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 for a nonlocal fractional differential equation. (English) Zbl 1220.34006

Summary: We study the following singular boundary value problem of a nonlocal fractional differential equation

D 0 + α u(t)+q(t)f(t,u(t))=0,0<t<1,n-1<αn,u(0)=u ' (0)==u (n-2) (0)=0,u(1)= 0 1 u(s)dA(s),

where α2, D 0 + α is the standard Riemann-Liouville derivative, 0 1 u(s)dA(s) is given by the Riemann-Stieltjes integral with a signed measure, q may be singular at t=0 and/or t=1,f(t,x) may also have a singularity at x=0. Existence and multiplicity of positive solutions are obtained by means of fixed point index theory in cones.

MSC:
34A08Fractional differential equations
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
References:
[1]Podlubny, I.: Fractional differential equations, Mathematics in science and engineering 198 (1999) · Zbl 0924.34008
[2]Kilbas, A. A.; Srivastava, H. M.; Nieto, J. J.: Theory and applicational differential equations, (2006)
[3]Agrawal, O. P.: Formulation of Euler–Lagrange equations for fractional variational problems, J. math. Anal. appl. 272, 368-379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[4]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[5]Bhaskar, T. Gnana; Lakshmikantham, V.; Leela, S.: Fractional differential equations with a Krasnoselskii–Krein type condition, Nonlinear anal. 3, 734-737 (2009) · Zbl 1181.34008 · doi:10.1016/j.nahs.2009.06.010
[6]Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. J.: Fractional order differential equations on an unbounded domain, Nonlinear anal. 72, 580-586 (2010) · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106
[7]Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal. 72, 916-924 (2010) · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033
[8]Jiang, D.; Yuan, C.: The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear anal. 72, 710-719 (2010) · Zbl 1192.34008 · doi:10.1016/j.na.2009.07.012
[9]Ahmad, B.; Alsaedi, A.: Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro–differential equations of mixed type, Nonlinear anal. 3, 501-509 (2009) · Zbl 1179.45008 · doi:10.1016/j.nahs.2009.03.007
[10]Benchohraa, M.; Hamania, S.; Ntouyas, S. K.: Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear anal. 71, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[11]El-Shahed, M.; Nieto, J. J.: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. math. Appl. 59, 3438-3443 (2010) · Zbl 1197.34003 · doi:10.1016/j.camwa.2010.03.031
[12]Ahmad, B.; Nieto, J. J.: Existence results for higher order fractional differential inclusions with nonlocal boundary conditions, Nonlinear stud. 17, 131-138 (2010) · Zbl 1206.34010
[13]Salem, H. A. H.: On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies, Comput. math. Appl. 224, 565-572 (2009) · Zbl 1176.34070 · doi:10.1016/j.cam.2008.05.033
[14]Li, C. F.; Luo, X. N.; Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. math. Appl. 59, 1363-1375 (2010) · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029
[15]Agarwal, R. P.; O’regan, D.; Stanek, S.: Positive solutions for Dirichlet problem of singular nonlinear fractional differential equations, J. math. Anal. appl. 371, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034
[16]Goodrich, C. S.: Existence of positive solution to a class of fractional differential equations, Appl. math. Lett. 23, 1050-1055 (2010) · Zbl 1204.34007 · doi:10.1016/j.aml.2010.04.035
[17]Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal. 70, 2521-2529 (2009) · Zbl 1169.34302 · doi:10.1016/j.na.2008.03.037
[18]Yuan, C.: Multiple positive solutions for (n-1,1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. theory differ. Equ., No. 36 (2010) · Zbl 1210.34008 · doi:emis:journals/EJQTDE/2010/201036.html
[19]Feng, M.; Zhang, X.; Ge, W.: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. value probl. (2011)
[20]Allison, J.; Kosmatov, N.: Multi-point boundary value problems of fractional order, Commun. appl. Anal. 12, No. 4, 451-458 (2008) · Zbl 1184.34012
[21]Ma, R.: Existence of solutions of nonlinear m-point boundary value problems, J. math. Anal. appl. 256, 556-567 (2001) · Zbl 0988.34009 · doi:10.1006/jmaa.2000.7320
[22]Eloe, P. W.; Ahmad, B.: Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions, Appl. math. Lett. 18, 521-527 (2005) · Zbl 1074.34022 · doi:10.1016/j.aml.2004.05.009
[23]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
[24]Webb, J. R. L.; Infante, G.: Positive solutions of nonlocal boundary value problems: a unified approach, J. London math. Soc. 74, 673-693 (2006) · Zbl 1115.34028 · doi:10.1112/S0024610706023179
[25]Webb, J. R. L.; Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear differ. Equ. appl. 15, 45-67 (2008) · Zbl 1148.34021 · doi:10.1007/s00030-007-4067-7
[26]Webb, J. R. L.: Nonlocal conjugate type boundary value problems of higher order, Nonlinear anal. 71, 1933-1940 (2009) · Zbl 1181.34025 · doi:10.1016/j.na.2009.01.033
[27]J.R.L. Webb, Higher order non-local (n-1,1) conjugate type boundary value problems, in: Mathematical Models in Engineering, Biology and Medicine, AIP Conf. Proc., in: Amer. Inst. Phys., vol. 1124, Melville, NY, 2009, pp. 332–341.
[28]Graef, J. R.; Moussaoui, T.: A class of nth-order BVPs with nonlocal conditions, Comput. math. Appl. 58, 1662-1671 (2009) · Zbl 1189.34033 · doi:10.1016/j.camwa.2009.07.009
[29]Hao, X.; Liu, L.; Wu, Y.; Sun, Q.: Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions, Nonlinear anal. 73, 1653-1662 (2010) · Zbl 1202.34038 · doi:10.1016/j.na.2010.04.074
[30]Yang, B.: Positive solutions of the (n-1,1) conjugate boundary value problem, Electron. J. Qual. theory differ. Equ., No. 53 (2010) · Zbl 1211.34032 · doi:emis:journals/EJQTDE/2010/201053.html
[31]Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)