| Title:
|
On a two-point boundary value problem for second order singular equations (English) |
| Author:
|
Lomtatidze, A. |
| Author:
|
Torres, P. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
53 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
19-43 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
second order singular equation |
| Keyword:
|
two-point boundary value problem |
| Keyword:
|
solvability |
| MSC:
|
34B10 |
| MSC:
|
34B16 |
| MSC:
|
34B18 |
| idZBL:
|
Zbl 1023.34011 |
| idMR:
|
MR1961996 |
| . |
| Date available:
|
2009-09-24T10:58:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127778 |
| . |
| Reference:
|
[1] J. A. Ackroyd: On the laminar compressible boundary layer with stationary origin on a moving flat wall.Proc. Cambridge Phil. Soc. 63 (1967), 871–888. Zbl 0166.45804 |
| Reference:
|
[2] R. P. Agarwal and D. O’Regan: Singular boundary value problems for superlinear second order ordinary and delay differential equations.J. Differential Equations 130 (1996), 333–355. MR 1410892, 10.1006/jdeq.1996.0147 |
| Reference:
|
[3] J. E. Bouillet and S. M. Gomes: An equation with singular nonlinearity related to diffusion problems in one dimension.Quart. Appl. Math. 42 (1985), 395–402. MR 0766876, 10.1090/qam/766876 |
| Reference:
|
[4] J. V. Baxley: A singular nonlinear boundary value problem: membrane response of a spherical cap.SIAM J. Appl. Math. 48 (1988), 497–505. Zbl 0642.34014, MR 0941097, 10.1137/0148028 |
| Reference:
|
[5] L. E. Bobisud, D. O’Regan and W. D. Royalty: Solvability of some nonlinear boundary value problems.Nonlinear Anal. 12 (1988), 855–869. MR 0960631, 10.1016/0362-546X(88)90070-3 |
| Reference:
|
[6] A. J. Callegary and M. B. Friedman: An analytic solution of a nonlinear singular boundary value problem in the theory of viscous fluids.J. Math. Anal. Appl. 21 (1968), 510–529. MR 0224331, 10.1016/0022-247X(68)90260-6 |
| Reference:
|
[7] A. J. Callegary and A. Nachman: Some singular nonlinear differential equations arising in boundary layer theory.J. Math. Anal. Appl. 64 (1978), 96–105. MR 0478973, 10.1016/0022-247X(78)90022-7 |
| Reference:
|
[8] A. J. Callegary and A. Nachman: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids.SIAM J. Appl. Math. 38 (1980), 275–281. MR 0564014, 10.1137/0138024 |
| Reference:
|
[9] D. R. Dunninger and J. C. Kurtz: A priori bounds and existence of positive solutions for singular nonlinear boundary value problems.SIAM J. Math. Anal. 17 (1986), 595–609. MR 0838243, 10.1137/0517044 |
| Reference:
|
[10] J. A. Gatika, V. Oliker and P. Waltman: Singular nonlinear boundary value problems for second order ordinary differential equations.J. Differential Equations 79 (1989), 62–78. MR 0997609, 10.1016/0022-0396(89)90113-7 |
| Reference:
|
[11] Z. Guo: Solvability of some singular nonlinear boundary value problems and existence of positive radial solutions of some nonlinear elliptic problems.Nonlinear Anal. 16 (1991), 781–790. Zbl 0737.35024, MR 1097131, 10.1016/0362-546X(91)90083-D |
| Reference:
|
[12] J. Janus and A. Myjak: A generalized Emden-Fowler equation with a negative exponent.Nonlinear Anal. 23 (1994), 953–970. MR 1304238, 10.1016/0362-546X(94)90193-7 |
| Reference:
|
[13] P. Habets and F. Zanolin: Upper and lower solutions for a generalized Emden–Fowler equation.J. Math. Anal. Appl. 181 (1994), 684–700. MR 1264540, 10.1006/jmaa.1994.1052 |
| Reference:
|
[14] P. Habets and F. Zanolin: Positive solutions for a class of singular boundary value problems.Boll. Un. Mat. Ital. A 9 (1995), 273–286. MR 1336236 |
| Reference:
|
[15] I. T. Kiguradze: On some singular boundary value problems for nonlinear differential equations of the second order.Differentsial’nye Uravneniya 4 (1968), 1753–1773. (Russian) MR 0245893 |
| Reference:
|
[16] I. T. Kiguradze and A. G. Lomtatidze: On certain boundary value problems for second order linear ordinary differential equations with singularities.J. Math. Anal. Appl. 101 (1984), 325–347. MR 0748576, 10.1016/0022-247X(84)90107-0 |
| Reference:
|
[17] I. T. Kiguradze and B. L. Shekhter: 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. MR 0925830 |
| Reference:
|
[18] Yu. A. Klokov and A. I. Lomakina: One a boundary value problem with singularities on the ends of the segment.Differ. Equations Latv. Mat. Ezhegodnik 17 (1976), 179–186. |
| Reference:
|
[19] A. Lomtatidze: Positive solutions of boundary value problems for second order ordinary differential equations with singular points.Differentsial’nye Uravneniya 23 (1987), 1685–1692. MR 0928850 |
| Reference:
|
[20] A. Lomtatidze: Existence of conjugate points for second order linear differential equations.Georgian Math. J. 2 (1995), 93–98. Zbl 0820.34018, MR 1310503, 10.1007/BF02257736 |
| Reference:
|
[21] C. D. Luning and W. L. Perry: Positive solutions of negative exponent generalized Emden–Fowler boundary value problems.SIAM J. Math. Anal. 12 (1981), 874–879. MR 0635240, 10.1137/0512073 |
| Reference:
|
[22] N. F. Morozov: On analytic structure of a solution of the membrane equation.Dokl. Akad. Nauk SSSR 152 (1963), 78–80. |
| Reference:
|
[23] N. F. Morozov and L. S. Srubshchik: Application of Chaplygin’s method to investigation of the membrane equation.Differentsial’nye Uravneniya 2 (1966), 425–427. (Russian) |
| Reference:
|
[24] L. S. Srubshchik and V. I. Yudovich: Asymptotics of equation of large deflection of circular symmetrically loaded plate.Sibirsk. Mat. Zh. 4 (1963), 657–672. (Russian) |
| Reference:
|
[25] S. Taliaferro: A nonlinear singular boundary value problem.Nonlinear Anal. 3 (1979), 897–904. Zbl 0421.34021, MR 0548961, 10.1016/0362-546X(79)90057-9 |
| Reference:
|
[26] A. Tineo: Existence theorems for a singular two-point Dirichlet problem.Nonlinear Anal. 19 (1992), 323–333. Zbl 0900.34019, MR 1178406, 10.1016/0362-546X(92)90177-G |
| Reference:
|
[27] J. Wang: Solvability of singular nonlinear two-point boundary value problems.Nonlinear Anal. 24 (1995), 555–561. Zbl 0876.34017, MR 1315694, 10.1016/0362-546X(95)93091-H |
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