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)
Sequential fractional differential equations with three-point boundary conditions. (English) Zbl 1268.34006
Summary: We study a nonlinear three-point boundary value problem of sequential fractional differential equations of order $\alpha +1$ with $1<\alpha \le 2$. The expression for Green’s function of the associated problem involving the classical gamma function and the generalized incomplete gamma function is obtained. Some existence results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented. Existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results.

34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
Full Text: DOI
[1] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · Zbl 0818.26003
[2] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[3] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006) · Zbl 1092.45003
[4] , Advances in fractional calculus: theoretical developments and applications in physics and engineering (2007) · Zbl 1116.00014
[5] Ahmad, B.; Nieto, J. J.: Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. appl. Anal. (2009) · Zbl 1186.34009
[6] Ahmad, B.; Nieto, J. 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
[7] Ahmad, B.: Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. math. Lett. 23, 390-394 (2010) · Zbl 1198.34007 · doi:10.1016/j.aml.2009.11.004
[8] Nieto, J. J.: Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. math. Lett. 23, 1248-1251 (2010) · Zbl 1202.34019 · doi:10.1016/j.aml.2010.06.007
[9] Baleanu, D.; Mustafa, O. G.; Agarwal, R. P.: An existence result for a superlinear fractional differential equation, Appl. math. Lett. 23, 1129-1132 (2010) · Zbl 1200.34004 · doi:10.1016/j.aml.2010.04.049
[10] Zhang, S.: Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. math. Appl. 59, 1300-1309 (2010) · Zbl 1189.34050 · doi:10.1016/j.camwa.2009.06.034
[11] 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
[12] Sztonyk, P.: Regularity of harmonic functions for anisotropic fractional Laplacians, Math. nachr. 283, No. 2, 89-311 (2010) · Zbl 1194.47044 · doi:10.1002/mana.200711116
[13] Wang, G.; Ahmad, B.; Zhang, L.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear anal. 74, 792-804 (2011) · Zbl 1214.34009 · doi:10.1016/j.na.2010.09.030
[14] Bhalekar, S.; Daftardar-Gejji, V.; Baleanu, D.: Fractional Bloch equation with delay, Comput. math. Appl. 61, 1355-1365 (2011) · Zbl 1217.34123 · doi:10.1016/j.camwa.2010.12.079
[15] Ahmad, B.; Agarwal, Ravi P.: On nonlocal fractional boundary value problems, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 18, 535-544 (2011) · Zbl 1230.26003 · http://online.watsci.org/abstract_pdf/2011v18/v18n4a-pdf/9.pdf
[16] Ahmad, B.; Nieto, J. J.; Pimentel, J.: Some boundary value problems of fractional differential equations and inclusions, Comput. math. Appl. 62, 1238-1250 (2011) · Zbl 1228.34011 · doi:10.1016/j.camwa.2011.02.035
[17] Tomovski, Z.; Hilfer, R.; Srivastava, H. M.: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral transforms spec. Funct. 21, 797-814 (2010) · Zbl 1213.26011 · doi:10.1080/10652461003675737
[18] Konjik, S.; Oparnica, L.; Zorica, D.: Waves in viscoelastic media described by a linear fractional model, Integral transforms spec. Funct. 22, 283-291 (2011) · Zbl 1225.26009 · doi:10.1080/10652469.2010.541039
[19] Keyantuo, V.; Lizama, C.: A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications, Math. nachr. 284, 494-506 (2011) · Zbl 1221.34012 · doi:10.1002/mana.200810158
[20] M.D. Riva, S. Yakubovich, On a Riemann--Liouville fractional analog of the Laplace operator with positive energy, Integral Transforms Spec. Funct. (in press) doi:10.1080/10652469.2011.576832.
[21] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[22] Wei, Z.; Dong, W.: Periodic boundary value problems for Riemann--Liouville sequential fractional differential equations, Electron. J. Qual. theory differ. Equ. 87, 1-13 (2011) · Zbl 06528091
[23] Wei, Z.; Li, Q.; Che, J.: Initial value problems for fractional differential equations involving Riemann--Liouville sequential fractional derivative, J. math. Anal. appl. 367, 260-272 (2010) · Zbl 1191.34008 · doi:10.1016/j.jmaa.2010.01.023
[24] Klimek, M.: Sequential fractional differential equations with Hadamard derivative, Commun. nonlinear sci. Numer. simul. 16, 4689-4697 (2011) · Zbl 1242.34009
[25] Baleanu, D.; Mustafa, O. G.; Agarwal, R. P.: On lp-solutions for a class of sequential fractional differential equations, Appl. math. Comput. 218, 2074-2081 (2011) · Zbl 1235.34008
[26] Bai, C.: Impulsive periodic boundary value problems for fractional differential equation involving Riemann--Liouville sequential fractional derivative, J. math. Anal. appl. 384, 211-231 (2011) · Zbl 1234.34005
[27] Bressan, A.: Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, (2000) · Zbl 0997.35002
[28] Akyildiz, F. T.; Bellout, H.; Vajravelu, K.; Van Gorder, R. A.: Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear anal. RWA 12, 2919-2930 (2011) · Zbl 1231.35155 · doi:10.1016/j.nonrwa.2011.02.017
[29] Ahmad, B.; Nieto, J. J.; Alsaedi, A.; El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear anal. RWA 13, 599-606 (2012) · Zbl 1238.34008
[30] Polyanin, A. D.; Zaitsev, V. F.: Handbook of nonlinear partial differential equations, (2004) · Zbl 1053.35001
[31] I. Thompson, A note on the real zeros of the incomplete gamma function, Integral Transforms Spec. Funct. (in press) doi:10.1080/10652469.2011.597391.
[32] Sebastian, N.: A generalized gamma model associated with a Bessel function, Integral transforms spec. Funct. 22, 631-645 (2011) · Zbl 1230.33007 · doi:10.1080/10652469.2010.536413
[33] Krasnoselskii, M. A.: Two remarks on the method of successive approximations, Uspekhi mat. Nauk 10, 123-127 (1955)