# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 47H07 Monotone and positive operators on ordered topological linear spaces
Full Text:
##### References:
 [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