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

Janus, J.; Myjak, J.

Nonlinear Anal., Theory Methods Appl. 23, No. 8, 953-970 (1994).

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.

Reviewer: W.Šeda (Bratislava)

MSC:

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

Keywords:

generalized Emden-Fowler equation; boundary value problems; numerical discussion; unique positive solution

References:

[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