana Univ. Math. J., 22 (1973), 65-74. doi: 10.1512/iumj.1973.22.22008. Google Scholar

[10]

C. Conley, Isolated Invariant Sets and The Morse Index, CBMS regional conference series in mathematics No. 38., Providence: Amer. Math. Soc., 1978. Google Scholar

[11]

M. ContiS. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560. doi: 10.1016/j.aim.2004.08.006. Google Scholar

[12]

E. C. M. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36. Google Scholar

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fucik Spectrum for the p-Laplacian, J. Differ. Equ., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645. Google Scholar

[14]

E. N. Dancer, Bifurcation theory in real Banach space, Proc. London Math. Soc., (3) 23 (1971), 699-734. doi: 10.1112/plms/s3-23.4.699. Google Scholar

[15]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., (3) 26 (1973), 359-384. doi: 10.1112/plms/s3-26.2.359. Google Scholar

[16]

E. N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. London Math. Soc., (3) 27 (1973), 747-765. doi: 10.1112/plms/s3-27.4.747. Google Scholar

[17]

E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 76 (1977), 283-300. doi: 10.1017/S0308210500019648. Google Scholar

[18]

E. N. Dancer, On the existence of solutions of certain asymptotically homogeneous problems, Math. Z., 177 (1981), 33-48. doi: 10.1007/BF01214337. Google Scholar

[19]

E. N. Dancer, Symmetries, degree, homotopy indices and asymptotically homogeneous problems, Nonlinear Anal., 6 (1982), 667-686. doi: 10.1016/0362-546X(82)90037-2. Google Scholar

[20]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. Google Scholar

[21]

E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743. doi: 10.1090/S0002-9947-1984-0743741-4. Google Scholar

[22]

E. N. Dancer, On positive solutions of some pairs of differential equations, II, J. Differential Equations, 60 (1985), 236-258. doi: 10.1016/0022-0396(85)90115-9. Google Scholar

[23]

E. N. Dancer, A new degree for $S^1$-invariant gradient mappings and applications, Ann. Inst. H. Poincare Anal. Non Linaire, 2 (1985), 329-370. doi: 10.1016/S0294-1449(16)30396-1. Google Scholar

[24]

E. N. Dancer, Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann., 272 (1985), 421-440. doi: 10.1007/BF01455568. Google Scholar

[25]

E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math., 371 (1986), 46-66. doi: 10.1515/crll.1986.371.46. Google Scholar

[26]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations, 74 (1988), 120-156. doi: 10.1016/0022-0396(88)90021-6. Google Scholar

[27]

E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285 (1989), 647-669. doi: 10.1007/BF01452052. Google Scholar

[28]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, Ⅱ, J. Differential Equations, 87 (1990), 316-339. doi: 10.1016/0022-0396(90)90005-A. Google Scholar

[29]

E. N. Dancer and J. F. Toland, Degree theory for orbits of prescribed period of flows with the first integral, Proc. London Math. Soc., 60 (1990), 549-580. doi: 10.1112/plms/s3-60.3.549. Google Scholar

[30]

E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math., 419 (1991), 125-139. Google Scholar

[31]

E. N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. Methods Nonlinear Anal., 1 (1993), 139-150. doi: 10.12775/TMNA.1993.011. Google Scholar

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