boundary value problems for ordinary differential equations and systems; periodic and bounded solutions; asymptotic of solutions of nonautonomous differential equations; oscillation theory.
Subject:
A method of a priori estimates of solutions of one-sided differential inequalities has been developed, on the basis of which the theory of boundary value problems of Cauchy–Nicoletti, Vallee Poussin, Kneser and periodic type has been constructed for ordinary differential equations and systems with nonintegrable singularities. Optimal existence and uniqueness criteria of periodic and bounded solutions have been established for higher order nonautonomous differential equations. A complete investigation has been carried out of asymptotic properties of solutions of a wide class of nonautonomous differential equations. In particular, the problem on the dimension of the space of vanishing at infinity solutions of a linear homogeneous equation of an arbitrary order has been solved; asymptotic formulas have been obtained for solutions of linear and quasilinear equations; necessary and sufficient conditions have been found for essentially nonlinear equations to have proper, singular, oscillatory and monotone solutions of various types, and asymptotic estimates of these solutions have been established.
Biography
Graduated from Faculty of Mechanics and Mathematics of Tbilisi State University in 1960 (chair of differential and integral equations). Ph.D. thesis was defended in 1963. D.Sci. thesis was defended in 1972. A list of my scientific works contains more than 120 titles. Since 1966 — leader of the Tbilisi seminar on qualitative theory of differential equations. Since 1989 — leader of the seminar on differential equations and mathematical physics at A. Razmadze Mathematical Institute. Since 1994 — managing editor of the "Georgian Mathematical Journal" and chairman of editorial board of the journal "Memoirs on Differential Equations and Mathematical Physics".
In 1975 I was awarded the A. Razmadze prize of the Georgian Academy of Sciences for the monograph "Some singular boundary value problems for ordinary differential equations". In 1979 I was elected a corresponding member of the Georgian Academy of Sciences. Since 1993 I have been a full member (academician) of the Georgian Academy of Sciences.
Main publications:
Kiguradze I. T., Chanturia T. A. Asymptotic properties of solutions of nonaotonomous ordinary differential equations. Kluwer Academic Publishers, Dordrecht–Boston–London, 1993.
Kiguradze I. On periodic solutions of n-th order ordinary differential equations // Nonlinear Anal., 2000, 40(1–8), 309–321.
Kiguradze I., Kusano T. On periodic solutions of even-order ordinary differential equations // Ann. Mat. Pura Appl., 2001, 180(3), 285–301.
S. R. Baslandze, I. T. Kiguradze, “On the unique solvability of a periodic boundary value problem for third-order linear differential equations”, Differ. Uravn., 42:2 (2006), 153–158; Differ. Equ., 42:2 (2006), 165–171
2004
2.
I. T. Kiguradze, S. V. Mukhigulashvili, “On Nonlinear Boundary Value Problems for Two-Dimensional Differential Systems”, Differ. Uravn., 40:6 (2004), 747–755; Differ. Equ., 40:6 (2004), 797–806
2003
3.
I. T. Kiguradze, “On Boundary Value Problems for Linear Differential Systems with Singularities”, Differ. Uravn., 39:2 (2003), 198–209; Differ. Equ., 39:2 (2003), 212–225
2002
4.
I. T. Kiguradze, N. L. Partsvaniya, I. P. Stavroulakis, “Oscillatory Properties of Higher-Order Advance Functional-Differential Equations”, Differ. Uravn., 38:8 (2002), 1030–1041; Differ. Equ., 38:8 (2002), 1095–1107
2001
5.
I. T. Kiguradze, “Blow-Up Kneser Solutions of Nonlinear Higher-Order Differential Equations”, Differ. Uravn., 37:6 (2001), 735–743; Differ. Equ., 37:6 (2001), 768–777
2000
6.
I. T. Kiguradze, T. Kusano, “Conditions for the existence and uniqueness of periodic solutions of nonautonomous differential equations”, Differ. Uravn., 36:10 (2000), 1301–1306; Differ. Equ., 36:10 (2000), 1436–1442
1999
7.
I. T. Kiguradze, T. Kusano, “On periodic solutions of higher-order nonautonomous ordinary differential equations”, Differ. Uravn., 35:1 (1999), 72–78; Differ. Equ., 35:1 (1999), 71–77
1998
8.
I. T. Kiguradze, I. P. Stavroulakis, “On the existence of proper oscillating solutions of advanced differential equations”, Differ. Uravn., 34:6 (1998), 751–757; Differ. Equ., 34:6 (1998), 748–754
1997
9.
I. T. Kiguradze, “On systems of ordinary differential equations and differential inequalities with multipoint boundary conditions”, Differ. Uravn., 33:5 (1997), 646–652; Differ. Equ., 33:5 (1997), 649–655
10.
I. T. Kiguradze, B. Puzha, “Theorems of Conti–Opial type for nonlinear functional-differential equations”, Differ. Uravn., 33:2 (1997), 185–194; Differ. Equ., 33:2 (1997), 184–193
11.
I. T. Kiguradze, Z. P. Sokhadze, “On the Cauchy problem for evolution singular functional-differential equations”, Differ. Uravn., 33:1 (1997), 48–59; Differ. Equ., 33:1 (1997), 47–58
1996
12.
I. T. Kiguradze, “On a singular Cauchy problem for systems of linear ordinary differential equations”, Differ. Uravn., 32:2 (1996), 171–179; Differ. Equ., 32:2 (1996), 173–180
1995
13.
I. T. Kiguradze, Z. P. Sokhadze, “On the uniqueness of the solution of the Cauchy problem for functional-differential equations”, Differ. Uravn., 31:12 (1995), 1977–1988; Differ. Equ., 31:12 (1995), 1947–1958
1992
14.
I. T. Kiguradze, “An oscillation criterion for a class of ordinary differential equations”, Differ. Uravn., 28:2 (1992), 207–219; Differ. Equ., 28:2 (1992), 180–190
1991
15.
I. T. Kiguradze, D. I. Chichua, “On the Kneser problem for functional-differential equations”, Differ. Uravn., 27:11 (1991), 1879–1892; Differ. Equ., 27:11 (1991), 1322–1335
1987
16.
I. T. Kiguradze, B. L. Shekhter, “Singular boundary value problems for second-order ordinary differential equations”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 30 (1987), 105–201; J. Soviet Math., 43:2 (1988), 2340–2417
17.
I. T. Kiguradze, “Boundary value problems for systems of ordinary differential equations”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 30 (1987), 3–103; J. Soviet Math., 43:2 (1988), 2259–2339
1986
18.
I. T. Kiguradze, “Periodic solutions of systems of nonautonomous ordinary differential equations”, Mat. Zametki, 39:4 (1986), 562–575; Math. Notes, 39:4 (1986), 308–315
1985
19.
D. V. Izyumova, I. T. Kiguradze, “Oscillatory properties of a class of differential equations with deviating argument”, Differ. Uravn., 21:4 (1985), 588–596
20.
I. T. Kiguradze, “Solvability of the de la Vallée-Poussin boundary value problem”, Differ. Uravn., 21:3 (1985), 391–398
21.
I. T. Kiguradze, “A priori estimates of solutions of higher order nonlinear differential inequalities”, Differ. Uravn., 21:2 (1985), 198–213
22.
I. T. Kiguradze, “Bounded and periodic solutions of linear differential equations of higher order”, Mat. Zametki, 37:1 (1985), 48–62; Math. Notes, 37:1 (1985), 28–36
1984
23.
I. T. Kiguradze, “Behavior of the solutions of higher-order ordinary differential equations in the neighborhood of a singular point”, Differ. Uravn., 20:10 (1984), 1703–1709
24.
D. G. Bitsadze, I. T. Kiguradze, “Stability of the solution set of nonlinear boundary value problems”, Differ. Uravn., 20:9 (1984), 1495–1501
1979
25.
I. T. Kiguradze, I. Rahunkova, “Solvability of a nonlinear problem of Kneser's type”, Differ. Uravn., 15:10 (1979), 1754–1765
1977
26.
I. T. Kiguradze, “The periodic boundary value problem for a two-dimensional differential system”, Differ. Uravn., 13:6 (1977), 996–1007
1976
27.
I. T. Kiguradze, B. Puzha, “Certain boundary value problems for a system of ordinary differential equations”, Differ. Uravn., 12:12 (1976), 2139–2148
1974
28.
I. T. Kiguradze, N. R. Lezhava, “Certain nonlinear two-point boundary value problems”, Differ. Uravn., 10:12 (1974), 2147–2161
29.
N. V. Gogiberidze, I. T. Kiguradze, “On the question of the disconjugacy of second order singular linear differential equations”, Differ. Uravn., 10:11 (1974), 2064–2067
30.
I. T. Kiguradze, “Conditions for the conjugacy of the solutions of nonlinear ordinary differential equations. I, II”, Differ. Uravn., 10:9 (1974), 1586–1594
31.
I. T. Kiguradze, “Conditions for the conjugacy of the solutions of nonlinear ordinary differential equations. I, II”, Differ. Uravn., 10:8 (1974), 1387–1399
32.
I. T. Kiguradze, N. R. Lezhava, “The problem of the solvability of nonlinear two-point boundary-value problems”, Mat. Zametki, 16:3 (1974), 479–490; Math. Notes, 16:3 (1974), 873–880
1971
33.
I. T. Kiguradze, “On the periodic solutions of a system of ordinary differential equations with singularities”, Dokl. Akad. Nauk SSSR, 198:2 (1971), 286–289
34.
M. A. Kakabadze, I. T. Kiguradze, “A certain boundary value problem for a system of ordinary differential equations”, Differ. Uravn., 7:9 (1971), 1611–1616
1970
35.
I. T. Kiguradze, “A certain singular boundary value problem for an ordinary differential equation of order $n$”, Dokl. Akad. Nauk SSSR, 192:5 (1970), 973–975
36.
I. T. Kiguradze, T. A. Chanturiya, “A remark on the asymptotic behavior of the solutions of the equation $u"+a(t)u=0$”, Differ. Uravn., 6:6 (1970), 1115–1117
1969
37.
I. T. Kiguradze, “The singular Nicoletti problem”, Dokl. Akad. Nauk SSSR, 186:4 (1969), 769–772
38.
I. T. Kiguradze, “The singular two-point boundary value problem”, Differ. Uravn., 5:11 (1969), 2002–2016
39.
I. T. Kiguradze, “On monotone solutions of nonlinear ordinary differential equations of order $n$.”, Izv. Akad. Nauk SSSR Ser. Mat., 33:6 (1969), 1373–1398; Math. USSR-Izv., 3:6 (1969), 1293–1317
40.
I. T. Kiguradze, “Conditions for non-oscillation of singular linear differential equations of second order”, Mat. Zametki, 6:5 (1969), 633–639; Math. Notes, 6:5 (1969), 843–847
1968
41.
I. T. Kiguradze, “Monotonic solutions of $n$th-order nonlinear ordinary differential equations”, Dokl. Akad. Nauk SSSR, 181:5 (1968), 1054–1057
42.
I. T. Kiguradze, “Some singular boundary value problems for second order nonlinear ordinary differential equations”, Differ. Uravn., 4:10 (1968), 1753–1773
43.
D. V. Izyumova, I. T. Kiguradze, “Some remarks on the solutions of the equation $u"+a(t)f(u)=0$”, Differ. Uravn., 4:4 (1968), 589–605
1967
44.
I. T. Kiguradze, “A priori estimates for the derivatives of bounded functions satisfying second-order differential inequalities”, Differ. Uravn., 3:7 (1967), 1043–1052
1965
45.
I. T. Kiguradze, “On the Cauchy problem for singular systems of ordinary differential equations”, Differ. Uravn., 1:10 (1965), 1271–1291
46.
I. T. Kiguradze, “On the question of variability of solutions of nonlinear differential equations”, Differ. Uravn., 1:8 (1965), 995–1006
47.
I. T. Kiguradze, “The asymptotic behavior of the solutions of a nonlinear differential equation of Emden–Fowler type”, Izv. Akad. Nauk SSSR Ser. Mat., 29:5 (1965), 965–986
1964
48.
I. T. Kiguradze, “On the oscillatory character of solutions of the equation $\frac{d^mu}{dt^m}+a(t)|u|^n\operatorname{sign}u=0$”, Mat. Sb. (N.S.), 65(107):2 (1964), 172–187
1962
49.
I. T. Kiguradze, “The capability of certain solutions of ordinary differential equations to oscillate”, Dokl. Akad. Nauk SSSR, 144:1 (1962), 33–36
2010
50.
A. V. Arutyunov, Yu. N. Bibikov, F. P. Vasil'ev, I. V. Gaishun, S. V. Emel'yanov, V. A. Il'in, I. T. Kiguradze, S. K. Korovin, L. D. Kudryavtsev, A. B. Kurzhanskii, S. A. Mazanik, E. K. Makarov, A. A. Martynyuk, E. F. Mishchenko, E. I. Moiseev, Yu. S. Osipov, V. A. Pliss, N. Kh. Rozov, V. A. Sadovnichy, A. M. Samoilenko, T. K. Shemjakina, “Nikolai Alekseevich Izobov (A tribute in honor of his 70th birthday)”, Differ. Uravn., 46:1 (2010), 3–8; Differ. Equ., 46:1 (2010), 1–7
2002
51.
I. V. Gaishun, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, L. D. Kudryavtsev, V. P. Maksimov, N. Kh. Rozov, T. K. Shemjakina, “N. V. Azbelev (A tribute on the occasion of his 80th birthday)”, Differ. Uravn., 38:7 (2002), 867–869; Differ. Equ., 38:7 (2002), 917–919
2000
52.
V. N. Abrashin, I. V. Gaishun, V. A. Il'in, I. T. Kiguradze, S. K. Korovin, L. D. Kudryavtsev, A. B. Kurzhanskii, A. A. Martynyuk, V. M. Millionshchikov, Yu. A. Mitropol'skii, V. A. Pliss, N. Kh. Rozov, A. A. Samarskii, A. M. Samoilenko, T. K. Shemjakina, “Nikolai Alekseevich Izobov”, Differ. Uravn., 36:1 (2000), 3–11; Differ. Equ., 36:1 (2000), 1–11
1999
53.
D. V. Anosov, V. S. Vladimirov, I. V. Gaishun, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, V. A. Kondratiev, L. D. Kudryavtsev, Yu. A. Mitropol'skii, E. F. Mishchenko, O. A. Oleinik, V. A. Pliss, M. I. Rakhimberdiev, N. Kh. Rozov, A. M. Samoilenko, A. F. Filippov, T. K. Shemjakina, “Vladimir Mikhaǐlovich Millionshchikov (on the occasion of his sixtieth birthday)”, Differ. Uravn., 35:10 (1999), 1299–1312; Differ. Equ., 35:10 (1999), 1311–1327
1998
54.
N. A. Izobov, V. A. Il'in, I. T. Kiguradze, V. A. Kondratiev, A. B. Kurzhanskii, V. M. Millionshchikov, O. A. Oleinik, M. I. Rakhimberdiev, T. K. Shemjakina, “Nikolaǐ Khristovich Rozov (on the occasion of his sixtieth birthday)”, Differ. Uravn., 34:2 (1998), 147–148; Differ. Equ., 34:2 (1998), 151–152
1997
55.
A. F. Andreev, I. V. Gaishun, V. I. Gromak, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, N. N. Krasovskii, L. D. Kudryavtsev, N. A. Lukashevich, I. P. Martynov, V. A. Pliss, L. A. Cherkas, T. K. Shemjakina, “Nikolaǐ Pavlovich Erugin (on the ninetieth anniversary of his birth)”, Differ. Uravn., 33:5 (1997), 579–582; Differ. Equ., 33:5 (1997), 583–586
56.
N. A. Izobov, V. A. Il'in, I. T. Kiguradze, L. D. Kudryavtsev, V. P. Maksimov, T. K. Shemjakina, “Nikolaǐ Viktorovich Azbelev (on the occasion of his seventy-fifth birthday)”, Differ. Uravn., 33:4 (1997), 435–439; Differ. Equ., 33:4 (1997), 433–438
1995
57.
L. M. Berkovich, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, V. A. Pliss, N. Kh. Rozov, “Otakar Boruvka [1899–1995]”, Differ. Uravn., 31:10 (1995), 1770–1771; Differ. Equ., 31:10 (1995), 1744–1745
1990
58.
N. V. Azbelev, È. I. Grudo, N. A. Izobov, I. T. Kiguradze, V. A. Kondratiev, V. M. Millionshchikov, N. Kh. Rozov, “Obituary: Teǐmuraz Ambros'evich Chanturiya”, Differ. Uravn., 26:12 (1990), 2190–2191; Differ. Equ., 26:12 (1990), 1648–1649
1988
59.
A. F. Andreev, A. V. Bitsadze, N. P. Erugin, V. I. Zubov, N. A. Izobov, V. A. Il'in, I. T. Kiguradze, N. N. Krasovskii, L. D. Kudryavtsev, V. M. Millionshchikov, V. A. Pliss, A. A. Samarskii, K. S. Sibirskii, A. N. Tikhonov, “Yuriǐ Stanislavovich Bogdanov”, Differ. Uravn., 24:6 (1988), 1091–1097; Differ. Equ., 24:6 (1988), 692–699
1985
60.
T. V. Burchuladze, N. P. Vekua, T. G. Gegelia, N. P. Erugin, I. T. Kiguradze, L. D. Kudryavtsev, A. G. Sveshnikov, “Viktor Dmitrievich Kupradze”, Differ. Uravn., 21:11 (1985), 2018–2019
1982
61.
N. P. Erugin, N. A. Izobov, I. T. Kiguradze, L. D. Kudryavtsev, V. P. Maksimov, “Nikolaǐ Viktorovich Azbelev (on the occasion of his sixtieth birthday)”, Differ. Uravn., 18:4 (1982), 725–730