A generalized Emden-Fowler equation with a negative exponent. (English) Zbl 0819.34016

The existence of a unique positive solution to the BVP \(x'' + f(t)x^{- \sigma} = h(t)\), \(0 < t < 1\), \(x(0) = a\), and either (1) \(x(1) = b\), or \(x'(1) = c\), or \(\alpha x'(1) + \beta x(1) = c\), is established under the assumptions that \(f,h \in C(0,1)\), \(f(t) > 0\), \(t \in (0,1)\), \(G > 0\), \(a \geq 0\), \(b \geq 0\) and, in case (1), \(\int^ 1_ 0 t(1-t) (f(t) + | h(t) |) dt < \infty\). The other cases require more complicated assumptions.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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[1] Luning, C. D.; Perry, W. L., Positive solutions of negative exponent generalized Emden-Fowler boundary value problems, SIAM J. math. Analysis, 12, 874-879 (1981) · Zbl 0478.34021
[2] Wong, J. S.W., On the generalized Emden-Fowler equation, SIAM Rev., 17, 339-360 (1975) · Zbl 0295.34026
[3] Nachman, A.; Callegari, A., A nonlinear boundary value problem in the theory of pseudoplastic fluids, SIAM J. appl. Math., 40, 275-281 (1980) · Zbl 0453.76002
[4] Callegari, A.; Nachman, A., Some singular nonlinear differential equations arising in boundary layer theory, J. math. Analysis Applic., 64, 96-105 (1978) · Zbl 0386.34026
[5] Whitman, G. B., Linear and Nonlinear Waves (1973), Wiley-Interscience: Wiley-Interscience New York
[6] Carelman, T., Problémes mathématiques dans la théorie cinétique de gas (1957), Almquist-Wiksells: Almquist-Wiksells Uppsala
[7] Baxley, J. V., A singular nonlinear boundary value problem: membrane response of a spherical cap, SIAM J. appl. Math., 48, 497-505 (1988) · Zbl 0642.34014
[8] Goldberg, M. A., An iterative solution for rotationally symmetric nonlinear membrane problems, Int. J. Non-Linear Mech., 1, 169-178 (1966) · Zbl 0143.45901
[9] Na, Y., Computational Methods in Engineering Boundary Value Problems (1979), Academic Press: Academic Press New York · Zbl 0456.76002
[10] Na, T. Y.; Kurajian, G. M.; Chiou, J. P., General solution of the problem of large deflection of an annular membrane under pressure, Aeronaut. Q. London, 27, 195-200 (1976)
[11] Nachman, A.; Taliaferro, S., Mass transfer into boundary layers for power law fluids, Proc. R. Soc. Lond. A, 365, 313-326 (1979) · Zbl 0414.76003
[12] O’Regan, D., Positive solutions to singular and non-singular second-order boundary value problem, J. math. Analysis Applic., 142, 40-52 (1989) · Zbl 0689.34015
[13] Perrone, N.; Kao, R., A general nonlinear relaxation iteration technique for solving nonlinear problems in mechanics, J. Appl. Mech., 38, 371-378 (1971)
[14] Pifko, A. B.; Goldberg, M. A., Iterative and power series solutions for the large deflection of an annular membrane, AIAA J., 2, 1340-1342 (1964)
[15] Schmitt, K., Boundary value problems for quasilinear second order elliptic equations, Nonlinear Analysis, 2, 263-309 (1978) · Zbl 0378.35022
[16] Taliaferro, S., A nonlinear singular boundary value problem, Nonlinear Analysis, 3, 897-904 (1979) · Zbl 0421.34021
[17] Van Duijn, C. J.; Gomes, S. M.; Hongfei, Z., On a class of similarity solutions of the equation \(u_t\) = (|\(u|^{m−1}u_{x\) · Zbl 0701.35090
[18] Bobisud, L. E.; O’Regan, D.; Royalty, W. D., Solvability of some nonlinear boundary value problems, Nonlinear Analysis, 12, 855-869 (1988) · Zbl 0653.34015
[19] Granas, A.; Guenther, R. B.; Lee, J. W., Nonlinear boundary value problems for ordinary differential equations, Diss. Math. (1985) · Zbl 0476.34017
[20] Gatica, J. A.; Oliker, V.; Waltman, P., Singular nonlinear boundary value problems for second-order ordinary differential equations, J. diff. Eqns, 79, 62-78 (1989) · Zbl 0685.34017
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