On a two-point boundary value problem for second order singular equations

Alexander Lomtatidze; P. Torres

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 1, page 19-43
  • ISSN: 0011-4642

Abstract

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The problem on the existence of a positive in the interval ] a , b [ solution of the boundary value problem u ' ' = f ( t , u ) + g ( t , u ) u ' ; u ( a + ) = 0 , u ( b - ) = 0 is considered, where the functions f and g ] a , b [ × ] 0 , + ∞ [ → ℝ satisfy the local Carathéodory conditions. The possibility for the functions f and g to have singularities in the first argument (for t = a and t = b ) and in the phase variable (for u = 0 ) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.

How to cite

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Lomtatidze, Alexander, and Torres, P.. "On a two-point boundary value problem for second order singular equations." Czechoslovak Mathematical Journal 53.1 (2003): 19-43. <http://eudml.org/doc/30756>.

@article{Lomtatidze2003,
abstract = {The problem on the existence of a positive in the interval $\mathopen ]a,b\mathclose [$ solution of the boundary value problem \[ u^\{\prime \prime \}=f(t,u)+g(t,u)u^\{\prime \};\quad u(a+)=0, \quad u(b-)=0 \] is considered, where the functions $f$ and $g\:\mathopen ]a,b\mathclose [\times \mathopen ]0,+\infty \mathclose [ \rightarrow \mathbb \{R\}$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.},
author = {Lomtatidze, Alexander, Torres, P.},
journal = {Czechoslovak Mathematical Journal},
keywords = {second order singular equation; two-point boundary value problem; solvability; second-order singular equation; two-point boundary value problem; solvability},
language = {eng},
number = {1},
pages = {19-43},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a two-point boundary value problem for second order singular equations},
url = {http://eudml.org/doc/30756},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Lomtatidze, Alexander
AU - Torres, P.
TI - On a two-point boundary value problem for second order singular equations
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 1
SP - 19
EP - 43
AB - The problem on the existence of a positive in the interval $\mathopen ]a,b\mathclose [$ solution of the boundary value problem \[ u^{\prime \prime }=f(t,u)+g(t,u)u^{\prime };\quad u(a+)=0, \quad u(b-)=0 \] is considered, where the functions $f$ and $g\:\mathopen ]a,b\mathclose [\times \mathopen ]0,+\infty \mathclose [ \rightarrow \mathbb {R}$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.
LA - eng
KW - second order singular equation; two-point boundary value problem; solvability; second-order singular equation; two-point boundary value problem; solvability
UR - http://eudml.org/doc/30756
ER -

References

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  1. On the laminar compressible boundary layer with stationary origin on a moving flat wall, Proc. Cambridge Phil. Soc. 63 (1967), 871–888. (1967) Zbl0166.45804
  2. 10.1006/jdeq.1996.0147, J.  Differential Equations 130 (1996), 333–355. (1996) MR1410892DOI10.1006/jdeq.1996.0147
  3. 10.1090/qam/766876, Quart. Appl. Math. 42 (1985), 395–402. (1985) MR0766876DOI10.1090/qam/766876
  4. 10.1137/0148028, SIAM J.  Appl. Math. 48 (1988), 497–505. (1988) Zbl0642.34014MR0941097DOI10.1137/0148028
  5. 10.1016/0362-546X(88)90070-3, Nonlinear Anal. 12 (1988), 855–869. (1988) MR0960631DOI10.1016/0362-546X(88)90070-3
  6. 10.1016/0022-247X(68)90260-6, J.  Math. Anal. Appl. 21 (1968), 510–529. (1968) MR0224331DOI10.1016/0022-247X(68)90260-6
  7. 10.1016/0022-247X(78)90022-7, J.  Math. Anal. Appl. 64 (1978), 96–105. (1978) MR0478973DOI10.1016/0022-247X(78)90022-7
  8. 10.1137/0138024, SIAM J.  Appl. Math. 38 (1980), 275–281. (1980) MR0564014DOI10.1137/0138024
  9. 10.1137/0517044, SIAM J.  Math. Anal. 17 (1986), 595–609. (1986) MR0838243DOI10.1137/0517044
  10. 10.1016/0022-0396(89)90113-7, J.  Differential Equations 79 (1989), 62–78. (1989) MR0997609DOI10.1016/0022-0396(89)90113-7
  11. 10.1016/0362-546X(91)90083-D, Nonlinear Anal. 16 (1991), 781–790. (1991) Zbl0737.35024MR1097131DOI10.1016/0362-546X(91)90083-D
  12. 10.1016/0362-546X(94)90193-7, Nonlinear Anal. 23 (1994), 953–970. (1994) MR1304238DOI10.1016/0362-546X(94)90193-7
  13. 10.1006/jmaa.1994.1052, J.  Math. Anal. Appl. 181 (1994), 684–700. (1994) MR1264540DOI10.1006/jmaa.1994.1052
  14. Positive solutions for a class of singular boundary value problems, Boll. Un. Mat. Ital.  A 9 (1995), 273–286. (1995) MR1336236
  15. On some singular boundary value problems for nonlinear differential equations of the second order, Differentsial’nye Uravneniya 4 (1968), 1753–1773. (Russian) (1968) MR0245893
  16. 10.1016/0022-247X(84)90107-0, J.  Math. Anal. Appl. 101 (1984), 325–347. (1984) MR0748576DOI10.1016/0022-247X(84)90107-0
  17. Singular boundary value problems for second order ordinary differential equations, In: Curent Problems in Mathematics: Newest Results, Vol. 3, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 3-103. (1987) MR0925830
  18. One a boundary value problem with singularities on the ends of the segment, Differ. Equations Latv. Mat. Ezhegodnik 17 (1976), 179–186. (1976) 
  19. Positive solutions of boundary value problems for second order ordinary differential equations with singular points, Differentsial’nye Uravneniya 23 (1987), 1685–1692. (1987) MR0928850
  20. 10.1007/BF02257736, Georgian Math. J. 2 (1995), 93–98. (1995) Zbl0820.34018MR1310503DOI10.1007/BF02257736
  21. 10.1137/0512073, SIAM J.  Math. Anal. 12 (1981), 874–879. (1981) MR0635240DOI10.1137/0512073
  22. On analytic structure of a solution of the membrane equation, Dokl. Akad. Nauk SSSR 152 (1963), 78–80. (1963) 
  23. Application of Chaplygin’s method to investigation of the membrane equation, Differentsial’nye Uravneniya 2 (1966), 425–427. (Russian) (1966) 
  24. Asymptotics of equation of large deflection of circular symmetrically loaded plate, Sibirsk. Mat. Zh. 4 (1963), 657–672. (Russian) (1963) 
  25. 10.1016/0362-546X(79)90057-9, Nonlinear Anal. 3 (1979), 897–904. (1979) Zbl0421.34021MR0548961DOI10.1016/0362-546X(79)90057-9
  26. 10.1016/0362-546X(92)90177-G, Nonlinear Anal. 19 (1992), 323–333. (1992) Zbl0900.34019MR1178406DOI10.1016/0362-546X(92)90177-G
  27. 10.1016/0362-546X(95)93091-H, Nonlinear Anal. 24 (1995), 555–561. (1995) Zbl0876.34017MR1315694DOI10.1016/0362-546X(95)93091-H

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