Previous |  Up |  Next


second order singular equation; two-point boundary value problem; solvability
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.
[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
[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. DOI 10.1006/jdeq.1996.0147 | MR 1410892
[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
[4] J. V.  Baxley: A singular nonlinear boundary value problem: membrane response of a spherical cap. SIAM J.  Appl. Math. 48 (1988), 497–505. DOI 10.1137/0148028 | MR 0941097 | Zbl 0642.34014
[5] L. E.  Bobisud, D.  O’Regan and W. D.  Royalty: Solvability of some nonlinear boundary value problems. Nonlinear Anal. 12 (1988), 855–869. DOI 10.1016/0362-546X(88)90070-3 | MR 0960631
[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. DOI 10.1016/0022-247X(68)90260-6 | MR 0224331
[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. DOI 10.1016/0022-247X(78)90022-7 | MR 0478973
[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. DOI 10.1137/0138024 | MR 0564014
[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. DOI 10.1137/0517044 | MR 0838243
[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. DOI 10.1016/0022-0396(89)90113-7 | MR 0997609
[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. DOI 10.1016/0362-546X(91)90083-D | MR 1097131 | Zbl 0737.35024
[12] J.  Janus and A.  Myjak: A generalized Emden-Fowler equation with a negative exponent. Nonlinear Anal. 23 (1994), 953–970. DOI 10.1016/0362-546X(94)90193-7 | MR 1304238
[13] P.  Habets and F.  Zanolin: Upper and lower solutions for a generalized Emden–Fowler equation. J.  Math. Anal. Appl. 181 (1994), 684–700. DOI 10.1006/jmaa.1994.1052 | MR 1264540
[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
[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
[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. DOI 10.1016/0022-247X(84)90107-0 | MR 0748576
[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
[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.
[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
[20] A.  Lomtatidze: Existence of conjugate points for second order linear differential equations. Georgian Math. J. 2 (1995), 93–98. DOI 10.1007/BF02257736 | MR 1310503 | Zbl 0820.34018
[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. DOI 10.1137/0512073 | MR 0635240
[22] N. F.  Morozov: On analytic structure of a solution of the membrane equation. Dokl. Akad. Nauk SSSR 152 (1963), 78–80.
[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)
[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)
[25] S.  Taliaferro: A nonlinear singular boundary value problem. Nonlinear Anal. 3 (1979), 897–904. DOI 10.1016/0362-546X(79)90057-9 | MR 0548961 | Zbl 0421.34021
[26] A.  Tineo: Existence theorems for a singular two-point Dirichlet problem. Nonlinear Anal. 19 (1992), 323–333. DOI 10.1016/0362-546X(92)90177-G | MR 1178406 | Zbl 0900.34019
[27] J.  Wang: Solvability of singular nonlinear two-point boundary value problems. Nonlinear Anal. 24 (1995), 555–561. DOI 10.1016/0362-546X(95)93091-H | MR 1315694 | Zbl 0876.34017
Partner of
EuDML logo