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)
On the natural solution of an impulsive fractional differential equation of order $q\in (1,2)$. (English) Zbl 1248.35226
Summary: This paper is motivated from some recent papers treating the impulsive Cauchy problems for some differential equations with fractional order $q\in (1,2)$. A better definition of solution for impulsive fractional differential equation is given. We build up an effective way to find natural solution for such problems. Then sufficient conditions for existence of the solutions are established by applying fixed point methods. Four examples are given to illustrate the results.

35R12Impulsive partial differential equations
35R11Fractional partial differential equations
37C25Fixed points, periodic points, fixed-point index theory
Full Text: DOI
[1] Diethelm, K.: The analysis of fractional differential equations. Lecture notes in mathematics, (2010) · Zbl 1215.34001
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[3] Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009) · Zbl 1188.37002
[4] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993) · Zbl 0789.26002
[5] Michalski, M. W.: Derivatives of non-integer order and their applications, Dissertationes mathematicae (1993) · Zbl 0880.26007
[6] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[7] Tarasov, V. E.: Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2011)
[8] 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 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[9] Benchohra, M.; Seba, D.: Impulsive fractional differential equations in Banach spaces, Electron J qual theory differ eq spec ed I 2009, No. 8, 1-14 (2009) · Zbl 1189.26005 · emis:journals/EJQTDE/sped1/108.pdf
[10] Balachandran, K.; Kiruthika, S.: Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron J qual theory differ eq 2010, No. 4, 1-12 (2010) · Zbl 1201.34091 · emis:journals/EJQTDE/2010/201004.pdf
[11] Ahmad, B.; Sivasundaram, S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear anal hybrid syst 3, 251-258 (2009) · Zbl 1193.34056 · doi:10.1016/j.nahs.2009.01.008
[12] Ahmad, B.; Sivasundaram, S.: Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear anal hybrid syst 4, 134-141 (2010) · Zbl 1187.34038 · doi:10.1016/j.nahs.2009.09.002
[13] Ahmad, B.; Wang, G.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Comput math appl 59, 1341-1349 (2010) · Zbl 1228.34012
[14] Tian, Y.; Bai, Z.: Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comput math appl 59, 2601-2609 (2010) · Zbl 1193.34007 · doi:10.1016/j.camwa.2010.01.028
[15] Cao, J.; Chen, H.: Some results on impulsive boundary value problem for fractional differential inclusions, Electron J qual theory differ eq 2010, No. 11, 1-24 (2010)
[16] Wang, G.; Ahmad, B.; Zhang, L.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear anal theory methods appl 74, 792-804 (2011) · Zbl 1214.34009
[17] Wang, G.; Zhang, L.; Song, G.: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear anal theory methods appl 74, 974-982 (2011) · Zbl 1223.34091 · doi:10.1016/j.na.2010.09.054
[18] Wang, G.; Ahmad, B.; Zhang, L.: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput math appl 59, 1389-1397 (2010) · Zbl 1228.34021 · doi:10.1016/j.camwa.2011.04.004
[19] Wang, X.: Impulsive boundary value problem for nonlinear differential equations of fractional order, Comput math appl 62, 2383-2391 (2011) · Zbl 1231.34009 · doi:10.1016/j.camwa.2011.07.026
[20] Cao, J.; Chen, H.: Impulsive fractional differential equations with nonlinear boundary conditions, Math comput model 55, 303-311 (2012) · Zbl 1255.34006
[21] Yang, L.; Chen, H.: Nonlocal boundary value problem for impulsive differential equations of fractional order, Adv differ eq 2011 (2011) · Zbl 1219.34011 · doi:10.1155/2011/404917
[22] Fec&breve, M.; Kan; Zhou, Y.; Wang, J.: On the concept and existence of solution for impulsive fractional differential equations, Commun nonlinear sci numer simul 17, 3050-3060 (2011)
[23] Wang, J.; Fec&breve, M.; Kan; Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynam part differ eq 8, 345-361 (2011)
[24] Wang, J.; Zhou, Y.; Wei, W.: Impulsive fractional evolution equations and optimal controls in infinite dimensional spaces, Topol methods nonlinear anal 38, 17-43 (2011) · Zbl 1237.26008
[25] Ye, H.; Gao, J.; Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation, J math anal appl 328, 1075-1081 (2007) · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
[26] Wei, W.; Xiang, X.; Peng, Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls, Optimization 55, 141-156 (2006) · Zbl 1101.45002 · doi:10.1080/02331930500530401
[27] Wang, J.; Lv, L.; Zhou, Y.: Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces, J appl math comput 38, 209-224 (2012) · Zbl 1296.34032
[28] Tarasov VE. Theoretical physics models with integro-differentiation of fractional order. IKI, RCD (in Russian); 2011.
[29] Westerlund, S.: Dead matter has memory! causal consulting, (2002)