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)
Nonlocal conjugate type boundary value problems of higher order. (English) Zbl 1181.34025
In this interesting paper, the author studies the nonlocal boundary value problem $$ \gathered u^{(n)}(t)+g(t)f(t,u(t))=0, \;t \in (0,1),\\ u^{(k)}(0) = 0, \; 0 \leq k \leq n-2, \;\; u(1) = \alpha[u], \endgathered$$ where $\alpha[\cdot]$ is a linear functional on $C[0,1]$ given by a Riemann-Stieltjes integral, namely $$ \alpha[u]=\int_0^1 u(s) dA(s), $$ with $dA$ a {\it signed} measure. This formulation is quite general and covers classical $m$-point boundary conditions and integral conditions as special cases. The author proves, under suitable growth conditions on the nonlinearity $f$, existence of multiple positive solutions. Interesting features of this paper are that the theory is illustrated with explicit examples, including a 4-point problem with coefficients with both signs, and that all the constants that appear in the theoretical results are explicitly determined. The methodology involves classical fixed point index theory and makes extensive use of the results in {\it J. R. L. Webb} and {\it G. Infante} [NoDEA, Nonlinear Differ. Equ. Appl. 15, No. 1--2, 45--67 (2008; Zbl 1148.34021)], {\it J. R. L. Webb} and {\it K. Q. Lan} [Topol. Methods Nonlinear Anal. 27, 91--115 (2006; Zbl 1146.34020)], {\it J. R. L. Webb} and {\it G. Infante} [J. Lond. Math. Soc., II. Ser. 74, No. 3, 673--693 (2006; Zbl 1115.34028)].

MSC:
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
34B27Green functions
47N20Applications of operator theory to differential and integral equations
WorldCat.org
Full Text: DOI
References:
[1] Eloe, P. W.; Henderson, J.: Positive solutions for (n-1,1) conjugate boundary value problems. Nonlinear anal. 28, 1669-1680 (1997) · Zbl 0871.34015
[2] Eloe, P. W.; Ahmad, B.: Positive solutions of a nonlinear n-th order boundary value problem with nonlocal conditions. Appl. math. Lett. 18, 521-527 (2005) · Zbl 1074.34022
[3] Guo, Y.; Ji, Y.; Zhang, J.: Three positive solutions for a nonlinear nth-order m-point boundary value problem. Nonlinear anal. 68, 3485-3492 (2008) · Zbl 1156.34311
[4] Hao, X.; Liu, L.; Wu, Y.: Positive solutions for nonlinear nth-order singular nonlocal boundary value problems. Bound. value probl., 10 (2007) · Zbl 1148.34015
[5] W. Jiang, Multiple positive solutions for nth-order m-point boundary value problems with all derivatives, Nonlinear Anal. 68, 1064--1072 · Zbl 1136.34025
[6] Pang, C.; Dong, W.; Wei, Z.: Green’s function and positive solutions of nth order m-point boundary value problem. Appl. math. Comput. 182, 1231-1239 (2006) · Zbl 1111.34024
[7] Yang, J.; Wei, Z.: Positive solutions of nth order m-point boundary value problem. Appl. math. Comput. 202, 715-720 (2008) · Zbl 1151.34022
[8] Webb, J. R. L.; Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. Nodea nonlinear differential equations appl. 15, 45-67 (2008) · Zbl 1148.34021
[9] Webb, J. R. L.; Infante, G.: Positive solutions of nonlocal boundary value problems: A unified approach. J. London math. Soc. (2) 74, 673-693 (2006) · Zbl 1115.34028
[10] Webb, J. R. L.; Infante, G.; Franco, D.: Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions. Proc. roy. Soc. Edinburgh 138A, 427-446 (2008) · Zbl 1167.34004
[11] Krasnosel’skiĭ, M. A.; Zabreĭko, P. P.: Geometrical methods of nonlinear analysis. (1984)
[12] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045
[13] Lan, K. Q.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities. J. differential equations 148, 407-421 (1998) · Zbl 0909.34013
[14] Webb, J. R. L.; Lan, K. Q.: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. methods nonlinear anal. 27, 91-116 (2006) · Zbl 1146.34020
[15] Martin, R. H.: Nonlinear operators and differential equations in Banach spaces. (1976) · Zbl 0333.47023
[16] Nussbaum, R. D.: Eigenvectors of order preserving linear operators. J. London math. Soc. (2) 58, 480-496 (1996) · Zbl 0968.47010
[17] Infante, G.; Webb, J. R. L.: Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations. Proc. math. Soc. edinb. 49, 637-656 (2006) · Zbl 1115.34026
[18] Lan, K. Q.: Multiple positive solutions of semilinear differential equations with singularities. J. London math. Soc. 63, 690-704 (2001) · Zbl 1032.34019
[19] Nussbaum, R. D.: Periodic solutions of some nonlinear integral equations. Dynamical systems (Proc. Internat. sympos., univ. Florida, Gainesville, fla., 1976), 221-249 (1977)
[20] Lan, K. Q.: Multiple positive solutions of conjugate boundary value problems with singularities. Appl. math. Comput. 147, 461-474 (2004) · Zbl 1054.34032
[21] Agarwal, R. P.; O’regan, D.: Multiplicity results for singular conjugate, focal, and N,P problems. J. differential equations 170, 142-156 (2001) · Zbl 0978.34018
[22] Kong, L.; Wang, J.: The Green’s function for k,n-k conjugate boundary value problems and its applications. J. math. Anal. appl. 255, 404-422 (2001) · Zbl 0991.34023