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Global comparison of finite-dimensional reduction schemes in smooth variational problems. (English. Russian original) Zbl 1036.49019
Math. Notes 67, No. 5, 631-638 (2000); translation from Mat. Zametki 67, No. 5, 745-754 (2000).
The usage of various finite-dimensional reduction schemes in problems of variational calculus makes it necessary to compare results obtained by different schemes and, in particular, to compare analytic and topological properties of the pair of functions corresponding to one another of the action functions in a specified pair of schemes. In variational boundary problems reduction schemes are related to the interpretation of the problem as a functional-operator equation $$f(x)=0$$ where $$f$$ is a potential operator of the zero index acting in a given Banach space. From this equation it is passed to the equivalent extremal problem about minimization of a smooth functional $$V(x)$$ which is potential of $$f$$ on a Banach space or a Banach manifold $$X$$. This variational problem can fairly often be reduced to a similar minimization problem in a finite-dimensional space $$\mathbb{R}^n$$
In this paper a criterion for global smooth equivalence of a pair of key functions corresponding to a smooth functional in the calculus of variations for a given pair of finite-dimensional reduction schemes is established. Here the main condition is the possibility to deform the reduction schemes into each other preserving the coactivity of the key functions. The theorem proved concerns global smooth equivalence of the key functions calculated by the Lyapunov-Schmidt and Morse-Bott reduction schemes in the two-point boundary value problem for a mechanical system.

##### MSC:
 49J40 Variational inequalities 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 49J53 Set-valued and variational analysis 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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##### References:
 [1] M. A. Krasnosel’skii, N. A. Bobylev, and É. M. Mukhamadiev, ”A scheme for the study of degenerate extremals of functionals in the classical calculus of variations,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],240, No. 3, 530–533 (1978). [2] C. C. Conley and E. Zehnder, ”The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol’d,”Invent. Math.,73, 33–49 (1983). · Zbl 0516.58017 · doi:10.1007/BF01393824 [3] S. V. Bolotin, ”Periodic solutions of systems with gyroscopic forces,”Prikl. Mat. Mekh. [J. Appl. Math. Mekh.],51, No. 4, 686–687 (1987). · Zbl 0672.70007 [4] Yu. I. Sapronov, ”Finite-dimensional reductions in smooth extremal problems,”Uspekhi Mat. Nauk [Russian Math. Surveys],51, No. 1, 101–132 (1996). · Zbl 0888.58010 [5] S. L. Tsarev, ”Comparison of key functions for smooth functionals,” in:Stochastic and Global Analysis, Abstracts of Int. Conf, VSU, Voronezh (1997), p. 68. · Zbl 0920.68061 [6] Yu. I. Sapronov, ”Nonlocal finite-dimensional reductions in variational boundary value problems,”Mat. Zametki [Math. Notes],49, No. 1, 94–103 (1991). · Zbl 0724.58025 [7] Yu. I. Sapronov, ”Existence and comparison of finite-dimensional reductions for smooth functionals,” in:Global and Stochastic Analysis [in Russian], Izd. VGU, Voronezh (1995), pp. 69–90. · Zbl 0908.65044 [8] H. Poincaré,Selected Works in Three Volumes. Volume II: New Methods in Celestial Mechanics. Topology. Number Theory [Russian translation], Nauka, Moscow (1972). [9] V. I. Arnol’d, A. N. Varchenko, and C. M. Gusein-Zade,Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts [in Russian], Nauka, Moscow (1982); English transl.: Birkhauser, Boston (1985). [10] T. Poston and I. Stewart,Catastrophe Theory and Its Applications, Pitman, London (1978). · Zbl 0382.58006 [11] E. P. Popov,Nonlinear Problems of the Statics of Thin Rods [in Russian], OGIZ, Moscow (1948). [12] O. N. Levchenko and Yu. I. Sapronov, ”Morse-Bott reduction for a symmetric Kirchhoff rood,” in:Methods and Applications of Global Analysis, Voronezh Univ. Press (1993), pp. 95–100. · Zbl 0864.73030 [13] Yu. G. Borisovich, V. G. Zvyagin, and Yu. I. Sapronov, ”Nonlinear Fredholm maps and Leray-Schauder theory,”Uspekhi Mat. Nauk [Russian Math. Surveys],32, No. 4, 3–54 (1977). · Zbl 0383.58002
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