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)
Existence results for boundary value problems of nonlinear fractional differential equations. (English) Zbl 1231.34007
Summary: We consider the existence of solutions for the nonlinear fractional differential equation with the boundary value conditions where and are the standard Caputo derivative with $1<\alpha \le 2$, $r\ne 0$. By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations.

34A08Fractional differential equations
45J05Integro-ordinary differential equations
Full Text: DOI
[1] Glockle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68, 46-53 (1995)
[2] Hilfer, R.: Applications of fractional calculus in physics, (2000) · Zbl 0998.26002
[3] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F.: Relaxation in filled polymers: a fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995)
[4] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[5] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. calc. Appl. anal. 5, 367-386 (2002) · Zbl 1042.26003
[6] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J.: North-holland mathematics studies, Theory and applications of fractional differential equations 204 (2006) · Zbl 1092.45003
[7] Lakshmikantham, V.; Leela, S.; Vasundhara, J.: Theory of fractional dynamic systems, (2009) · Zbl 1188.37002
[8] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivaes theory and applications, (1993) · Zbl 0818.26003
[9] Agarwal, R. P.; Benchohra, M.; Hamani, S.: A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. Math. 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[10] Lakshmikanthama, 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
[11] Lakshmikanthama, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[12] Agarwal, Ravi P.; O’regan, Donal; Stanek, Svatoslav: Positive solutions for Dirichlet problems 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
[13] Mcrae, F. A.: Monotone iterative technique and existence results for fractional differential equtions, Nonlinear anal. 71, 6093-6096 (2009) · Zbl 1260.34014
[14] Benchohra, M.; Hamani, S.; Ntouyas, S. K.: Boundary value problems for differential equational with fractional order and nonlocal condtions, Nonlinear anal. 71, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[15] Agarwal, R. P.; Zhou, Yong; He, Yunyun: Existence of fractional neutral functional differential equations, Comput. math. Appl. 59, 1095-1100 (2010) · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010
[16] Ahmad, Bashir; Nieto, Juan J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. Appl. 58, 1838-1843 (2009) · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[17] Shuqin Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differential Equations, 2006, (36), 1--12. · Zbl 1096.34016 · emis:journals/EJDE/Volumes/2006/36/abstr.html
[18] Rehman, Mujeeb Ur; Khan, Rahmat Ali: Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. math. Lett., 1038-1044 (2010) · Zbl 1214.34007 · doi:10.1016/j.aml.2010.04.033
[19] Balachandran, Krishnan; Trujillo, Juan J.: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear anal. 72, 4587-4593 (2010) · Zbl 1196.34007 · doi:10.1016/j.na.2010.02.035
[20] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J. math. Anal. appl. 338, No. 2, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[21] Baleanu, Dumitru; Mustafa, Octavian G.: On the global existence of solutions to a class of fractional differential equations, Comput. math. Appl. 59, 1835-1841 (2010) · Zbl 1189.34006 · doi:10.1016/j.camwa.2009.08.028
[22] El-Shahed, Moustafa: 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