On a Numerical Method for Constructing a Positive Solution of the Two-Point Boundary-Value Problem for a Second-Order Nonlinear Differential Equation

E. I. Abduragimov

Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 803-812.

Résumé

An iterative method is proposed for finding an approximation to the positive solution of the two-point boundary-value problem $$ y''+c(x)y^m=0,\quad 01,\qquad y(0)=y(1)=0, $$ where $m=\mathrm{const}>1$ and $c(x)$ is a continuous nonnegative function on $[0,1]$. The convergence of this method is proved. An error estimate is also obtained.

Détail

Keywords: second-order nonlinear differential equation, two-point boundary-value problem, elliptic differential equation, Cauchy problem, Green function.

Bibliographie

[1] M. A. Krasnoselskii, Polozhitelnye resheniya operatornykh uravnenii. Glavy nelineinogo analiza, Sovremennye problemy matematiki, Fizmatgiz, M., 1962 | MR | Zbl

[2] M. A. Krasnoselskii, P. P. Zabreiko, Geometricheskie metody nelineinogo analiza, Nelineinyi analiz i ego prilozheniya, Nauka, M., 1975 | MR | Zbl

[3] S. I. Pokhozhaev, “Ob odnoi zadache L. V. Ovsyannikova”, Zhurn. prikl. mekhaniki i tekhn. fiziki, 1989, no. 2, 5–10 | MR

[4] S. I. Pokhozhaev, “Ob odnom konstruktivnom metode variatsionnogo ischisleniya”, DAN SSSR, 298:6 (1988), 1330–1333 | MR | Zbl

[5] S. I. Pokhozhaev, “O tselykh radialnykh resheniyakh nekotorykh kvazilineinykh ellipticheskikh uravnenii”, Matem. sb., 183:11 (1992), 3–18 | MR | Zbl

[6] K. Schmitt, “Boundary value problems for quasilinear second-order elliptic equations”, Nonlinear Anal., 2:3 (1978), 263–309 | MR | Zbl

[7] T. S. Kozhevnikova, “O razreshimosti nekotorykh nelineinykh ellipticheskikh zadach”, Matem. zametki, 38:6 (1985), 852–859 | MR | Zbl

[8] E. I. Galakhov, “Polozhitelnye resheniya kvazilineinogo ellipticheskogo uravneniya”, Matem. zametki, 78:2 (2005), 202–211 | MR | Zbl

[9] K.-S. Cheng, J.-T. Lin, “On the elliptic equations $\Delta u=K(x)u^\sigma$ and $\Delta u=K(x)e^{2u}$”, Trans. Amer. Math. Soc., 304:2 (1987), 639–668 | MR | Zbl

[10] Yu. L. Gaponenko, “O polozhitelnykh resheniyakh nelineinykh kraevykh zadach”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem., kibernet., 1983, no. 4, 8–12 | MR | Zbl

[11] C. Bandle, M. K. Kwong, “Semilinear elliptic problems in annular domains”, Z. Angew. Math. Phys., 40:2 (1989), 245–257 | DOI | MR | Zbl

[12] N. Kawano, J. Satsuma, S. Yotsutani, “On the positive solutions of an Emden-type elliptic equation”, Proc. Japan Acad. Ser. A Math. Sci., 61:6 (1985), 186–189 | MR | Zbl

[13] M. K. Kwong, Y. Li, “Uniqueness of radial solutions of semilinear elliptic equations”, Trans. Amer. Math. Soc., 333:1 (1992), 339–363 | DOI | MR | Zbl

[14] E. I. Abduragimov, “O edinstvennosti polozhitelnogo resheniya odnoi nelineinoi dvukhtochechnoi kraevoi zadachi”, Izv. vuzov. Matem., 2002, no. 6, 3–6 | MR | Zbl

[15] E. I. Abduragimov, “Edinstvennost polozhitelnogo resheniya odnoi nelineinoi dvukhtochechnoi kraevoi zadachi i chislennyi metod ego nakhozhdeniya”, Izv. vuzov. Matem., 1998, no. 11, 3–7 | MR | Zbl

[16] B. Gidas, J. Spruck, “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm. Pure Appl. Math., 34:4 (1981), 525–598 | DOI | MR | Zbl

[17] R. Kajikiya, “Norm estimates for radially symmetric solutions of semilinear elliptic equations”, Trans. Amer. Math. Soc., 374:4 (1995), 1163–1199 | DOI | MR | Zbl

[18] P. Donato, “Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains”, Manuscripta Math., 65:2 (1989), 147–165 | DOI | MR | Zbl

[19] C. Cosner, C. Schmitt, “A priori bounds for positive solutions of a semilinear elliptic equation”, Proc. Amer. Math. Soc., 95:1 (1985), 47–50 | MR | Zbl

[20] R. Gardner, L. Peletier, “The set of positive solutions of semilinear equations in large balls”, Proc. Roy. Soc. Edinburgh Sect. A, 104:1-2 (1986), 52–72 | MR | Zbl