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)
Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. (English) Zbl 1206.34009
Consider the existence of a positive solution for the singular fractional boundary value problem $$D^\alpha u(t)+ f(t,u(t),D^\mu u(t))=0,\,u(0)=u(1)=0,$$ where $1<\alpha<2$, $\mu>0$ with $\alpha-\mu\geq 1,$ $D^\alpha$ is the standard Riemann-Liouville fractional derivative, the function $f$ is positive, satisfies the Carathéodory conditions on $ [0,1]\times (0,\infty)\times {\mathbb R}$ and $f(t,x,y)$ is singular at $x=0$. The proofs are based on regularization and sequential techniques and the results are obtained by means of fixed point theorem of cone compression type due to [{\it M. A. Krasnosel’skij}, Positive solutions of operator equations. Groningen: The Netherlands: P.Noordhoff Ltd. (1964; Zbl 0121.10604)].

34A08Fractional differential equations
34B18Positive solutions of nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] Agarwal, R. P.; Benchohra, M.; Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. Math. (2008) · Zbl 1198.26004
[2] Agarwal, R. P.; Belmekki, M.; Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. difference equ. (2009) · Zbl 1182.34103 · doi:10.1155/2009/981728
[3] Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[4] Benchohra, M.; Hamani, 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
[5] Bonilla, B.; Rivero, M.; Rodriguez-Germá, L.; Trujillo, J. J.: Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. Comput. 187, 79-88 (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[6] Daftardar-Gejji, V.; Bhalekar, S.: Boundary value problems for multi-term fractional differential equations, J. math. Anal. appl. 345, 754-765 (2008) · Zbl 1151.26004 · doi:10.1016/j.jmaa.2008.04.065
[7] Guo, D. J.; Lakshmikantham, V.: Nonlinear problems in abstract cones, Notes and reports math. Sci. eng. 5 (1988) · Zbl 0661.47045
[8] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[9] Krasnosel’skii, M. A.: Positive solutions of operator equations, (1964) · Zbl 0121.10604
[10] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[11] Podlubny, I.: Fractional differential equations, Math. sci. Eng. 198 (1999) · Zbl 0924.34008
[12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[13] Shi, A.; Zhang, S.: Upper and lower solutions method and a fractional differential boundary value problem, Electron. J. Qual. theory differ. Equ. (2009) · Zbl 1183.34009 · emis:journals/EJQTDE/2009/200930.html
[14] Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. math. Lett. 22, 64-69 (2009) · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001
[15] Su, X.; Liu, L.: Existence of solution for boundary value problem of nonlinear fractional differential equation, Appl. math. J. chinese univ. Ser. B 22, No. 3, 291-298 (2007) · Zbl 1150.34005 · doi:10.1007/s11766-007-0306-2
[16] Yang, A.; Ge, W.: Positive solutions for boundary value problems of n-dimension nonlinear fractional differential system, Bound. value probl. (2008) · Zbl 1167.34314 · doi:10.1155/2008/437453
[17] Zhang, S.: Existence of solutions for a boundary value problem of fractional order, Acta math. Sci. 26, 220-228 (2006) · Zbl 1106.34010 · doi:10.1016/S0252-9602(06)60044-1