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 multiplicity of positive solutions for an elastic beam equation. (English) Zbl 1249.34082

Summary: This paper investigates the boundary value problem for an elastic beam equation of the form

u '''' (t)=q(t)f(t,u(t),u ' (t),u '' (t),u ''' (t)),0<t<1

with the boundary conditions

u(0)=u ' (1)=u '' (0)=u ''' (1)=0·

The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. By using Leray-Schauder nonlinear alternative, Leray-Schauder fixed point theorem and a fixed point theorem due to R. I. Avery and A. C. Peterson [Comput. Math. Appl. 42, No. 3–5, 313–322 (2001; Zbl 1005.47051)], we establish some results on the existence and the multiplicity of positive solutions to the boundary value problem.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
References:
[1]R P Agarwal, D O’Regan, P J Y Wong. Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic, Boston, 1999.
[2]R P Agarwal. On fourth-order boundary value problems arising in beam analysis, Differ Integral Equ, 1989, 2: 91–110.
[3]R I Avery, A C Peterson. Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput Math Appl, 2001, 42(3–5): 313–322. · Zbl 1005.47051 · doi:10.1016/S0898-1221(01)00156-0
[4]Z Bai. The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal, 2007, 67: 1704–1709. · Zbl 1122.34010 · doi:10.1016/j.na.2006.08.009
[5]Z Bai, H Wang. On positive solutions of some nonliear fourth-order beam equations, J Math Anal Appl, 2002, 270: 357–368. · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9
[6]K Deimling. Nonlinear Functional Analysis, Springer, 1985.
[7]J Ehme, P W Eloe, J Henderson. Upper and lower solution methods for fully nonlinear boundary value problems, J Differential Equations, 2001, 180: 51–64. · Zbl 1019.34015 · doi:10.1006/jdeq.2001.4056
[8]J R Graef, B Yang. Existence and nonexistence of positive solutions of fourth-order nonlinear boundary value problems, Appl Anal, 2000, 74(1–2): 201–214. · Zbl 1031.34025 · doi:10.1080/00036810008840810
[9]C P Gupta. Existence and uniqueness theorems for the bending of an elastic beam equation, Appl Anal, 1988, 26: 289–304. · Zbl 0611.34015 · doi:10.1080/00036818808839715
[10]D Guo, V Lakshmikantham. Nonlinear Problems in Abstract Cones, Academic Press, 1988.
[11]Y Li. Positive solutions of fourth-order boundary value problems with two parameters, J Math Anal Appl, 2003, 281: 477–484. · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8
[12]Y Li. On the existence of positive solutions for the bending elastic beam equations, Appl Math Comput, 2007, 189: 821–827. · Zbl 1118.74032 · doi:10.1016/j.amc.2006.11.144
[13]B Liu. Positive solutions of fourth-order two-point boundary value problems, Appl Math Comput, 2004, 148: 407–420. · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3
[14]R Ma, H Wang. On the existence of positive solutions of fourth-order ordinary differential equations, Appl Anal, 1995, 59: 225–231. · Zbl 0841.34019 · doi:10.1080/00036819508840401
[15]C Pang, Z Wei. Positive solutions and multiplicity of fourth-order boundary value problems with two parameters, Acta Math Sinica (Chin Ser), 2006, 49(3): 625–632.
[16]Y Sun. Symmetric positive solutions for a fourth-order nonlinear differential equation with nonlocal boundary conditions, Acta Math Sinica (Chin Ser), 2007, 50(3): 547–556.
[17]Z Wei. Positive solutions to singular boundary value problems of a class of fourth order sublinear differential equations, Acta Math Sinica (Chin Ser), 2005, 48(4): 727–738.
[18]Y R Yang. Triple positive solutions of a class of fourth-order two-point boundary value problems, Appl Math Lett, 2010, 23: 366–370. · Zbl 1193.34048 · doi:10.1016/j.aml.2009.10.012
[19]Q Yao. On the positive solutions of a nonlinear fourth-order boundary value problem with two parameters, Appl Anal, 2004, 83: 97–107. · Zbl 1051.34018 · doi:10.1080/00036810310001632817
[20]Q Yao. Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl Math Lett, 2004, 17: 237–243. · Zbl 1072.34022 · doi:10.1016/S0893-9659(04)90037-7
[21]X Zhang. Existence and iteration of monotone positive solutions for an elastic beam equation with corner, Nonlinear Anal Real World Appl, 2009, 10(4): 2097–2103. · Zbl 1163.74478 · doi:10.1016/j.nonrwa.2008.03.017
[22]Z Zhao. A necessary and sufficient condition for the existence of positive solutions of fourth-order singular superlinear differential equations, Acta Math Sinica (Chin Ser), 2007, 50(6): 1425–1434.