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Oriented degree of Fredholm maps: finite-dimensional reduction method. (English. Russian original) Zbl 1384.47007

J. Math. Sci., New York 204, No. 5, 543-714 (2015); translation from Sovrem. Mat., Fundam. Napravl. 44, 3-171 (2012).
The paper contains a complete and detailed description of the oriented degree of zero-index Fredholm maps and their compact single-valued perturbations. The authors carefully introduce all needed notions and facts (in particular, a brief theory of linear Fredholm maps, Fredholm structures on manifolds, the concept of their orientation; also, in three appendices: notions of general and differential topology and of function spaces), all steps of the construction with detailed considerations concerning properties of the degree in each step (the finite-dimensional method is used), and examples of applications in linear and nonlinear (elliptic) differential equations.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47H11 Degree theory for nonlinear operators
58B15 Fredholm structures on infinite-dimensional manifolds
58J20 Index theory and related fixed-point theorems on manifolds
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
58C30 Fixed-point theorems on manifolds
35G15 Boundary value problems for linear higher-order PDEs
35J60 Nonlinear elliptic equations
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[1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions. I [Russian translation], Inostrannaya Literatura, Moscow (1962). · Zbl 0104.32305
[2] P. S. Aleksandrov, Introduction to Set Theory and General Topology [in Russian], Nauka, Moscow (1977).
[3] P. A. Aleksandrjan and È. A. Mirzahanjan, General Topology [in Russian], Vyssh. Shkola, Moscow (1979).
[4] D. Arlt, “Zusammenziehbarkeit der allgemeinen linearen Gruppe des Raumes <Emphasis Type=”Italic“>c0 der Nullfolgen,” Invent. Math., 1, 36-44 (1966). · Zbl 0143.36403 · doi:10.1007/BF01389697
[5] D. V. Beklemishev, Supplementary Chapters in Linear Algebra [in Russian], Nauka, Moscow (1983). · Zbl 0532.15002
[6] P. Benevieri and M. Furi, “A simple notion of orientablity for Fredholm maps of index zero between Banach manifolds and degree theory,” Ann. Sci. Math. Quebec, 22, No. 2, 131-148 (1998). · Zbl 1058.58502
[7] P. Benevieri and M. Furi, “On the concept of orientability for Fredholm maps betweeen real Banach manifolds,” Topol. Methods Nonlinear Anal., 16, 279-306 (2000). · Zbl 1007.47026
[8] P. Benevieri and M. Furi, “A degree theory for locally compact perturbations of Fredholm maps in Banach spaces,” Abstr. Appl. Anal., Art ID 64764 (2006). · Zbl 1090.47049
[9] E. Berkson, “Some metrics on the subspaces of a Banach space,” Pacific J. Math., 13, 7-22 (1963). · Zbl 0118.10402 · doi:10.2140/pjm.1963.13.7
[10] R. Bonic and J. Frampton, “Smooth functions on Banach manifolds,” J. Math. Mech., 15, No. 3, 877-897 (1966). · Zbl 0143.35202
[11] Yu. Borisovich, N. Bliznyakov, Ya. Izrailevich, and T. N. Fomenko, Introduction to Topology [in Russian], Vyssh. Shkola, Moscow (1980). · Zbl 0478.57001
[12] Yu.G. Borisovich, V.G. Zvyagin, and Yu. I. Sapronov, “Nonlinear Fredholm mappings, and Leray-Schauder theory,” Uspekhi Mat. Nauk, 32, No.4(196), 3-54 (1977). · Zbl 0383.58002
[13] R. Caccioppoli, “Sulle corrispondenze funzionali inverse diramate: teoria generale e applicazioni ad alcune equazioni non lineari e al problema di Plateau. I, II” Rend. Accad. Lincei., 24, 258-263, 416-421 (1936). · Zbl 0016.36103
[14] H. Cartan, Differential Calculus. Differential Forms [Russian translation], Mir, Moscow (1971).
[15] A. V. Chernavsky and S.V. Matveev, Foundations of the Topology of Manifolds [in Russian], Kuban. Gos. Univ., Krasnodar (1974).
[16] J. Dieudonné, Foundations of Modern Analysis [Russian translation], Mir, Moscow (1964). · Zbl 0122.29702
[17] V.T. Dmitrienko and V.G. Zvyagin, “Homotopy classification of a class of continuous mappings,” Mat. Zametki, 31, No. 5, 801-812 (1982). · Zbl 0493.47028
[18] A. Douady, “Un espace de Banach dont le groupe linéaire n’est pas connexe,” Indag. Math., 68, 787-789 (1965). · Zbl 0178.26403
[19] B. A. Dubrovin, S.P. Novikov, and A.T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979).
[20] R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
[21] J. Eells and K.D. Elworthy, “Open embeddings of certain Banach manifolds,” Ann. of Math., 91, No. 3, 465-485 (1970). · Zbl 0198.28804 · doi:10.2307/1970634
[22] K. D. Elworthy and A. J. Tromba, “Differential structures and Fredholm maps on Banach manifolds,” Proc. Sympos. Pure Math. (Global Analysis), 15, 45-94 (1970). · doi:10.1090/pspum/015/0264708
[23] K. D. Elworthy and A. J. Tromba, “Degree theory on Banach manifolds,” Proc. Sympos. Pure Math., 18, 86-94 (1970). · doi:10.1090/pspum/018.1/0277009
[24] L. C. Evans, Partial differential equations, American Mathematical Society, Providence, RI (2011).
[25] D.K. Faddeev, B. Z. Vulikh, and N.N. Ural’tseva, Selected Chapters of Analysis and Higher Algebra. Textbook [in Russian], Izdat. Leningrad. Gos. Univ., Leningrad (1981). · Zbl 0518.00001
[26] G. Fichtengoltz and L. Kantorovitch, “Sur les opérations dans l’espace des fonctions bornées,” Studia Math., 5, 69-98 (1934). · JFM 60.1074.05
[27] P. M. Fitzpatric, J. Pejsachovicz, and P. J. Rabier, “The degree of proper <Emphasis Type=”Italic“>C2-Fredholm mappings. I,” J. Reine Angew. Math., 427, 1-33 (1992).
[28] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin (1974). · Zbl 0289.47029
[29] K. Gęba, “On the Homotopy Groups of <Emphasis Type=”Italic“>GLc(<Emphasis Type=”Italic“>E)<Emphasis Type=”Italic“>,” Bull. de L’Académie Polonaise des Sciences. Série des Sciences Math., Astr. et Phys., 16, No. 9, 699-702 (1968). · Zbl 0164.43901
[30] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (2001). · Zbl 1042.35002
[31] A.Ya. Helemskii, Lectures and Exercises on Functional Analysis, American Mathematical Society, Providence, RI (2006). · Zbl 1123.46001
[32] M. Hirsch, Differential Topology [Russian translation], Mir, Moscow (1979).
[33] C. A. Isnard, “The topological degree on Banach manifolds,” Global Analysis and its Applications (Internat. Summer Course, Trieste; IAEA, Vienna), 2 291-313 (1972).
[34] M. I. Kadec and B. S.Mityagin, “Complemented subspaces in Banach spaces,” Uspekhi Mat. Nauk, 28, No. 6(174), 77-94 (1973). · Zbl 0287.46029
[35] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977). · Zbl 0555.46001
[36] J. L. Kelley, General Topology [Russian translation], Nauka, Moscow (1968).
[37] A. N. Kolmogorov and S.V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972). · Zbl 0235.46001
[38] M. A. Krasnosel’skiĭ and P. P. Zabreĭko, Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).
[39] S. G. Kreĭn, Linear Equations in a Banach Space [in Russian], Nauka, Moscow (1971).
[40] N. H. Kuiper, “The homotopy type of the unitary group of Hilbert space,” Topology, 3, 19-30 (1965). · Zbl 0129.38901 · doi:10.1016/0040-9383(65)90067-4
[41] S. Lang, Introduction to Differentiable Manifolds [Russian translation], Platon, Volgograd (1996).
[42] E.B. Leach and J.H.M. Whitfield, “Differentiable functions and rough norms on Banach spaces,” Proc. Amer. Math. Soc., 33, No. 1, 120-126 (1972). · Zbl 0236.46051 · doi:10.1090/S0002-9939-1972-0293394-4
[43] J. Leray and J. Schauder, “Topologie et équations fonctionnelles,” Ann. Sci. Ecole Norm. Sup., Ser.3, 51, 45-78 (1934). · Zbl 0009.07301
[44] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, 1. Sequence Spaces, Springer-Verlag, Berlin (1977). · Zbl 0362.46013
[45] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. 1 [Russian translation], Mir, Moscow (1971). · Zbl 0212.43801
[46] J. Mawhin, “A Leray-Schauder degree: a half century of extension and applications,” Topol. Meth. Nonlin. Anal., 14, 195-228 (1999). · Zbl 0957.47045
[47] V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad. Univ., Leningrad (1985).
[48] J. Milnor and A. Wallace, Differential Topology: a First Course [Russian translation], Mir, Moscow (1972).
[49] C. Miranda, Partial Differential Equations of Elliptic Type [Russian translation], Inostrannaya Literatura, Moscow (1957).
[50] J.R. Munkres, “Elementary differential topology,” Characteristic Classes by J. W. Milnor and J. Stasheff [Russian translation], Mir, Moscow (1979), 270-359.
[51] G. Neubauer, On a Class of Sequence Spaces with Contractible Linear Group, Notes. University of California, Berkeley (1957).
[52] S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).
[53] L. Nirenberg, Lectures on Nonlinear Functional Analysis [Russian translation], Mir, Moscow (1977).
[54] R. Palais et. al., Seminar on the Atiyah-Singer Index Theorem [Russian translation], Mir, Moscow (1970). · Zbl 0202.23103
[55] J. Peetre, “Another approach to elliptic boundary problems,” Comm. Pure Appl. Math., 14, 711-731 (1961). · Zbl 0104.07303 · doi:10.1002/cpa.3160140404
[56] J. Pejsachowicz and P. J. Rabier, “Degree theory for <Emphasis Type=”Italic“>C1 Fredholm mappings of index 0<Emphasis Type=”Italic“>,” J. Anal. Math., 76, 289-319 (1998). · Zbl 0932.47046 · doi:10.1007/BF02786939
[57] P. Rabier and M. F. Salter, “A degree theory for compact perturbations of proper <Emphasis Type=”Italic“>C1 Fredholm mappings of index 0<Emphasis Type=”Italic“>,” Abstr. Appl. Anal., No. 7, 707-731 (2005). · Zbl 1117.47049 · doi:10.1155/AAA.2005.707
[58] P. H. Rabinowitz, “A global theorem for nonlinear eigenvalue problems and applications,” Contrib. Nonlin. Funct. Anal., Academic Press, N. Y., 11-36 (1971). · Zbl 0774.47035
[59] N.M. Ratiner, “On the theory of the degree of Fredholm mappings of a manifold,” Equations on Manifolds [in Russian], Voronezh. Gos. Univ., Voronezh, 126-129 (1982). · Zbl 0118.10402
[60] N. M. Ratiner, On solvability of nonlinear boundary-value problems for systems of second-order ordinary differential equations [in Russian], Preprint, Voronezh (1985). · Zbl 0143.36403
[61] N.M. Ratiner, “On the application of degree theory to the study of an oblique derivative problem,” Russian Math. (Iz. VUZ), 45, No. 4, 41-50 (2001).
[62] K. Rektoris, Variational Methods in Mathematical Physics and Engineering [Russian translation], Mir, Moscow (1985).
[63] G. Restrepo, “Differentiable norms in Banach spaces,” Bull. Amer. Math. Soc., 70, 413-414 (1964). · Zbl 0173.41304 · doi:10.1090/S0002-9904-1964-11121-6
[64] A. Sard, “The measure of the critical values of differentiable maps,” Bull. Amer. Math. Soc., 48, 883-890 (1942). · Zbl 0063.06720 · doi:10.1090/S0002-9904-1942-07811-6
[65] Yu. I. Sapronov, “The local invertibility of nonlinear Fredholm mappings,” Funktsional. Anal. i Prilozhen., 5, No. 4, 38-43 (1971).
[66] Yu. I. Sapronov, “Regular perturbations of a Fredholm mapping and the theorem on the odd field,” Voronež. Gos. Univ. Trudy Mat. Fak., No. 10, 82-88 (1973). · Zbl 0143.35301
[67] Yu. I. Sapronov, “On the theory of the degree of nonlinear Fredholm mappings,” Voronež. Gos. Univ. Trudy Naučn.-Issled. Inst. Mat. VGU, 11, 92-101 (1973).
[68] L. Schwartz, Analysis. Vol. 1 [Russian translation], Mir, Moscow (1972).
[69] M.A. Shubin, Lectures on Equations of Mathematical Physics [in Russian], MCCME, Moscow (2003).
[70] S. Smale, “An infinite dimensional version of Sard’s theorem,” Amer. J. Math., 87, 861-867 (1965). · Zbl 0143.35301 · doi:10.2307/2373250
[71] S.L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Izdat. Leningrad. Gos. Univ., Leningrad (1950).
[72] A. S. Švarc, “On the homotopic topology of Banach spaces,” Dokl. Akad. Nauk SSSR, 154, 61-63 (1964).
[73] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [Russian translation], Mir, Moscow (1980).
[74] A. J. Tromba, “On the number of simply connected minimal surfaces spanning a curve,” Mem. Amer. Math. Soc., 12, No. 194, (1977). · Zbl 0389.53003
[75] A. J. Tromba, “Degree theory on oriented infinite dimensional varieties and Morse number of minimal surfaces. Part I: <Emphasis Type=”Italic“>n > 4<Emphasis Type=”Italic“>,” Trans. Amer. Math. Soc., 290, No. 1, 385-413 (1985). · Zbl 0604.58008
[76] V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971). · Zbl 0231.35002
[77] V. G. Zvyagin, “The degree of mappings that commute with <Emphasis Type=”Italic“>p-periodic diffeomorphisms,” Voroneˇz. Gos. Univ. Trudy Mat. Fak., No. 10, 45-52 (1973).
[78] V. G. Zvyagin, Investigation of Topological Characteristics of Nonlinear Operators, PhD Thesis, Voronezh, 1974. · Zbl 0774.47035
[79] V. G. Zvyagin, “The existence of a continuous branch for the eigenfunctions of a nonlinear elliptic boundary value problem,” Differ. Uravn., 13, No. 8, 1524-1527 (1977). · Zbl 0367.35045
[80] V.G. Zvyagin, “On the number of solutions for certain boundary value problems,” Lecture Notes in Math., 1334, 157-172 (1988). · doi:10.1007/BFb0080427
[81] V.G. Zvyagin, “The degree of Fredholm mappings that are equivariant with respect to the actions of a circle and a torus,” Russian Math. Surveys, 45, No. 2, 229-230 (1990). · Zbl 0722.47052 · doi:10.1070/RM1990v045n02ABEH002346
[82] V.G. Zvyagin, “The oriented degree of a class of perturbations of Fredholm mappings and the bifurcation of the solutions of a nonlinear boundary value problem with noncompact perturbations,” Math. USSR-Sb., 74, No. 2, 487-512 (1993). · Zbl 0774.47035 · doi:10.1070/SM1993v074n02ABEH003358
[83] V. G. Zvyagin, “On a degree theory for equivariant Φ0 <Emphasis Type=”Italic“>C1 <Emphasis Type=”Italic“>BH-mappings,” Dokl. Akad. Nauk, 364, No. 2, 155-157 (1999).
[84] V. G. Zvyagin, È.M. Muhamadiev, and Yu. I. Sapronov, “On the question of the computation of the degree of nonlinear Fredholm mappings,” Voronež. Gos. Univ. Trudy Mat. Fak., No. 10, 25-44 (1973).
[85] V.G. Zvyagin and N.M. Ratiner, “The degree of completely continuous perturbations of Fredholm maps and its application to the bifurcation of solutions,” Dokl. Akad. Nauk Ukrain. SSR Ser. A, No. 6, 8-11 (1989). · Zbl 0693.47046
[86] V.G. Zvyagin and N.M. Ratiner, “Oriented degree of Fredholm maps of non-negative index and its applications to global bifurcation of solutions,” Lecture Notes in Math., 1520, 111-137 (1992). · doi:10.1007/BFb0084718
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