# 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 nonlinear singular third-order two-point boundary value problem. (English) Zbl 1107.34019
Summary: We are concerned with the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem $$u'''(t)+ \lambda a(t)f\bigl(u(t)\bigr)=0,\quad 0<t<1,\quad u(0)=u'(0)=u''(1)=0,$$ where $\lambda$ is a positive parameter. Under various assumptions on $a$ and $f$, we establish intervals of the parameter $\lambda$ which yield the existence of at least one, at least two, and infinitely many positive solutions of the boundary value problem by using Krasnoselskii’s fixed-point theorem of cone expansion-compression type.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 34B24 Sturm-Liouville theory
Full Text:
##### References:
 [1] Agarwal, R. P.; Bohner, M.; Wong, P. J. Y.: Positive solutions and eigenvalues of conjugate boundary value problems. Proc. Edinburgh math. Soc. 42, 349-374 (1999) · Zbl 0934.34008 [2] Agarwal, R. P.; O’regan, D.; Wong, P. J. Y.: Positive solutions of differential, difference, and integral equations. (1999) [3] Anderson, D.: Multiple positive solutions for a three-point boundary value problem. Math. comput. Modelling 27, No. 6, 49-57 (1998) · Zbl 0906.34014 [4] Anderson, D.; Avery, R. I.: Multiple positive solutions to a third-order discrete focal boundary value problem. Comput. math. Appl. 42, 333-340 (2001) · Zbl 1001.39022 [5] Cabada, A.: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. J. math. Anal. appl. 185, 302-320 (1994) · Zbl 0807.34023 [6] Cabada, A.: The method of lower and upper solutions for third-order periodic boundary value problems. J. math. Anal. appl. 195, 568-589 (1995) · Zbl 0846.34019 [7] Cabada, A.; Grossinbo, M. R.; Minhos, F.: On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions. J. math. Anal. appl. 285, 174-190 (2003) [8] Cabada, A.; Heikkilä, S.: Extremality and comparison results for discontinuous third order functional initial-boundary value problems. J. math. Anal. appl. 255, 195-212 (2001) · Zbl 0976.34009 [9] Cabada, A.; Heikkilä, S.: Uniqueness, comparison and existence results for third order initial-boundary value problems. Comput. math. Appl. 41, 607-618 (2001) · Zbl 0991.34015 [10] Cabada, A.; Lois, S.: Existence of solution for discontinuous third order boundary value problems. J. comput. Appl. math. 110, 105-114 (1999) · Zbl 0936.34015 [11] Davis, J. M.; Henderson, J.: Triple positive symmetric solutions for a lidstone boundary value problem. Differential equations dynam. Systems 7, 321-330 (1999) · Zbl 0981.34014 [12] 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 [13] Gregus, M.: Third order linear differential equations. Math. appl. (1987) [14] Gregus, M.: Two sorts of boundary-value problems of nonlinear third order differential equations. Arch. math. 30, 285-292 (1994) [15] Grossinho, M. R.; Minhös, F.: Existence result for some third order separated boundary value problems. Nonlinear anal. 47, 2407-2418 (2001) · Zbl 1042.34519 [16] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045 [17] Krasnoselskii, M. A.: Positive solutions of operator equations. (1964) [18] Leggett, R. W.; Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana univ. Math. J. 28, 673-688 (1979) · Zbl 0421.47033 [19] Omari, P.; Trombetta, M.: Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems. Appl. math. Comput. 50, 1-21 (1992) · Zbl 0760.65078 [20] Rachunkova, I.: On some three-point problems for third-order differential equations. Math. bohem. 117, 98-110 (1992) [21] Rusnak, J.: Constructions of lower and upper solutions for a nonlinear boundary value problem of the third order and their applications. Math. slovaca 40, 101-110 (1990) · Zbl 0731.34016 [22] Rusnak, J.: Existence theorems for a certain nonlinear boundary value problem of the third order. Math. slovaca 37, 351-356 (1987) · Zbl 0631.34022 [23] Senkyrik, M.: Method of lower and upper solutions for a third-order three-point regular boundary value problem. Acta univ. Palack. olomuc. Fac. rerum natur. Math. 31, 60-70 (1992) [24] Senkyrik, M.: Existence of multiple solutions for a third-order three-point regular boundary value problem. Math. bohem. 119, 113-321 (1994) [25] Yao, Q.: The existence and multiplicity of positive solutions for a third-order three-point boundary value problem. Acta math. Appl. sinica 19, No. 1, 117-122 (2003) · Zbl 1048.34031 [26] Yosida, K.: Functional analysis. (1978) · Zbl 0365.46001 [27] Zhao, W.: Existence and uniqueness of solutions for third order nonlinear boundary value problems. Tohoku math. J. 44, No. 2, 545-555 (1992) · Zbl 0774.34019