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 multi-point boundary value problems with multivalued operators at resonance. (English) Zbl 1185.34019

This paper is concerned with a class of multi-point boundary value problems with multi valued operators. Using a fixed point theorem for multi valued operators obtained by D. O’Regan and M. Zima [Nonlinear Anal., Theory Methods Appl. 68, No. 10 (A), 2879–2888 (2008; Zbl 1152.47041)], the authors establish the existence of positive solutions for such problem.

Moreover, an example is provided to demonstrate the applications of this result.

MSC:
34A60Differential inclusions
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34B45Boundary value problems for ODE on graphs and networks
47N20Applications of operator theory to differential and integral equations
34B18Positive solutions of nonlinear boundary value problems for ODE
References:
[1]Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)
[2]Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
[3]Mawhin, J.: Topological degree methods in nonlinear boundary value problems. In: NSFCBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1979)
[4]O’Regan, D., Zima, M.: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 87, 233–244 (2006) · Zbl 1109.47051 · doi:10.1007/s00013-006-1661-6
[5]O’Regan, D., Zima, M.: Leggett-Williams theorems for coincidences of multivalued operators. Nonlinear Anal. 68, 2879–2888 (2008) · Zbl 1152.47041 · doi:10.1016/j.na.2007.02.034
[6]Ge, W.: Boundary Value Problems for Ordinary Nonlinear Differential Equations. Science Press, Beijing (2007). (In Chinese)
[7]Petryshyn, W.V.: On the solvability of xx+λ Fx in quasinormal cones with T and F k-set contractive. Nonlinear Anal. 5, 585–591 (1981) · Zbl 0474.47028 · doi:10.1016/0362-546X(81)90105-X
[8]Santanilla, J.: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 105, 357–371 (1985) · Zbl 0576.34018 · doi:10.1016/0022-247X(85)90053-8
[9]Gaines, R.E., Santanilla, J.: A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 12, 669–678 (1982) · Zbl 0508.34030 · doi:10.1216/RMJ-1982-12-4-669
[10]Graef, J.R., Kong, L.: Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems. Nonlinear Anal. 68, 1529–1552 (2008)
[11]Tian, Y., Ge, W.: Positive solutions for multi-point boundary value problem on the half-line. J. Math. Anal. Appl. 325, 1339–1349 (2007) · Zbl 1110.34018 · doi:10.1016/j.jmaa.2006.02.075
[12]Kosmatov, N.: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 68, 2158–2171 (2008) · Zbl 1138.34006 · doi:10.1016/j.na.2007.01.038
[13]Yang, A., Ge, W.: Existence of symmetric solutions for a fourth-order multi-point boundary value problem with a p-Laplacian at resonance. J. Appl. Math. Comput. (2008). doi: 10.1007/s12190-008-0131-7 . In press
[14]Wei, Z., Pang, C.: The method of lower and upper solutions for fourth order singular m-point boundary value problems. J. Math. Anal. Appl. 322, 675–692 (2006) · Zbl 1112.34010 · doi:10.1016/j.jmaa.2005.09.064
[15]Ma, R.: Positive solutions for second-order three-point boundary value problems. Appl. Math. Lett. 14, 1–5 (2001) · Zbl 0989.34009 · doi:10.1016/S0893-9659(00)00102-6
[16]Yao, Q.: Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem. Comput. Math. Appl. 47, 1195–1200 (2004) · Zbl 1062.34024 · doi:10.1016/S0898-1221(04)90113-7
[17]Pang, H., Feng, M., Ge, W.: Existence and monotone iteration of positive solutions for a three-point boundary value problem. Appl. Math. Lett. 21, 656–661 (2008) · Zbl 1152.34313 · doi:10.1016/j.aml.2007.07.019
[18]Webb, J.R.L.: Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (Suppl.) 895–903 (2005)
[19]Lan, K.Q.: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 63, 690–704 (2001) · Zbl 1032.34019 · doi:10.1112/S002461070100206X
[20]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, 91–115 (2006)