zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
On discrete fourth-order boundary value problems with three parameters. (English) Zbl 1189.39006
The article deals with the following discrete nonlinear fourth-order boundary value problem $$\Delta^4 u(t - 2) + \eta \Delta^2 u(t - 1) - \xi u(t) = \lambda f(t,u(t)), \quad t \in {\Bbb Z}[a + 1,b+1],$$ $$u(a) = \Delta^2 u(a - 1) = 0, \qquad u(b + 2) = \Delta^2 u(b + 1) = 0,$$ where $\Delta u(t) = u(t + 1) - u(t)$. Using the classical critical point theory and theory of monotone operators in Banach spaces, the authors describe the intervals for $\lambda$ such that the boundary value problem under consideration has no nontrivial solutions, has a unique solution, has at least one nontrivial solution, has at least two nontrivial solutions, has at least three nontrivial solutions, has at least $k$ ($k \in {\Bbb Z}[1,b - a]$) distinct pairs of nontrivial solutions.

39A12Discrete version of topics in analysis
46N10Applications of functional analysis in optimization and programming
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Agarwal, R. P.: Difference equations and inequalities: theory, methods, and applications, (2000) · Zbl 0952.39001
[2] Agarwal, R. P.; O’regan, D.; Wong, P. J. Y.: Positive solutions of differential, difference and integral equations, (1999)
[3] Agarwal, R. P.; Perera, K.; O’regan, D.: Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear anal. 58, 69-73 (2004) · Zbl 1070.39005 · doi:10.1016/j.na.2003.11.012
[4] Wong, P. J. Y.; Xie, L.: Three symmetric solutions of lidstone boundary value problems for difference and partial difference equations, Comput. math. Appl. 45, 1445-1460 (2003) · Zbl 1057.39020 · doi:10.1016/S0898-1221(03)00102-0
[5] Yu, J. S.; Guo, Z. M.: On boundary value problems for a discrete generalized Emden--Fowler equation, J. differential equations 231, 18-31 (2006) · Zbl 1112.39011 · doi:10.1016/j.jde.2006.08.011
[6] Zhang, G.; Yang, Z. L.: Positive solutions of a general discrete boundary value problem, J. math. Anal. appl. 339, 469-481 (2008) · Zbl 1132.39011 · doi:10.1016/j.jmaa.2007.07.011
[7] Zhang, B. G.; Kong, L. J.; Sun, Y. J.; Deng, X. H.: Existence of positive solutions for BVPs of fourth-order difference equations, Appl. math. Comput. 131, 583-591 (2002) · Zbl 1025.39006 · doi:10.1016/S0096-3003(01)00171-0
[8] He, Z. M.; Yu, J. S.: On the existence of positive solutions of fourth-order difference equations, Appl. math. Comput. 161, 139-148 (2005) · Zbl 1068.39008 · doi:10.1016/j.amc.2003.12.016
[9] Wang, D. B.; Guan, W.: Three positive solutions of boundary value problems for p-Laplacian difference equations, Comput. math. Appl. 55, 1943-1949 (2008) · Zbl 1147.39008 · doi:10.1016/j.camwa.2007.08.033
[10] Li, Y. X.: Positive solutions of fourth-order boundary value problems with two parameters, J. math. Anal. appl. 281, 477-484 (2003) · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8
[11] Yao, Q. L.: Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear anal. 63, 237-246 (2005) · Zbl 1082.34025 · doi:10.1016/j.na.2005.05.009
[12] Li, F. Y.; Zhang, Q.; Liang, Z. P.: Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear anal. 62, 803-816 (2005) · Zbl 1076.34015 · doi:10.1016/j.na.2005.03.054
[13] Liu, X. L.; Li, W. T.: Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. math. Anal. appl. 327, 362-375 (2007) · Zbl 1109.34015 · doi:10.1016/j.jmaa.2006.04.021
[14] Yang, Y.; Zhang, J. H.: Existence of solutions for some fourth-order boundary value problems with parameters, Nonlinear anal. 69, 1364-1375 (2008) · Zbl 1166.34012 · doi:10.1016/j.na.2007.06.035
[15] Han, G. D.; Xu, Z. B.: Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear anal. 68, 3646-3656 (2008) · Zbl 1145.34008 · doi:10.1016/j.na.2007.04.007
[16] Guo, D. J.: Nonlinear functional analysis, (2001)
[17] Zhang, G. Q.: Critical point theory and its applications, (1986)
[18] Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS regional conf. Ser. in math. 65 (1986) · Zbl 0609.58002
[19] Brezis, H.; Nirenberg, L.: Remarks on finding critical points, Comm. pure appl. Math. 44, 939-963 (1991) · Zbl 0751.58006 · doi:10.1002/cpa.3160440808
[20] Clark, D. C.: A variant of the liusternik--schnirelman theory, Indiana univ. Math. J. 22, 65-74 (1972) · Zbl 0228.58006 · doi:10.1512/iumj.1972.22.22008
[21] Ricceri, B.: On a three critical points theorem, Arch. math. 75, 220-226 (2000) · Zbl 0979.35040 · doi:10.1007/s000130050496
[22] Ricceri, B.: Existence of three solutions for a class of elliptic eigenvalue problems, Math. comput. Modelling 32, 1485-1495 (2000) · Zbl 0970.35089 · doi:10.1016/S0895-7177(00)00220-X
[23] Graef, J. R.; Kong, L. J.; Wang, H. Y.: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem, J. differential equations 245, 1185-1197 (2008) · Zbl 1203.34028 · doi:10.1016/j.jde.2008.06.012
[24] Liu, X. L.; Li, W. T.: Existence and uniqueness of positive periodic solutions of functional differential equations, J. math. Anal. appl. 293, 28-39 (2004) · Zbl 1057.34094 · doi:10.1016/j.jmaa.2003.12.012