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)
Exact null controllability for fractional nonlocal integrodifferential equations via implicit evolution system. (English) Zbl 1251.93029
Summary: We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.

MSC:
93B05Controllability
34A08Fractional differential equations
45J05Integro-ordinary differential equations
WorldCat.org
Full Text: DOI
References:
[1] D. Baleanu, R. P. Agarwal, O. G. Mustafa, and M. C. Sulschi, “A symptotic integration of some nonlinear differential equations with fractional time derivative,” Journal of Physics A, vol. 44, pp. 1-9, 2011.
[2] D. Baleanu and A. K. Golmankhaneh, “On electromagnetic field in fractional space,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 288-292, 2010. · Zbl 1196.35224 · doi:10.1016/j.nonrwa.2008.10.058
[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[4] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[5] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Technical University of Kosice, Kosice, Slovak Republic, 1999. · Zbl 0924.34008
[6] B. Zhu, “The space fractional diffusion equation with Feller’s operator,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 9971-9977, 2011. · Zbl 1220.65129 · doi:10.1016/j.amc.2011.04.065
[7] E. Hernández, D. O’Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,” Nonlinear Analysis, vol. 73, no. 10, pp. 3462-3471, 2010. · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035
[8] D. B\ualeanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835-1841, 2010. · Zbl 1189.34006 · doi:10.1016/j.camwa.2009.08.028
[9] B. Ahmad and R. P. Agarwal, “On nonlocal fractional boundary value problems,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 18, no. 4, pp. 535-544, 2011. · Zbl 1230.26003 · http://online.watsci.org/abstract_pdf/2011v18/v18n4a-pdf/9.pdf
[10] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. · Zbl 1248.26011
[11] B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, article 36, 9 pages, 2011. · Zbl 1275.45004
[12] B. Ahmad and S. K. Ntouyas, “A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order,” Electronic Journal of Qualitative Theory of Differential Equations, no. 22, 15 pages, 2011. · Zbl 1241.26004
[13] X. B. Shu, Y. Z. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis, vol. 74, pp. 2003-2011, 2011. · Zbl 1227.34009 · doi:10.1016/j.na.2010.11.007
[14] R. Almeida and D. F. M. Torres, “Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1490-1500, 2011. · Zbl 1221.49038 · doi:10.1016/j.cnsns.2010.07.016
[15] X. Fu, “Controllability of neutral functional differential systems in abstract space,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 281-296, 2003. · Zbl 1175.93029 · doi:10.1016/S0096-3003(02)00253-9
[16] X. Fu, “Controllability of abstract neutral functional differential systems with unbounded delay,” Applied Mathematics and Computation, vol. 151, no. 2, pp. 299-314, 2004. · Zbl 1044.93008 · doi:10.1016/S0096-3003(03)00342-4
[17] A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442-1450, 2011. · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075
[18] L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494-505, 1991. · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[19] L. Byszewski, “Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 33, no. 5, pp. 413-426, 1998. · Zbl 0933.34064 · doi:10.1016/S0362-546X(97)00594-4
[20] A. Debbouche, “Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems,” Advances in Difference Equations, vol. 5, pp. 1-10, 2011. · Zbl 1267.45020
[21] A. Debbouche, “Fractional evolution integro-differential systems with nonlocal conditions,” Advances in Dynamical Systems and Applications, vol. 5, no. 1, pp. 49-60, 2010.
[22] G. M. N’Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 5, pp. 1873-1876, 2009. · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[23] Z. Yan, “Controllability of semilinear integrodifferential systems with nonlocal conditions,” International Journal of Computational and Applied Mathematics, vol. 2, no. 3, pp. 221-236, 2007.
[24] K. Deng, “Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions,” Journal of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 630-637, 1993. · Zbl 0798.35076 · doi:10.1006/jmaa.1993.1373
[25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. · Zbl 0516.47023
[26] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433-440, 2002. · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[27] A. Debbouche and M. M. El-Borai, “Weak almost periodic and optimal mild solutions of fractional evolution equations,” Electronic Journal of Differential Equations, vol. 46, pp. 1-8, 2009. · Zbl 1171.34331 · emis:journals/EJDE/Volumes/2009/46/abstr.html · eudml:130580
[28] S. D. Zaidman, Abstract Differential Equations, Pitman, San Francisco, Calif, USA, 1979. · Zbl 0465.34002
[29] R. Sakthivel, Q. H. Choi, and S. M. Anthoni, “Controllability result for nonlinear evolution integrodifferential systems,” Applied Mathematics Letters, vol. 17, no. 9, pp. 1015-1023, 2004. · Zbl 1072.93005 · doi:10.1016/j.aml.2004.07.003
[30] R. Sakthivel, S. M. Anthoni, and J. H. Kim, “Existence and controllability result for semilinear evolution integrodifferential systems,” Mathematical and Computer Modelling, vol. 41, no. 8-9, pp. 1005-1011, 2005. · Zbl 1129.93005 · doi:10.1016/j.mcm.2004.03.007