# 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)
Multiplicity of positive solutions for singular three-point boundary value problems at resonance. (English) Zbl 1188.34032
The paper deals with the problem of existence of positive solutions for the three-point boundary value problem $$x''+m^{2}x=f(t,x)+e(t),\quad t\in (0,1),$$ with conditions $$x'(0)= 0, \qquad x(\eta)=x(1),$$ where $m \in (0,\frac{\pi}{2})$ and $\eta \in (0,1)$ are given. The problem is assumed to be with singular nonlinear perturbations at resonance. The proof is based on a nonlinear Leray-Schauder principle and the fixed point theorem in cones for completely continuous operators.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 34B16 Singular nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations 34B10 Nonlocal and multipoint boundary value problems for ODE
Full Text:
##### References:
 [1] Infante, G.; Webb, J. R. L.: Loss of positivity in a nonlinear scalar heat equation. Nodea nonlinear differential equations appl. 13, 249-261 (2006) · Zbl 1112.34017 [2] Il’in, V.; Moiseev, E.: Non-local boundary value problem of the first kind for a strum--Liouville operator in its differential and finite difference aspects. Differ. equ. 23, 803-810 (1987) [3] Gupta, C.: Existence theorems for a second order m-point boundary value problem at resonance. Int. J. Math. math. Sci. 18, 705-710 (1995) · Zbl 0839.34027 [4] Feng, M.; Ge, W.: Positive solutions for a class of m-point singular boundary value problems. Math. comput. Modelling 46, 375-383 (2007) · Zbl 1142.34012 [5] Gupta, C.: Solvability of a multi-point boundary value problem at resonance. Results math. 28, 270-276 (1995) · Zbl 0843.34023 [6] Khan, R. A.; Webb, J. R. L.: Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear anal. 64, 1356-1366 (2006) · Zbl 1101.34005 [7] Ma, R.: Positive solutions of some three-point boundary value problem. Electron. J. Differential equations 34, 1-8 (1999) · Zbl 0926.34009 [8] Webb, J. R. L.: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear anal. 47, 4319-4332 (2001) · Zbl 1042.34527 [9] Mawhin, J.: Topological degree and boundary value problems for nonlinear differential equations. Lecture notes in mathematics 1537, 74-142 (1993) · Zbl 0798.34025 [10] Zhang, Q.; Jiang, D.: Upper and lower solutions method and a second order three-point singular boundary value problem. Comput. math. Appl. 56, 1059-1070 (2008) · Zbl 1155.34305 [11] Han, X.: Positive solution for a three-point boundary value problem at resonance. J. math. Anal. appl. 336, 556-568 (2007) · Zbl 1125.34014 [12] Infante, G.; Zima, M.: Positive solutions of multi-point boundary value problems at resonance. Nonlinear anal. 69, 2458-2465 (2008) · Zbl 1203.34041 [13] Agarwal, R. P.; O’regan, D.: Existence theory for single and multiple solutions to singular positone boundary value problems. J. differential equations 175, 393-414 (2001) · Zbl 0999.34018 [14] Chu, J.; Nieto, J. J.: Impulsive periodic solutions of first-order singular differential equations. Bull. lond. Math. soc. 40, 143-150 (2008) · Zbl 1144.34016 [15] Chu, J.; Torres, P. J.; Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. differential equations 239, 196-212 (2007) · Zbl 1127.34023 [16] Del Pino, M. A.; Manásevich, R. F.; Montero, A.: T-periodic solutions for some second order differential equations with singularities. Proc. roy. Soc. Edinburgh sect. A 120, 231-243 (1992) · Zbl 0761.34031 [17] Jiang, D.; Chu, J.; Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. differential equations 211, 282-302 (2005) · Zbl 1074.34048 [18] Rachunková, I.; Tvrdý, M.; Vrkoc\breve{}, I.: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. differential equations 176, 445-469 (2001) · Zbl 1004.34008 [19] Torres, P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. differential equations 190, 643-662 (2003) · Zbl 1032.34040 [20] Torres, P. J.: Weak singularities May help periodic solutions to exist. J. differential equations 232, 277-284 (2007) · Zbl 1116.34036 [21] Zhang, M.: A relationship between the periodic and the Dirichlet equations. Proc. roy. Soc. Edinburgh sect. 128A, 1099-1114 (1998) · Zbl 0918.34025 [22] Gordon, W. B.: Conservative dynamical systems involving strong forces. Trans. amer. Math. soc. 204, 113-135 (1975) · Zbl 0276.58005 [23] O’regan, D.: Existence theory for nonlinear ordinary differential equations. (1997) [24] Krasnosel’skii, M. A.: Positive solution of operator equations. (1964) [25] Lü, H.; O’regan, D.; Agarwal, R. P.: Upper and lower solutions for the singular p-Laplacian with sign changing nonlinearities and nonlinear boundary data. J. comput. Appl. math. 181, 442-466 (2005)