×

zbMATH — the first resource for mathematics

Deformations of functionals and bifurcations of extremals. (English. Russian original) Zbl 1136.47041
Math. Notes 81, No. 1, 61-71 (2007); translation from Mat. Zametki 81, No. 1, 70-82 (2006).
The present paper is devoted to the bifurcation-point problem for multivalued operators of monotonic type. The paper contains an analysis of two illustrative examples, which are of interest on their own.
The authors summary is: “We study homology characteristics of critical values and extremals of Lipschitz functionals defined on bounded closed convex subsets of a reflexive space that are invariant under deformations. Sufficient conditions for the existence of a bifurcation point of a multivalued potential operator (the switch principle for the typical number of an extremal) are established.”

MSC:
47J15 Abstract bifurcation theory involving nonlinear operators
47H11 Degree theory for nonlinear operators
49J40 Variational inequalities
47H50 Potential operators (MSC2000)
PDF BibTeX Cite
Full Text: DOI
References:
[1] M. A. Krasnoselskii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis (Nauka, Moscow, 1975) [in Russian].
[2] S. I. Pokhozhaev, ”The solvability of nonlinear equations with odd operators,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.] 1(3), 66–73 (1967). · Zbl 0165.49502
[3] F. E. Browder, ”Nonlinear elliptic boundary value problems and the generalized topological degree,” Bull. Amer. Math. Soc. 76(5), 999–1005 (1970). · Zbl 0201.18401
[4] I. V. Skrypnik, Nonlinear Higher-Order Elliptic Equations (Naukova dumka, Kiev, 1973) [in Russian].
[5] N. A. Bobylev, S. V. Emelyanov, and S. K. Korovin, Geometric Methods in Variational Problems (Magistr, Moscow, 1998) [in Russian].
[6] V. S. Klimov, ”On topological characteristics of nonsmooth functionals,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.] 62(5), 969–984 (1998). · Zbl 0924.47036
[7] V. S. Klimov, ”Type numbers of critical points of nonsmooth functionals,” Mat. Zametki [Math. Notes] 72(5), 693–705 (2002).
[8] V. S. Klimov and N. V. Senchakova, ”On the relative rotation of multivalued potential vector fields,” Mat. Sb. [Math. USSR-Sb.] 182(10), 131–144 (1991). · Zbl 0754.47042
[9] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis, and V. V. Obukhovskii, ”Topological methods in the theory of fixed points of multivalued mappings,” Uspekhi Mat. Nauk [Russian Math. Surveys] 35(1), 59–126 (1980).
[10] A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovskii, ”Ljusternik’s theorem and the theory of the extremum,” Uspekhi Mat. Nauk [Russian Math. Surveys] 35(6), 11–46 (1980). · Zbl 0464.49017
[11] F. Clarke, Optimization and Nonsmooth Analysis (John Wiley, Inc., New York, 1983). · Zbl 0582.49001
[12] A. Dold, Lectures on Algebraic Topology (Springer, Berlin, 1995). · Zbl 0872.55001
[13] W. Massey, Homology and Cohomology Theory (Marcel Dekker, Inc., New York, 1978).
[14] M. M. Postnikov, Introduction to Morse Theory (Nauka, Moscow, 1971) [in Russian].
[15] P. I. Plotnikov, ”Nonuniqueness of solutions of a problem on solitary waves, and bifurcations of critical points of smooth functionals,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.] 55(2), 339–366 (1991).
[16] Yu. I. Sapronov, ”Finite-dimensional reductions in smooth extremal problems,” Russian Math. Surveys 51(1), 97–127 (1996). · Zbl 0888.58010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.