## Boundary-value problems for systems of ordinary differential equations.(English. Russian original)Zbl 0782.34025

J. Sov. Math. 43, No. 2, 2259-2339 (1988); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30, 3-103 (1987).
See the review in Zbl 0631.34020.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations

### Keywords:

solvability; nonautonomous differential systems

Zbl 0631.34020
Full Text:

### References:

 [1] N. V. Azbelev, ?On some trends in generalizations of the differential equation,? Differents. Uravn.,21, No. 8, 1291?1304 (1985). [2] 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). · Zbl 0427.34058 [3] N. V. Azbelev and V. P. Maksimov, ?Equations with retarded argument,? Differents. Uravn.,18, No. 12, 2027?2050 (1982). [4] N. V. Azbelev and L. F. Rakhmatullina, ?Functional-differential equations,? Differents. Uravn.,14, No. 5, 771?797 (1978). · Zbl 0402.34054 [5] 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). · Zbl 0557.34011 [6] 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). · Zbl 0591.34015 [7] 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). · Zbl 0632.34001 [8] 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). · Zbl 0527.34011 [9] 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). [10] 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). [11] 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). [12] 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). [13] 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). · Zbl 0167.11502 [14] Sh. M. Gelashvili, ?On a boundary-value problem for systems of functional-differential equations,? Arch. Math.,20, No. 4, 157?168 (1984). · Zbl 0583.34054 [15] 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). · Zbl 0567.34011 [16] 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). · Zbl 0241.34015 [17] M. A. Kakabadze, ?On a problem with integral conditions for a system of ordinary differential equations,? Mat. ?as.24, No. 3, 225?237 (1974). · Zbl 0289.34025 [18] 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). · Zbl 0318.34026 [19] 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). [20] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, New York (1982). · Zbl 0484.46003 [21] B. V. Kvedaras, A. V. Kibenko, and A. I. Perov, ?On some boundary-value problems,? Lit. Mat. Sb.,5, No. 1, 69?84 (1965). [22] I. T. Kiguradze, ?On the singular Nicoletti problem,? Dokl. Akad. Nauk SSSR,86, No. 4, 769?772 (1969). · Zbl 0193.04901 [23] I. T. Kiguradze, ?On some nonlinear boundary-value problems (I),? Bul. Inst. Politehn. Iasi,16, No. 3?4, 57?65 (1970). [24] I. T. Kiguradze, ?On a boundary-value problem for a system of two differential equations,? Tr. Tbilissk. Univ.,1(137)A, 77?87 (1971). [25] 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). · Zbl 0233.34047 [26] I. T. Kiguradze, ?On some nonlinear boundary-value problems (II),? Bul. Inst. Politehn. Iasi,18, No. 3?4, 95?107 (1972). [27] I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilissk. Univ. Press (1975). [28] I. T. Kiguradze, ?On a periodic boundary-value problem for a two-dimensional differential system,? Differents. Uravn.,13, No. 6, 996?1007 (1977). · Zbl 0356.34020 [29] I. T. Kiguradze, ?On periodic solutions of a system of nonlinear ordinary differential equations,? Usp. Mat. Nauk,39, No. 4, 137?138 (1984). [30] 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). [31] 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). [32] L. T. Kiguradze, ?On periodic solutions of systems of nonautonomous ordinary differential equations,? Mat. Zametki,39, No. 4, 562?575 (1986). · Zbl 0652.34045 [33] I. T. Kiguradze and N. R. Lezhava, ?On some nonlinear two-point boundary-value problems,? Differents. Uravn.,10, No. 12, 2147?2161 (1974). [34] 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). [35] 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). [36] 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). [37] M. A. Krasnosel’skii, Translation along Trajectories of Differential Equations, American Mathematical Society Translations, Vol. 19 (1968). [38] M. A. Krasnosel’skii and S. G. Krein, ?On the averaging principle in nonlinear mechanics,? Usp. Mat. Nauk,10, No. 3, 147?152 (1955). [39] 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). · Zbl 0090.30001 [40] 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). [41] 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). [42] A. Ya. Lepin, ?Application of topological methods to nonlinear boundary-value problems for ordinary differential equations,? Differents. Uravn.,5, No. 8, 1390?1397 (1969). [43] 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). [44] 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). [45] G. N. Mil’shtein, ?On a boundary-value problem for a system of two differential equations,? Differents. Uravn.,1, No. 12, 1628?1639 (1965). [46] A. I. Perov, ?Periodic, almost-periodic, and bounded solutions of the differential equationdx/dt=f(t,x), Dokl. Akad. Nauk SSSR,132, No. 3, 531?534 (1960). [47] 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). · Zbl 0119.30101 [48] 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). [49] 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). [50] 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) [51] 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). [52] V. A. Pliss, Nonlocal Problems of the Theory of Oscillations, Academic Press, New York (1966). · Zbl 0151.12104 [53] V. D. Ponomarev, ?Existence of a solution of a boundary-value problem with functional boundary condition,? Differents. Uravn.9, No. 12, (1973). [54] V. D. Ponomarev, ?On local uniqueness of the solution of boundary-value problems,? Mat. Zametki,15, No. 6, 891?895 (1974). · Zbl 0339.34017 [55] B. Puzha, ?On a singular boundary-value problem for a system of ordinary differential equations,? Arch. Math.,13, No. 4, 207?226 (1977). [56] 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). [57] 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). [58] 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). [59] 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). [60] A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods of Studying the Solutions of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1986). [61] Yu. V. Trubnikov and A. I. Perov, Differential Equations with Monotonic Nonlinearities [in Russian], Nauka i Tekhnika, Minsk (1986). [62] F. Hartman, Ordinary Differential Equations, [Russian translation], Mir, Moscow (1970). [63] 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). [64] V. A. Chechik, ?Study of systems of ordinary differential equations with a singularity,? Tr. Mosk. Mat. Obshch.,8, 155?198 (1959). [65] 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). · Zbl 0572.34014 [66] 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). · Zbl 0492.34014 [67] A. I. Shindyapin, ?On a boundary-value problem for a singular equation,? Differents. Uravn.,20, No. 3, 450?455 (1984). · Zbl 0566.34052 [68] Z. Artstein, ?Continuous dependence on parameters: on the best possible results,? J. Diff. Eqs.,19, No. 2, 214?225 (1975). · Zbl 0342.34001 [69] R. Conti, ?Equazioni differenziali ordinarie quasilineari con condizioni lineari,? Ann. Mat. Pura ed Appl.,57, 49?61 (1962). · Zbl 0105.29402 [70] 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). · Zbl 0154.09101 [71] M. Fukuhara, ?Sur une généralisation d’un thórème de Kneser,? Proc. Jap. Acad.,29, 154?155 (1953). · Zbl 0051.29704 [72] R. E. Gaines and J. Mawhin, ?Ordinary differential equations with nonlinear boundary conditions,? J. Diff. Eqs.,26, No. 2, 200?222 (1977). · Zbl 0326.34021 [73] I. T. Kiguradze, ?On a singular problem of Cauchy-Nicoletti,? Ann. Mat. Pura ed Appl.,104, 151?175 (1975). · Zbl 0307.34003 [74] I. T. Kiguradze, ?On the modified problem of Cauchy-Nicoletti,? Ann. Mat. Pura ed Appl.,104, 177?186 (1975). · Zbl 0307.34004 [75] 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). · JFM 49.0302.03 [76] J. Kurzweil, ?Generalized ordinary differential equations and continuous dependence on a parameter,? Czechosl. Mat. J.,7, No. 3, 418?449 (1957). · Zbl 0090.30002 [77] J. Kurzweil, ?Generalized ordinary differential equations,? Czechosl. Mat. J.,8, No. 3, 360?388 (1958). [78] 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). · Zbl 0286.34025 [79] 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). · Zbl 0309.34012 [80] A. Lasota and C. Olech, ?An optimal solution of Nicoletti’s boundary-value problem,? Ann. Pol. Math.,18, No. 2, 131?139 (1966). · Zbl 0144.10301 [81] A. Lasota and Z. Opial, ?Sur les solutions périodiques des équations différentielles ordinaires,? Ann. Pol. Math.,16, No. 1, 69?94 (1964). · Zbl 0142.35303 [82] Z. Opial, ?Linear problems for systems of nonlinear differential equations,? J. Diff. Eqs.,3, No. 4, 580?594 (1967). · Zbl 0161.06102 [83] K. Schmitt, ?Periodic solutions of nonlinear differential systems,? J. Math. Anal. and Appl.,40, No. 1, 174?182 (1972). · Zbl 0215.44402 [84] ?. Schwabik and M. Tvrdy, ?Boundary-value problems for generalized linear differential equations,? Czechosl. Math. J.,29, No. 3, 451?477 (1979). [85] ?. Schwabik, M. Tvrdy, and O. Veivoda, Differential and Integral Equations: Boundary-Value Problems and Adjoints, Academia, Praha (1979). [86] S. Sedziwy, ?Periodic solutions of a system of nonlinear differential equations,? Proc. Amer. Math. Soc.,48, No. 2, 328?336 (1975). · Zbl 0331.34039 [87] B. L. Shekhter, ?On singular boundary-value problems for two-dimensional differential systems,? Arch. Math.,19, No. 1, 19?41 (1983). · Zbl 0535.34010 [88] Z. Vorel, ?Continuous dependence on parameters,? Nonlinear Anal., Theory, Meth., and Appl.,5, No. 4, 373?380 (1981). · Zbl 0462.34046 [89] 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). · Zbl 0228.34016
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