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 results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. (English) Zbl 1159.34020
The paper deals with the existence of positive solutions of the boundary value problem $$x^{(4)}-\lambda f(t,x)=\theta, \quad 0<t<1$$ in a Banach space $E$ under the boundary conditions $$x(0)=x(1)= \int_0^1g(s)x(s)\,ds \quad\text{and}\quad x''(0)=x''(1)= \int_0^1h(s)x''(s)\,ds.$$ The authors prove their results by using a cone in $E$ and applying the fixed point theory in cones for strict set contraction operators. An example is given to illustrate the result.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34G20Nonlinear ODE in abstract spaces
WorldCat.org
Full Text: DOI
References:
[1] Gallardo, J. M.: Second order differential operators with integral boundary conditions and generation of semigroups, Rocky mountain J. Math 30, 1265-1292 (2000) · Zbl 0984.34014 · doi:10.1216/rmjm/1021477351 · http://math.la.asu.edu/~rmmc/rmj/VOL30-4/CONT30-4/CONT30-4.html
[2] Karakostas, G. L.; Tsamatos, P. Ch.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems, Electron. J. Differential equations 30, 1-17 (2002) · Zbl 0998.45004 · emis:journals/EJDE/Volumes/2002/30/abstr.html
[3] Lomtatidze, A.; Malaguti, L.: On a nonlocal boundary-value problems for second order nonlinear singular differential equations, Georgian math. J. 7, 133-154 (2000) · Zbl 0967.34011
[4] Corduneanu, C.: Integral equations and applications, (1991) · Zbl 0714.45002
[5] Agarwal, R. P.; O’regan, D.: Infinite interval problems for differential, difference and integral equations, (2001)
[6] Agarwal, R. P.: On fourth-order boundary value problems arising in beam analysis, Differential integral equations 2, 91-110 (1989) · Zbl 0715.34032
[7] Gupta, C. P.: A generalized multi-point boundary value problem for second order ordinary differential equations, Appl. math. Comput. 89, 133-146 (1998) · Zbl 0910.34032 · doi:10.1016/S0096-3003(97)81653-0
[8] Gupta, C. P.; Ntouyas, S. K.; Tsamatos, P. Ch.: Solvability of an m-point boundary value problem for second order ordinary differential equations, J. math. Anal. appl. 189, 575-584 (1995) · Zbl 0819.34012 · doi:10.1006/jmaa.1995.1036
[9] Ma, H. L.: Positive solution for m-point boundary value problems of fourth order, J. math. Anal. appl. 321, 37-49 (2006) · Zbl 1101.34014 · doi:10.1016/j.jmaa.2005.08.015
[10] Ma, H. L.: Symmetric positive solutions for nonlocal boundary value problems of fourth order, Nonlinear anal. (2007)
[11] Feng, M. Q.; Zhang, X. M.: Multiple solutions of two-point boundary value problem of fourth-order ordinary differential equations in Banach space, Acta anal. Funct. appl. 6, 56-64 (2004) · Zbl 1102.34307
[12] Guo, D. J.; Lakshmikantham, V.: Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces, J. math. Anal. appl. 129, 211-222 (1988) · Zbl 0645.34014 · doi:10.1016/0022-247X(88)90243-0
[13] Guo, D. J.; Lakshmikantham, V.; Liu, X. Z.: Nonlinear integral equations in abstract spaces, (1996) · Zbl 0866.45004
[14] Guo, D. J.; Lakshmikantham, V.: Nonlinear problems in abstract, (1988) · Zbl 0661.47045
[15] Lakshmikanthan, V.; Leela, S.: Nonlinear differential equations in abstract spaces, (1981)
[16] Ma, R.; Wang, H.: Positive solutions of nonlinear three-point boundary-value problems, J. math. Anal. appl. 279, 216-227 (2003) · Zbl 1028.34014 · doi:10.1016/S0022-247X(02)00661-3
[17] Ma, R.; Wang, H.: On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal. 59, 225-231 (1995) · Zbl 0841.34019 · doi:10.1080/00036819508840401
[18] Bai, Z.; Wang, H.: On the positive solutions of some nonlinear fourth-order beam equations, J. math. Anal. appl. 270, 357-368 (2002) · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9
[19] Jiang, D. Q.; Gao, W. J.; Wan, A.: A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems, Appl. math. Comput. 132, 411-421 (2002) · Zbl 1036.34020 · doi:10.1016/S0096-3003(01)00201-6
[20] Wei, Z.; Pang, C.: Positive solutions of some singular m-point boundary value problems at non-resonance, Appl. math. Comput. 171, 433-449 (2005) · Zbl 1085.34017 · doi:10.1016/j.amc.2005.01.043
[21] Cheung, Wing-Sum; Ren, Jingli: Positive solutions for m-point boundary-value problems, J. math. Anal. appl. 303, 565-575 (2005) · Zbl 1071.34020 · doi:10.1016/j.jmaa.2004.08.056
[22] Ren, J.; Ge, W.: Existence of two solutions of nonlinear m-point boundary value problem, J. Beijing inst. Technol. 12, 97-100 (2003) · Zbl 1050.34010
[23] Zhang, G. W.; Sun, J. X.: Positive solutions of m-point boundary value problems, J. math. Anal. appl. 291, 406-418 (2004) · Zbl 1069.34037 · doi:10.1016/j.jmaa.2003.11.034
[24] Guo, Yanping; Liu, Xiujun; Qiu, Jiqing: Three positive solutions for higher order m point boundary value problems, J. math. Anal. appl. 289, 545-553 (2004) · Zbl 1046.34028 · doi:10.1016/j.jmaa.2003.08.038
[25] Zhang, Zhongxin; Wang, Junyu: The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J. comput. Appl. math. 147, 41-52 (2002) · Zbl 1019.34021 · doi:10.1016/S0377-0427(02)00390-4
[26] Feng, M. Q.; Ge, W. G.: Positive solutions for a class of m-point singular boundary value problems, Math. comput. Modelling 46, 375-383 (2007) · Zbl 1142.34012 · doi:10.1016/j.mcm.2006.11.009
[27] Feng, M. Q.; Ge, W. G.: Existence of positive solutions for singular eigenvalue problems, Electron. J. Differential equations 2006, No. 105, 1-9 (2006) · Zbl 1118.34014 · emis:journals/EJDE/Volumes/2006/105/abstr.html