## Abstract

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.

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## Literature cited

N. V. Azbelev, “On some trends in generalizations of the differential equation,” Differents. Uravn.,

**21**, No. 8, 1291–1304 (1985).N. V. Azbelev and V. P. Maksimov, “A priori estimates of solutions of the Cauchy problem and solvability of boundary-value problems for equations with retarded argument,” Differents. Uravn.,

**15**, No. 10, 1731–1747 (1979).N. V. Azbelev and V. P. Maksimov, “Equations with retarded argument,” Differents. Uravn.,

**18**, No. 12, 2027–2050 (1982).N. V. Azbelev and L. F. Rakhmatullina, “Functional-differential equations,” Differents. Uravn.,

**14**, No. 5, 771–797 (1978).M. T. Ashordiya, “On a many-point boundary-value problem for a system of generalized ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**115**, No. 1, 17–20 (1984).M. T. Ashordiya, “On a nonlinear boundary-value problem for a system of generalized ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**118**, No. 2, 261–264 (1985).M. T. Ashordiya, “On the structure of the set of solutions of the Cauchy problem for a system of generalized ordinary differential equations,” Proceedings of the Vekua Institute of Applied Mathematics, Tbilisi State University,

**17**, 5–16 (1986).D. G. Bitsadze and I. T. Kiguradze, “On well-posedness for boundary-value problems for systems of ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**111**, No. 2, 241–244 (1983).D. G. Bitsadze and I. T. Kiguradze, “On the stability of the set of solutions of nonlinear boundary-value problems,” Differents. Uravn.,

**20**, No. 9, 1495–1501 (1984).N. I. Vasil'ev, “Some boundary-value problems for a system of two first-order differential equations. I,” Latvian Mathematical Yearbook,

**5**, 11–24 (1969).N. I. Vasil'ev, “Some boundary-value problems for a system of two first-order differential equations. II.” Latvian Mathematical Yearbook,

**6**, 31–39 (1969).N. I. Vasil'ev and Yu. A. Klokov, Foundations of the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Zinatne, Riga, (1978).

R. V. Gamkrelidze and G. L. Kharatishvili, “Extremal problems in linear topological spaces,” Izv. Akad. Nauk SSSR, Ser. Mat.,

**33**, No. 4, 781–839, (1969).Sh. M. Gelashvili, “On a boundary-value problem for systems of functional-differential equations,” Arch. Math.,

**20**, No. 4, 157–168 (1984).Sh. M. Gelashvili and I. T. Kiguradze, “On a method of numerical solution of boundary-value problems for systems of ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**115**, No. 3, 469–472 (1984).G. N. Zhevlakov, Yu. V. Komlenko, and E. L. Tonkov, “On the existence of solutions of nonlinear ordinary differential equations with linear boundary conditions,” Differents. Uravn.

**4**, No. 10, 1814–1820 (1968).M. A. Kakabadze, “On a problem with integral conditions for a system of ordinary differential equations,” Mat. Čas.

**24**, No. 3, 225–237 (1974).M. A. Kakabadze, “On a singular boundary-value problem for a system of ordinary differential equations,” Dokl. Akad. Nauk SSSR,

**217**, No. 6, 1259–1262 (1974).M. A. Kakabadze and I. T. Kiguradze, “On a boundary-value problem for a system of ordinary differential equations,” Differents. Uravn.,

**7**, No. 9, 1611–1616 (1971).L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, New York (1982).

B. V. Kvedaras, A. V. Kibenko, and A. I. Perov, “On some boundary-value problems,” Lit. Mat. Sb.,

**5**, No. 1, 69–84 (1965).I. T. Kiguradze, “On the singular Nicoletti problem,” Dokl. Akad. Nauk SSSR,

**86**, No. 4, 769–772 (1969).I. T. Kiguradze, “On some nonlinear boundary-value problems (I),” Bul. Inst. Politehn. Iasi,

**16**, No. 3–4, 57–65 (1970).I. T. Kiguradze, “On a boundary-value problem for a system of two differential equations,” Tr. Tbilissk. Univ.,

**1(137)A**, 77–87 (1971).I. T. Kiguradze, “On periodic solutions of a system of ordinary differential equations with singularities,” Dokl. Akad. Nauk SSSR,

**198**, No. 2, 286–289 (1971).I. T. Kiguradze, “On some nonlinear boundary-value problems (II),” Bul. Inst. Politehn. Iasi,

**18**, No. 3–4, 95–107 (1972).I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilissk. Univ. Press (1975).

I. T. Kiguradze, “On a periodic boundary-value problem for a two-dimensional differential system,” Differents. Uravn.,

**13**, No. 6, 996–1007 (1977).I. T. Kiguradze, “On periodic solutions of a system of nonlinear ordinary differential equations,” Usp. Mat. Nauk,

**39**, No. 4, 137–138 (1984).I. T. Kiguradze, “On two-point boundary-value problems for systems of nonlinear ordinary differential equations,” In: Ninth International Conference on Nonlinear Oscillations. Vol. 1 [in Russian], 168–173, Naukova Dumka, Kiev (1984).

I. T. Kiguradze, “On many-point boundary-value problems for systems of ordinary differential equations,” Proceedings of the Extended Session of the Seminar of the Vekua Institute of Applied Mathematics, Tbilisi State Univ.,

**1**, No. 3, 54–60 (1985).L. T. Kiguradze, “On periodic solutions of systems of nonautonomous ordinary differential equations,” Mat. Zametki,

**39**, No. 4, 562–575 (1986).I. T. Kiguradze and N. R. Lezhava, “On some nonlinear two-point boundary-value problems,” Differents. Uravn.,

**10**, No. 12, 2147–2161 (1974).I. T. Kiguradze and B. Puzha, “On some boundary-value problems for a system of ordinary differential equations,” Differents. Uravn.,

**12**, No. 12, 2139–2148 (1976).Yu. A. Klokov, “Uniqueness of the solution of boundary-value problems for a system of two first-order differential equations,” Differents. Uravn.,

**8**, No. 8, 1377–1385 (1972).Yu. V. Komlenko, “A two-sided method of constructing a periodic solution of a system of ordinary first-order differential equations,” In: Problems of the Modern Theory of Periodic Motions [in Russian], 29–38, Izhevsk (1981).

M. A. Krasnosel'skii, Translation along Trajectories of Differential Equations, American Mathematical Society Translations, Vol. 19 (1968).

M. A. Krasnosel'skii and S. G. Krein, “On the averaging principle in nonlinear mechanics,” Usp. Mat. Nauk,

**10**, No. 3, 147–152 (1955).J. Kurzweil and Z. Vorel, “On continuous dependence of the solutions of differential equations on a parameter,” Czechosl. Mat. J.,

**7**, No. 4, 568–583 (1957).A. Yu. Levin, “Passage to the limit for nonsingular systems\(\dot X = A_n (t)X\),” Dokl. Akad. Nauk SSSR,

**176**, No. 4, 774–775 (1967).N. R. Lezhava, “On solvability of a nonlinear problem for a system of two differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**68**, No. 3, 545–547 (1972).A. Ya. Lepin, “Application of topological methods to nonlinear boundary-value problems for ordinary differential equations,” Differents. Uravn.,

**5**, No. 8, 1390–1397 (1969).A. Ya. Lepin and A. D. Myshkis, “On an approach to nonlinear boundary-value problems for ordinary differential equations,” Differents. Uravn.,

**3**, No. 11, 1882–1888 (1967).A. Ya. Lepin and V. D. Ponomarev, “Continuous dependence of the solution of boundary-value problems for ordinary differential equations,” Differents. Uravn.

**9**, No. 4, 626–629 (1973).G. N. Mil'shtein, “On a boundary-value problem for a system of two differential equations,” Differents. Uravn.,

**1**, No. 12, 1628–1639 (1965).A. I. Perov, “Periodic, almost-periodic, and bounded solutions of the differential equation

*dx/dt*=*f(t,x)*, Dokl. Akad. Nauk SSSR,**132**, No. 3, 531–534 (1960).A. I. Perov, “On a boundary-value problem for a system of two differential equations,” Dokl. Akad. Nauk SSSR,

**144**, No. 3, 493–496 (1962).A. I. Perov and A. V. Kibenko, “On a general method of studying boundary-value problems,” Izv. Akad. Nauk SSSR, Ser. Mat.,

**30**, No. 2, 249–264 (1966).N. N. Petrov, “Some sufficient conditions for continuous dependence of the solution of a differential equation on a parameter,” Vestnik Leningrad. Univ., Mat. Mekh. Astron., No. 19, 26–40 (1962).

N. N. Petrov, “On continuity of the solutions of differential equations over a parameter,” Vestnik Leningrad. Univ., Mat. Mekh. Astron., No. 7, 29–36 (1964)

N. N. Petrov, “Necessary conditions for continuity over a parameter for certain classes of equations,” Vestnik Leningrad. Univ., Mat. Mekh. Astron., No. 1, 47–53 (1965).

V. A. Pliss, Nonlocal Problems of the Theory of Oscillations, Academic Press, New York (1966).

V. D. Ponomarev, “Existence of a solution of a boundary-value problem with functional boundary condition,” Differents. Uravn.

**9**, No. 12, (1973).V. D. Ponomarev, “On local uniqueness of the solution of boundary-value problems,” Mat. Zametki,

**15**, No. 6, 891–895 (1974).B. Puzha, “On a singular boundary-value problem for a system of ordinary differential equations,” Arch. Math.,

**13**, No. 4, 207–226 (1977).B. Puzha, “On solvability of some boundary-value problems for systems of ordinary differential equations,” Scr. Fac. Sci. Natur. UJEP Brun., No. 8, 411–426 (1980).

F. Zh. Sadyrbaev, “On a two-point boundary-value problem for a system of ordinary first-order differential equations,” Latvian Mathematical Yearbook,

**23**, 131–136 (1979).F. Zh. Sadyrbaev, “On nonlinear boundary-value problems for a system of two ordinary first-order differential equations,” In: Functional Methods in the Equations of Mathematical Physics [in Russian], 59–62, Moscow (1980).

A. M. Samoilenko, “Study of a differential equation with ‘nonregular’ right-hand side,” Abhandlungen der deutschen Akademie der Wissenschaften in Berlin, Klasse Math. Phys. und Tech., No. l, 106–113 (1965).

A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods of Studying the Solutions of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1986).

Yu. V. Trubnikov and A. I. Perov, Differential Equations with Monotonic Nonlinearities [in Russian], Nauka i Tekhnika, Minsk (1986).

F. Hartman, Ordinary Differential Equations, [Russian translation], Mir, Moscow (1970).

A. Ya. Khokhryakov, “On the existence and estimation of the solution of a periodic boundary-value problem for a system of ordinary differential equations,” Differents. Uravn.,

**2**, No. 10, 1300–1306 (1966).V. A. Chechik, “Study of systems of ordinary differential equations with a singularity,” Tr. Mosk. Mat. Obshch.,

**8**, 155–198 (1959).B. L. Shekhter, “On a boundary-value problem for a system of ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,

**80**, No. 3, 541–544 (1975).B. L. Shekhter, “On a boundary-value problem for two-dimensional discontinuous differential systems,” Proceedings of the Vekua Institute of Applied Mathematics, Tbilisi State Univ.,

**8**, 79–161 (1980).A. I. Shindyapin, “On a boundary-value problem for a singular equation,” Differents. Uravn.,

**20**, No. 3, 450–455 (1984).Z. Artstein, “Continuous dependence on parameters: on the best possible results,” J. Diff. Eqs.,

**19**, No. 2, 214–225 (1975).R. Conti, “Equazioni differenziali ordinarie quasilineari con condizioni lineari,” Ann. Mat. Pura ed Appl.,

**57**, 49–61 (1962).R. Conti, “Recent trends in the theory of boundary-value problems for ordinary differential equations,” Boll. Unione Mat. Ital.,

**22**, No. 2, 135–178 (1967).M. Fukuhara, “Sur une généralisation d'un thórème de Kneser,” Proc. Jap. Acad.,

**29**, 154–155 (1953).R. E. Gaines and J. Mawhin, “Ordinary differential equations with nonlinear boundary conditions,” J. Diff. Eqs.,

**26**, No. 2, 200–222 (1977).I. T. Kiguradze, “On a singular problem of Cauchy-Nicoletti,” Ann. Mat. Pura ed Appl.,

**104**, 151–175 (1975).I. T. Kiguradze, “On the modified problem of Cauchy-Nicoletti,” Ann. Mat. Pura ed Appl.,

**104**, 177–186 (1975).H. Kneser, “Über die Lösungen eines Systems gewöhnlicher Differentialgleichungen, das der Lipschitzschen Bedingung nicht genügt,” Sitzungsberichte der preussischen Akademie der Wissenschaften, Phys.-Math. Klasse, 171–174 (1923).

J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,” Czechosl. Mat. J.,

**7**, No. 3, 418–449 (1957).J. Kurzweil, “Generalized ordinary differential equations,” Czechosl. Mat. J.,

**8**, No. 3, 360–388 (1958).A. Lasota, “Sur l'existence et l'unicité des solutions du problème aux limites de Nicoletti pour un système d'équations différentielles ordinaires,” Zesz. Nauk. Univ. Jagiell.,

**11**, 41–48 (1966).A. Lasota, “On two-point boundary-value problems for systems of ordinary nonlinear first-order differential equations,” Ann. Pol. Math.

**29**, No. 4, 391–396 (1975).A. Lasota and C. Olech, “An optimal solution of Nicoletti's boundary-value problem,” Ann. Pol. Math.,

**18**, No. 2, 131–139 (1966).A. Lasota and Z. Opial, “Sur les solutions périodiques des équations différentielles ordinaires,” Ann. Pol. Math.,

**16**, No. 1, 69–94 (1964).Z. Opial, “Linear problems for systems of nonlinear differential equations,” J. Diff. Eqs.,

**3**, No. 4, 580–594 (1967).K. Schmitt, “Periodic solutions of nonlinear differential systems,” J. Math. Anal. and Appl.,

**40**, No. 1, 174–182 (1972).Š. Schwabik and M. Tvrdy, “Boundary-value problems for generalized linear differential equations,” Czechosl. Math. J.,

**29**, No. 3, 451–477 (1979).Š. Schwabik, M. Tvrdy, and O. Veivoda, Differential and Integral Equations: Boundary-Value Problems and Adjoints, Academia, Praha (1979).

S. Sedziwy, “Periodic solutions of a system of nonlinear differential equations,” Proc. Amer. Math. Soc.,

**48**, No. 2, 328–336 (1975).B. L. Shekhter, “On singular boundary-value problems for two-dimensional differential systems,” Arch. Math.,

**19**, No. 1, 19–41 (1983).Z. Vorel, “Continuous dependence on parameters,” Nonlinear Anal., Theory, Meth., and Appl.,

**5**, No. 4, 373–380 (1981).P. Waltman, “Existence and uniqueness of solutions of boundary-value problems for two-dimensional systems of nonlinear differential equations,” Trans. Amer. Math. Soc.,

**153**, No. 1, 223–234 (1971).

## Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3–103, 1987.

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Kiguradze, I.T. Boundary-value problems for systems of ordinary differential equations.
*J Math Sci* **43**, 2259–2339 (1988). https://doi.org/10.1007/BF01100360

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DOI: https://doi.org/10.1007/BF01100360