New constructions of topological invariants (of topological degree type), covering the majority of previously known ones, are introduced and investigated for a broad class of set-valued mappings. The topological structure of the set of fixed points of set-valued maps (including connectedness, topological dimension, acyclicity, etc.) is investigated. The topological dimension of the set of solutions of Cauchy problem for differential inclusions in finite-dimensional spaces is investigated. Solvability and topological dimension of the set of solutions of operator equations in the form $a(x)=f(x)$, where $р$ is a surjective linear operator and $f$ is a copletely continuous operator, is investigated. For the infinite-dimensional Banach space a natural generalization of the classical Borsuk–Ulam finite-dimensional theorem is proved. A generalization of classical implicit function theorem is proved in the case where the Frechet derivative is a surjective operator.
Biography
Graduated from Mathematics Faculty of Voronezh State University in 1970 (department of Algebra and Topological Methods of Analysis). Ph.D. thesis was defended in 1970. A list of my works contains more than 60 titles.
Main publications:
Benkafadar N. M., Gel'man B. D. Generalized Local Degree for Multi-Valued Mappings // International Journal of Math., Game Theory and Algebra, 2000, 10(5), 413–434.
B. D. Gel'man, “On topological properties of the set of solutions of operator inclusions with a multi-valued Lipschitz right-hand side”, Izv. Vyssh. Uchebn. Zaved. Mat., 2021, no. 5, 11–15; Russian Math. (Iz. VUZ), 65:5 (2021), 4–7
2019
2.
B. D. Gel'man, “Karisti inequality and $\alpha$-contractive mappings”, Funktsional. Anal. i Prilozhen., 53:3 (2019), 84–88
3.
B. D. Gel'man, “On the Borsuk–Ulam theorem for Lipschitz mappings in an infinite-dimensional space”, Funktsional. Anal. i Prilozhen., 53:1 (2019), 79–83
2018
4.
B. D. Gel'man, “Periodic Trajectories and Coincidence Points of Tuples of Set-Valued Maps”, Funktsional. Anal. i Prilozhen., 52:2 (2018), 72–77; Funct. Anal. Appl., 52:2 (2018), 139–143
B. D. Gel'man, “The inequality of Karisti and generalized compression (the case of singular images)”, Tambov University Reports. Series: Natural and Technical Sciences, 23:122 (2018), 243–249
2017
6.
B. D. Gel'man, “A Hybrid Fixed-Point Theorem for Set-Valued Maps”, Mat. Zametki, 101:6 (2017), 832–842; Math. Notes, 101:6 (2017), 951–959
B. D. Gel'man, “How to Approach Nonstandard Boundary Value Problems”, Funktsional. Anal. i Prilozhen., 50:1 (2016), 38–46; Funct. Anal. Appl., 50:1 (2016), 31–38
8.
B. D. Gel'man, “A version of the infinite-dimensional Borsuk-Ulam theorem for multivalued maps”, Mat. Sb., 207:6 (2016), 79–92; Sb. Math., 207:6 (2016), 841–853
B. D. Gel'man, “The Solution Set of a Class of Equations with Surjective Operators”, Funktsional. Anal. i Prilozhen., 49:1 (2015), 74–78; Funct. Anal. Appl., 49:1 (2015), 60–63
B. D. Gel'man, V. V. Obukhovskii, “On fixed points of acyclic type multivalued maps”, Fundam. Prikl. Mat., 20:3 (2015), 47–59; J. Math. Sci., 225:4 (2017), 565–574
B. D. Gel'man, S. E. Zhukovskii, “Covering Maps of Spaces of Compact Subsets”, Mat. Zametki, 93:4 (2013), 530–536; Math. Notes, 93:4 (2013), 539–544
2012
13.
B. D. Gel'man, “On Local Solutions of Degenerate Differential Inclusions”, Funktsional. Anal. i Prilozhen., 46:1 (2012), 79–83; Funct. Anal. Appl., 46:1 (2012), 66–68
B. D. Gel'man, “On the acyclicity of the solution sets of operator equations”, Mat. Sb., 201:10 (2010), 47–58; Sb. Math., 201:10 (2010), 1449–1459
2009
15.
A. V. Arutyunov, B. D. Gel'man, “Minimum of a functional in a metric space and fixed points”, Zh. Vychisl. Mat. Mat. Fiz., 49:7 (2009), 1167–1174; Comput. Math. Math. Phys., 49:7 (2009), 1111–1118
B. D. Gel'man, “On the Cauchy Problem for a Class of Degenerate Differential Equations with Lipschitz Right-Hand Side”, Funktsional. Anal. i Prilozhen., 42:3 (2008), 78–81; Funct. Anal. Appl., 42:3 (2008), 227–229
B. D. Gel'man, “An Infinite-Dimensional Version of the Borsuk–Ulam Theorem”, Funktsional. Anal. i Prilozhen., 38:4 (2004), 1–5; Funct. Anal. Appl., 38:4 (2004), 239–242
B. D. Gel'man, “Rest points of generalized dynamical systems”, Mat. Zametki, 65:1 (1999), 28–36; Math. Notes, 65:1 (1999), 24–30
1997
23.
N. M. Benkafadar, B. D. Gel'man, “On the local degree for a class of multivalued field”, Izv. Vyssh. Uchebn. Zaved. Mat., 1997, no. 2, 24–34; Russian Math. (Iz. VUZ), 41:2 (1997), 22–31
24.
B. D. Gel'man, “Topological properties of the set of fixed points of a multivalued map”, Mat. Sb., 188:12 (1997), 33–56; Sb. Math., 188:12 (1997), 1761–1782
B. D. Gel'man, “The antipode theorem and points of coincidence”, Uspekhi Mat. Nauk, 51:1(307) (1996), 147–148; Russian Math. Surveys, 51:1 (1996), 142–143
1991
26.
B. D. Gel'man, V. V. Obukhovskii, “New results in the theory of multivalued mappings. II. Analysis and applications”, Itogi Nauki i Tekhn. Ser. Mat. Anal., 29 (1991), 107–159; J. Soviet Math., 64:2 (1993), 854–883
Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, V. V. Obukhovskii, “New results in the theory of multivalued mappings. I. Topological characteristics and solvability of operator relations”, Itogi Nauki i Tekhn. Ser. Mat. Anal., 25 (1987), 123–197; J. Soviet Math., 49:1 (1990), 800–855
Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, V. V. Obukhovskii, “Multivalued analysis and operator inclusions”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 29 (1986), 151–211; J. Soviet Math., 39:3 (1987), 2772–2811
Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, V. V. Obukhovskii, “Multivalued mappings”, Itogi Nauki i Tekhn. Ser. Mat. Anal., 19 (1982), 127–230; J. Soviet Math., 24:6 (1984), 719–791
Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis, V. V. Obukhovskii, “Topological methods in the fixed-point theory of multi-valued maps”, Uspekhi Mat. Nauk, 35:1(211) (1980), 59–126; Russian Math. Surveys, 35:1 (1980), 65–143
Yu. G. Borisovich, B. D. Gel'man, È. Muhamadiev, V. V. Obukhovskii, “The rotation of multivalued vector fields”, Dokl. Akad. Nauk SSSR, 187:5 (1969), 971–973
Yu. G. Borisovich, B. D. Gel'man, È. M. Muhamadiev, V. V. Obukhovskii, “Поправки к статье “О вращении многозначных векторных полей” (ДАН, 187, № 5, 1969 г.)”, Dokl. Akad. Nauk SSSR, 190:6 (1970), 750
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