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Positive solutions of superlinear semipositone singular Dirichlet boundary value problems. (English) Zbl 1097.34019
This interesting paper is devoted to the existence of positive solutions for the semipositone Dirichlet boundary value problem $$u''+f(t,u)+q(t)=0, \quad t\in (0,1),\qquad u(0)=u(1)=0, $$ where $f:(0,1)\times[0,\infty)\to [0,\infty)$ is continuous $q:(0,1)\to (-\infty, \infty)$ is Lebesgue integrable, $f$ may be singular at $t=0,1$ and $q$ can have finitely many singularities. The authors obtain sufficient conditions which imply that the considered BVP has at least one $C[0,1]\cap C^2(0,1)$ positive solution. The proof is based on the properties of the fixed-point index for positive completely continuous operators in a cone.

34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47H07Monotone and positive operators on ordered topological linear spaces
Full Text: DOI
[1] Erbe, L. H.; Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. amer. Math. soc. 120, 743-748 (1994) · Zbl 0802.34018
[2] Zhao, Z.: A necessary and sufficient condition for the existence of positive solutions to singular sublinear boundary value problems. Acta math. Sinica 41, 1025-1034 (1998) · Zbl 1022.34015
[3] Wei, Z.: Positive solutions of singular sublinear second order boundary value problems. J. systems sci. Math. sci. 10, 82-88 (1998) · Zbl 0902.34020
[4] Mao, A.; Xue, M.: Positive solutions of singular boundary value problems. Acta math. Sinica 44, 899-908 (2001) · Zbl 1024.34018
[5] Zhang, Y.: Positive solutions of singular enden -- Fowler boundary value problems. J. math. Anal. appl. 185, 215-222 (1994) · Zbl 0823.34030
[6] Ma, R.: Positive solutions of singular second order boundary value problems. Acta math. Sinica (Suppl.) 14, 691-698 (1998) · Zbl 0926.34011
[7] Aris, R.: Introduction to the analysis of chemical reactors. (1965)
[8] Wang, J.; Gao, W.: A note on singular nonlinear two-point boundary value problems. Nonlinear anal. 39, 281-287 (2000) · Zbl 0942.34018
[9] Agarwal, R. P.; O’regan, D.: A note on existence of nonnegative solutions to singular semi-positone problems. Nonlinear anal. 36, 615-622 (1999) · Zbl 0921.34027
[10] Ma, R.; Wang, R.; Ren, L.: Existence results for semipositone boundary value problems. Acta math. Sci. 21B, 189-195 (2001) · Zbl 1021.34017
[11] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cone. (1988) · Zbl 0661.47045