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Singular boundary-value problems for ordinary second-order differential equations

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Abstract

This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)

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Literature cited

  1. I. V. Amirkhanov and E. P. Zhidkov, “Some questions of existence and qualitative behavior of particle-like solutions,” Közp. fiz. kut. intéz. [Publ.], No. 82, 165–180 (1979).

    Google Scholar 

  2. N. K. Balabaev, V. D. Lakhno, and A. M. Molchanov, “Excited self-consistent electron states in homeopolar crystals,” Preprint, Scientific Center for Biological Research, Akad. Nauk SSSR, Pushchino (1983).

    Google Scholar 

  3. S. N. Bernshtein, “On the equations of the calculus of variations,” Usp. Mat. Nauk,8, No. 1, 32–74 (1940).

    Google Scholar 

  4. 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).

    Google Scholar 

  5. N. I. Vasil'ev and A. I. Lomakina, “On a two-point boundary-value problem with nonsummable singularity,” Differents. Uravn.,14, No. 2, 195–200 (1978).

    Google Scholar 

  6. G. D. Gaprindashvili, “On a boundary-value problem for systems of nonlinear ordinary differential equations with singularities,” Differents. Uravn.,20, No. 9, 1514–1523 (1984).

    Google Scholar 

  7. N. V. Gogiberidze and I. T. Kiguradze, “On the question of the nonoscillatory character of singular linear second-order differential equations,” Differents. Uravn.,10, No. 11, 2064–2067 (1974).

    Google Scholar 

  8. V. V. Gudkov, Yu. A. Klokov, A. Ya. Lepin, and V. D. Ponomarev, Two-Point Boundary-Value Problems for Ordinary Differential Equations [in Russian], Zinatne, Riga (1973).

    Google Scholar 

  9. E. P. Zhidkov and V. P. Shirikov, “On a boundary-value problem for ordinary second-order differential equations,” J. Vych. Mat. i Mat. Fiz.,4, No. 5, 804–816 (1964).

    Google Scholar 

  10. G. G. Kvinikadze, “On a singular boundary-value problem for nonlinear ordinary differential equations,” In: Ninth International Conference on Nonlinear Oscillations,” Vol. 1, 166–168, Naukova Dumka, Kiev (1984).

    Google Scholar 

  11. I. T. Kiguradze, “On the Cauchy problem for ordinary differential equations with singularities,” Soobshch. Akad. Nauk Gruz. SSR,37, No. 1, 19–24 (1965).

    Google Scholar 

  12. I. T. Kiguradze, “On the Cauchy problem for singular systems of ordinary differential equations,” Differents. Uravn.,1, No. 10, 1271–1291 (1965).

    Google Scholar 

  13. I. T. Kiguradze, “On a priori estimates of the derivatives of bounded functions satisfying second-order differential inequalities,” Differents. Uravn.,3, No. 7, 1043–1052 (1967).

    Google Scholar 

  14. I. T. Kiguradze, “On some singular boundary-value problems for nonlinear ordinary second-order differential equations,” Differents. Uravn.,4, No. 10, 1753–1773 (1968).

    Google Scholar 

  15. I. T. Kiguradze, “On a singular two-point boundary-value problem,” Differents. Uravn.,5, No. 11, 2002–2016 (1969).

    Google Scholar 

  16. I. T. Kiguradze, “On nonoscillating conditions for singular linear second-order differential equations,” Mat. Zametki,6, No. 5, 633–639 (1969).

    Google Scholar 

  17. I. T. Kiguradze, “On monontonic solutions ofnth-order nonlinear ordinary differential equations,” Izv. Akad. Nauk SSSR, Ser. Mat.,33, No. 6, 1373–1398 (1969).

    Google Scholar 

  18. I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University Press, Tbilisi (1975).

    Google Scholar 

  19. I. T. Kiguradze, “On the solvability of the Vallée-Poussin boundary-value problem,” Differents. Uravn.,21, No. 3, 391–397 (1985).

    Google Scholar 

  20. I. T. Kiguradze and I. Rachůnková, “On the solvability of a nonlinear problem of Kneser type,” Differents. Uravn.,15, No. 10, 1754–1765 (1979).

    Google Scholar 

  21. Yu. A. Klokov, “A method of solving a limiting boundary-value problem for a second-order ordinary differential equation,” Mat. Sb.,53, No. 2, 219–232 (1961).

    Google Scholar 

  22. Yu. A. Klokov, Boundary-Value Problems with Condition at Infinity for the Equations of Mathematical Physics [in Russian], RIIGVF, Riga (1963).

    Google Scholar 

  23. Yu. A. Klokov and A. I. Lomakina, “On a boundary-value problem with singularities at the endpoints of the interval,” Latv. Mat. Yearbook,17, 179–186 (1976).

    Google Scholar 

  24. N. L. Korshikova, “On the zeros of solutions of higher-order linear equations,” In: Differential Equations and Their Applications, 143–148, Moscow State University Press, Moscow (1984).

    Google Scholar 

  25. N. L. Korshikova, “On the zeros of the solutions of a class ofnth-order linear equations,” Differents. Uravn.,21, No. 5, 757–764 (1985).

    Google Scholar 

  26. M. A. Krasnosel'skii, “On a boundary-value problem,” Izv. Akad. Nauk SSSR, Ser. Mat.,20, No. 2, 241–252 (1956).

    Google Scholar 

  27. M. A. Krasnosel'skii and M. G. Krein, “On the averaging principle in nonlinear mechanics,” Usp. Mat. Nauk,10, No. 3, 147–152 (1955).

    Google Scholar 

  28. M. A. Krasnosel'skii, A. I. Perov, A. I. Povolotskii, and P. P. Zabreiko, Plane Vector Fields, Academic Press, New York (1966).

    Google Scholar 

  29. A. Ya. Lepin, “Existence of a solution of a nonlinear boundary-value problem for annth-order ordinary differential equation with a singularity at the endpoints,” Latv. Mat. Yearbook,4, 215–230 (1968).

    Google Scholar 

  30. L. A. Lepin, “Generalized solution and solvability of boundary-value problems for a second-order differential equation,” Differents. Uravn.18, No. 8, 1323–1330 (1982).

    Google Scholar 

  31. L. A. Lepin, “The method of lower and upper functions for second-order differential equations on open and half-open intervals,” Proceedings of the Extended Sessions of the Seminar of the Vekua Institute for Applied Mathematics,1, No. 3, 81–84 (1985).

    Google Scholar 

  32. A. A. Logunov and A. A. Vlasov, Minkowski Space as the Foundation of the Physical Theory of Gravitation, Moscow State University Press, Moscow (1984).

    Google Scholar 

  33. A. G. Lomtatidze, “On a boundary-value problem for linear differential equations with nonintegrable singularities,” Proc. Vekua Inst. Appl. Mat.,14, 136–144, (1983).

    Google Scholar 

  34. A. G. Lomtatidze, “On a singular boundary-value problem for linear second-order differential equations,” In: Boundary-Value Problems, Perm. Polyt. Inst., 46–50 (1984).

  35. A. G. Lomtatidze, “On solvability of boundary-value problems for nonlinear second-order ordinary differential equations with singularities,” Proceedings of the Extended Sessions of the Seminar of the Vekua Institute for Applied Mathematics,1, No. 3, 85–92 (1985).

    Google Scholar 

  36. A. G. Lomtatidze, “On oscillation properties of solutions of second-order linear differential equations,” Proceedings of the Seminar of the Vekua Institute of Applied Mathematics,19, 39–53 (1985).

    Google Scholar 

  37. A. G. Lomtatidze, “On a boundary-value problem for second-order nonlinear ordinary differential equations with singularities,” Differents. Uravn.,22, No. 3, 416–426 (1986).

    Google Scholar 

  38. A. G. Lomtatidze, “On a singular three-point boundary-value problem,” Proc. Vekua Inst. Appl. Mat.,17, 122–134 (1986).

    Google Scholar 

  39. A. G. Lomtatidze, “On positive solutions of singular boundary-value problems for nonlinear second-order ordinary differential equations,” Differents. Uravn.,22, No. 6, p. 1092 (1986).

    Google Scholar 

  40. E. I. Moiseev and V. A. Sadovnichii, On the Solution of a Nonlinear Equation in the Theory of Gravitation Based on Minkowski Space [in Russian], Moscow State University Press, Moscow (1984).

    Google Scholar 

  41. E. I. Moiseev and V. A. Sadovnichii, “A study of the solution of a nonlinear equation of the theory of gravitation,” Dokl. Akad. Nauk SSSR282, No. 4, 845–847 (1985).

    Google Scholar 

  42. E. I. Moiseev and V. A. Sadovnichii, “The solution of a boundary-value problem for a nonlinear eqaution of the theory of gravitation,” Dokl. Akad. Nauk SSSR,284, No. 4, 835–837 (1985).

    Google Scholar 

  43. E. I. Moiseev and V. A. Sadovnichii, On Boundary-Value Problems for a Nonlinear Equation of the Theory of Gravitation, Moscow State University Press, Moscow (1986).

    Google Scholar 

  44. N. F. Morozov, “On the analytic structure of the solution of the membrane equation,” Dokl. Akad. Nauk SSSR,152, No. 1, 78–80 (1963).

    Google Scholar 

  45. N. F. Morozov and L. S. Srubshchik, “Application of Chaplygin's method to the study of the membrane equation,” Differents. Uravn.,2, No. 3, 425–427 (1966).

    Google Scholar 

  46. A. D. Myshkis and G. V. Shcherbina, “On a limiting boundary-value problem not satisfying the Bernshtein condition and have applications in the theory of capillary phenomena,” Differents. Uravn.12, No. 6, 991–998 (1976).

    Google Scholar 

  47. A. I. Perov, “On a two-point boundary-value problem,” Dokl. Akad. Nauk SSSR,122, No. 6, 982–985 (1958).

    Google Scholar 

  48. 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).

    Google Scholar 

  49. A. I. Perov, “On the singular Cauchy problem,” Proceedings of the Seminar on Functional Analysis of Voronezh University,7, 104–107 (1963).

    Google Scholar 

  50. I. Rachůnková, “On the Kneser problem for systems of nonlinear ordinary differential equations,” Soobshch. Akad. Nauk Gruz. SSR,94, No. 3, 545–548 (1979).

    Google Scholar 

  51. G. Sansone, “Equazione Differenziali nel Campo Reale,” Vol. 2, Zanichelli, Bologna (1949).

    Google Scholar 

  52. L. S. Srubshchik and V. I. Yudovich, “The asymptotics of the equation for large bending of a round symmetrically loaded lamina,” Sib. Mat. J.,4, No. 3, 657–672 (1963).

    Google Scholar 

  53. G. S. Tabidze, “On approximate solution of a two-point singular boundary-value problem for a second-order ordinary differential equation,” Differents. Uravn.,10, No. 5, 851–859 (1974).

    Google Scholar 

  54. G. S. Tabidze, “On numerical solution of a two-point singular boundary-value problem,” Proc. Vekua Inst. Appl. Mat.,17, 153–179 (1986).

    Google Scholar 

  55. E. L. Tonkov, “On a second-order periodic equation,” Dokl. Akad. Nauk SSSR,184, No. 2, 296–299 (1969).

    Google Scholar 

  56. Philip Hartman, Ordinary Differential Equations, Wiley, New York (1964).

    Google Scholar 

  57. Ya. V. Tsepitis, “Solvability of a boundary-value problem for a second-order ordinary differential equation with nonsummable singularity,” Differents. Uravn.,19, No. 12, 2071–2075 (1983).

    Google Scholar 

  58. T. A. Chanturiya, “On a problem of Kneser type for a system of ordinary differential equations,” Mat. Zametki,15, No. 6, 897–906 (1974).

    Google Scholar 

  59. V. A. Chechik, “A study of systems of ordinary differential equations with a singularity,” Tr. Mosk. Mat. Obshch.8, 155–198 (1959).

    Google Scholar 

  60. B. L. Shekhter, “On the number of solutions of a two-point boundary-value problem for a second-order equation with discontinuous right-hand side,” Differents. Uravn.,11, No. 3, 484–497 (1975).

    Google Scholar 

  61. B. L. Shekhter, “On unique solvability of a linear two-point boundary-value problem,” Differents. Uravn.,11, No. 4, 687–693 (1975).

    Google Scholar 

  62. B. L. Shekhter, “On a two-point boundary-value problem for second-order ordinary differential equations with discontinuous right-hand side,” Trudy Tbilis. Univ.,A9, 19–31 (1975).

    Google Scholar 

  63. V. P. Shirikov, “The Cauchy problem and a boundary-value problem for some nonlinear second-order ordinary differential equations,” Dokl. Akad. Nauk SSSR,163, No. 4, 834–836 (1965).

    Google Scholar 

  64. G. V. Shcherbina, “On a boundary-value problem encountered in applications for a second-order differential equation on the half-line,” Dokl. Akad. Nauk SSSR178, No. 2, 314–316 (1968).

    Google Scholar 

  65. G. V. Shcherbina, “Sufficient conditions for solvability of a nonlinear boundary-value problem on the half-line,” Differents. Uravn.,11, No. 12, 2189–2195 (1975).

    Google Scholar 

  66. G. V. Shcherbina, “On a singular nonlinear boundary-value problem for a second-order equation with rapidly-growing right-hand side,” Differents. Uravn.,12, No. 2, 299–304 (1976).

    Google Scholar 

  67. J. A. Ackroyd, “On the laminar compressible boundary layer with stationary origin on a moving flat wall,” Proc. Cambridge Phil. Soc.,63, 871–888 (1967).

    Google Scholar 

  68. P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two-Point Boundary-Value Problems, Academic Press, New York (1968).

    Google Scholar 

  69. J. E. Bouillet and S. M. Gomes, “An equation with a singular nonlinearity related to diffusion problems in one dimension,” Quart. Appl. Math.,42, No. 4, 395–402 (1985).

    Google Scholar 

  70. A. J. Callegary and M. B. Friedman, “An analytical solution of a nonlinear singular boundary-value problem in the theory of viscous fluids,” J. Math. Anal. and Appl.,21, No. 3, 510–529 (1968).

    Google Scholar 

  71. A. J. Callegary and A. Nachman, “Some singular, nonlinear differential equations arising in boundry-slayer theory,” J. Math. Anal. and Appl.,64, No. 1, 96–105 (1978).

    Google Scholar 

  72. C. V. Coffman, “Nonlinear differential equations on cones in Banach spaces,” Pacific J. Math.,14, No. 1, 9–15 (1964).

    Google Scholar 

  73. W. A. Coppel, “Disconjugacy,” Lect. Notes Math.,220 (1971).

  74. R. Emden, Gaskugeln, Leipzig (1907).

  75. H. Epheser, “Über die Existenz der Lösungen von Randwertaufgaben mit gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung,” Math. Z.,61, No. 4, 435–454 (1955).

    Google Scholar 

  76. E. Fermi, “Un metodo statistico per la determinazione di alcune proprietà dell'atomo,” Rend. R. Acc. Naz. dei Lincei,6, 602–607 (1927).

    Google Scholar 

  77. P. Hartman and A. Wintner, “On the nonincreasing solutions ofy″=f(x,y,y'),” Amer. J. Math.,73, No. 2, 390–404 (1951).

    Google Scholar 

  78. P. Hartman and A. Wintner, “On monotone solutions of systems of nonlinear differential equations,” Amer. J. Math.,76, No. 4, 860–886 (1954).

    Google Scholar 

  79. P. Jamet, “On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems,” Numerical Math.,14, 355–378 (1970).

    Google Scholar 

  80. W. E. Johnson and L. M. Perko, “Interior and exterior boundary-value problems from the theory of the capillary tube,” Arch. Ration. Mech. and Anal.,29, No. 2, 125–143 (1968).

    Google Scholar 

  81. I. T. Kiguradze, “On the nonnegative nonincreasing solutions of nonlinear second-order differential equations,” Ann. Mat. Pura ed Appl.,81, 169–191 (1969).

    Google Scholar 

  82. I. T. Kiguradze, “On a singular boundary-value problem,” J. Math. Anal. and Appl.,30, No. 3, 475–489 (1970).

    Google Scholar 

  83. I. T. Kiguradze and A. G. Lomtatidze, “On certain boundary-value problems for second-order linear ordinary differential equations with singularities,” J. Math. Anal. and Appl.,101, No. 2, 325–347 (1984).

    Google Scholar 

  84. I. T. Kiguradze and I. Rachůnková, “On a certain nonlinear problem for two-dimensional differential systems,” Arch. Math. (Brno),16, No. 1, 15–37 (1980).

    Google Scholar 

  85. A. Kneser, “Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen be grossen reellen Werthen der Arguments. I,” J. reine und angew. Math.,116, 173–212 (1896).

    Google Scholar 

  86. I. I. Kolodner, “Heavy rotating string—a nonlinear eigenvalue problem,” Commun. Pure and Appl. Math.,8, No. 3, 395–408 (1955).

    Google Scholar 

  87. J. C. Kurtz, “A singular nonlinear boundary-value problem,” Rocky Mountain J. Math.,11, No. 2, 227–241, (1981).

    Google Scholar 

  88. M. Nagumo, “Über die Differentialgleichungy″=f(x,y,y'),” Proc. Phys.-Math. Soc. Japan,19, 861–866 (1937).

    Google Scholar 

  89. Z. Nehari, “On a nonlinear differential equation arising in nuclear physics,” Proc. Roy. Irish Acad.,62A, No. 9, 117–135 (1963).

    Google Scholar 

  90. Z. Opial, “Sur les intégrales bornées de l'équationu″=f(t,u,u'),” Ann. Polon. Math.,4, No. 3, 314–324 (1958).

    Google Scholar 

  91. Z. Opial, “Sur une inégalité de C. de la Vallée-Poussin dans la théorie de l'équation différentielle linéaire du second ordre,” Ann. Polon. Math.,6, No. 1, 87–91 (1959).

    Google Scholar 

  92. E. Picard, “Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires,” J. Math. Pures et Appl.,9, 217–271 (1893).

    Google Scholar 

  93. I. Rachůnková, “On a Kneser problem for a system of nonlinear ordinary differential equations,” Czech. Math. J.,31, No. 1, 114–126 (1981).

    Google Scholar 

  94. W. T. Reid, “Sturmian theory for ordinary differential equations,” Appl. Math. Sci.,31 (1980).

  95. G. H. Ryder, “Boundary-value problems for a class of nonlinear differential equations,” Pacif. J. Math.,22, No. 3, 477–503 (1967).

    Google Scholar 

  96. G. Sansone, “Su un'equazione differenziale non lineare delia fisica nucleare,” Symp. Math.,6, 3–139 (1970).

    Google Scholar 

  97. B. L. Shekhter, “On singular boundary-value problems for two-dimensional differential systems,” Arch. Math. (Brno),19, No. 1, 19–41 (1983).

    Google Scholar 

  98. B. L. Shekhter, “On existence and zeros of solutions of a nonlinear two-point boundary-value problem,” J. Math. Anal. and Appl.,97, No. 1, 1–20 (1983).

    Google Scholar 

  99. B. L. Shekhter, “On a boundary-value problem arising in nonlinear field theory,” Ann. Mat. Pura ed Appl. (in press) (1986).

  100. S. Taliaferro, “A nonlinear singular boundary-value problem,” Nonlinear Anal., Theory Meth. and Appl.,3, No. 6, 897–904 (1979).

    Google Scholar 

  101. L. H. Thomas, “The calculation of atomic fields,” Proc. Cambridge Phil. Soc.,23, 542–548 (1927).

    Google Scholar 

  102. L. Tonelli, “Sull'equazione differenzialey″=f(x,y,y'),” Ann. R. Scuola Norm. Super. Pisa, Sci. Fis. e Mat.,8, 75–88 (1939).

    Google Scholar 

  103. C. de la Vallée Poussin, “Sur l'équation différentielle linéaire du second order. Détermination d'une intégrale par deux valeurs assignées. Extension aux équations d'ordren,” J. Math. Pures et Appl.,8, No. 2, 125–144 (1929).

    Google Scholar 

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  1. I. T. Kiguradze
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  2. B. L. Shekhter
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105–201, 1987.

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Kiguradze, I.T., Shekhter, B.L. Singular boundary-value problems for ordinary second-order differential equations. J Math Sci 43, 2340–2417 (1988). https://doi.org/10.1007/BF01100361

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