# 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)
Existence and uniqueness of solutions for third-order nonlinear boundary value problems. (English) Zbl 1160.34312
Consider the boundary value problem $$y'''f(x,y,y',y''),\ 0<x<1,$$ $$k(y(0),y'(0))=0,\ g(y'(0),y''(0))=0,\tag *$$ $$h(y(0),y'(0),y''(0);\ y(1),y'(1),y''(1))=0,$$ where $k,g$ and $h$ are continuous. The authors provide sufficient conditions for the existence and uniqueness of a solution to (*).

##### MSC:
 34B15 Nonlinear boundary value problems for ODE
##### Keywords:
boundary value problem; existence; uniqueness
Full Text:
##### References:
 [1] Agarwal, R. P.: On boundary value problems for y‴=f(x,y,$y^{\prime}$,y″). Acad. sinica 12, No. 2, 153-157 (1984) · Zbl 0542.34015 [2] Agarwal, R. P.: Boundary value problems for higher order differential equations. (1986) · Zbl 0619.34019 [3] Agarwal, R. P.: Existence-uniqueness and iterative method for third-order boundary value problems. Comput. appl. Math. 17, 271-289 (1987) · Zbl 0617.34008 [4] Baxley, J. V.; Brown, S. E.: Existence and uniqueness for two-point boundary value problems. Proc. roy. Soc. Edinburgh sect. A 88, 219-234 (1981) · Zbl 0461.34011 [5] Cabada, A.; Grossinho, M. R.; Minhós, 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) · Zbl 1048.34033 [6] Du, Z. J.; Ge, W. G.; Lin, X. J.: Existence of solutions for a class of third-order nonlinear boundary value problems. J. math. Anal. appl. 294, 104-112 (2004) · Zbl 1053.34017 [7] Grossinho, M. R.; Minhós, F. M.: Existence result for some third order separated boundary value problems. Nonlinear anal. 47, 2047-2418 (2001) · Zbl 1042.34519 [8] Grossinho, M. R.; Minhós, F. M.; Santos, A. I.: Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control. J. math. Anal. appl. 309, 271-283 (2005) · Zbl 1089.34017 [9] Grossinho, M. R.; Minhós, F. M.; Santos, A. I.: Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities. Nonlinear anal. 62, 1235-1250 (2005) · Zbl 1097.34016 [10] Hartman, P.: Ordinary differential equations. (1964) · Zbl 0125.32102 [11] Henderson, J.: Best interval lengths for third order Lipschitz equations. SIAM J. Math. anal. 18, 293-305 (1987) · Zbl 0668.34017 [12] Society, Japanese Mathematical: Encyclopedia of mathematics. (1985) [13] Rovderova, E.: Third-order boundary value problem with nonlinear boundary conditions. Nonlinear anal. 25, 473-485 (1995) [14] Sperb, René: Maximum principles and their applications. (1981) · Zbl 0454.35001