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)
A class of fractional evolution equations and optimal controls. (English) Zbl 1214.34010

The authors, using the techniques of fractional calculus, a singular version of Gronwall’s inequality and the Leray-Schauder fixed point theorem for compact maps, study the existence of solutions of a fractional evolution equation of the type

D q x(t)=-Ax(t)+f(t,x(t)),tJ=[0,T],q(0,1),
x(t 0 )=x 0 ·

They introduce a notion of α-mild solution which is associated with a probability density function and a semigroup operator. Further, the existence of an optimal control for a Lagrange problem is proved. An example is given to demonstrate their results.

34A08Fractional differential equations
34G20Nonlinear ODE in abstract spaces
49J27Optimal control problems in abstract spaces (existence)
[1]Diethelm; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, Scientific computing in chemical engineering II–computational fluid dynamics, reaction engineering and molecular properties, 217-224 (1999)
[2]Gaul, L.; Klein, P.; Kempfle, S.: Damping description involving fractional operators, Mech. syst. Signal process. 5, 81-88 (1991)
[3]Glockle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68, 46-53 (1995)
[4]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[5]Mainardi, F.: Fractional calculus, some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997)
[6]Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmache, T. F.: Relaxation in filled polymers: a fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995)
[7]Kilbas, A. A.; Srivastava, Hari M.; Trujillo, J. Juan: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[8]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[9]Podlubny, I.: Fractional differential equations, (1999)
[10]Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009)
[11]Anguraj, A.; Karthikeyan, P.; N’guérékata, G. M.: Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces, Comm. math. Anal. 6, 31-35 (2009) · Zbl 1167.34387
[12]Balachandran, K.; Park, J. Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear anal. 71, 4471-4475 (2009) · Zbl 1213.34008 · doi:10.1016/j.na.2009.03.005
[13]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, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[14]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. calc. Appl. anal. 11, 35-56 (2008) · Zbl 1149.26010
[15]Chang, Yong-Kui; Kavitha, V.; Arjunan, M. Mallika: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear anal. 71, 5551-5559 (2009) · Zbl 1179.45010 · doi:10.1016/j.na.2009.04.058
[16]El-Borai, M. M.: Semigroup and some nonlinear fractional differential equations, Appl. math. Comput. 149, 823-831 (2004) · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[17]Henderson, J.; Ouahab, A.: Fractional functional differential inclusions with finite delay, Nonlinear anal. 70, 2091-2105 (2009) · Zbl 1159.34010 · doi:10.1016/j.na.2008.02.111
[18]Hu, Lanying; Ren, Yong; Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup forum 79, 507-514 (2009) · Zbl 1184.45006 · doi:10.1007/s00233-009-9164-y
[19]Jaradat, O. K.; Al-Omari, A.; Momani, S.: Existence of the mild solution for fractional semilinear initial value problems, Nonlinear anal. 69, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
[20]Liu, H.; Chang, J. C.: Existence for a class of partial differential equations with nonlocal conditions, Nonlinear anal. 70, 3076-3083 (2009) · Zbl 1170.34346 · doi:10.1016/j.na.2008.04.009
[21]N’guérékata, G. M.: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear anal. 70, 1873-1876 (2009) · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[22]Mophou, G. M.; N’guérékata, G. M.: Mild solutions for semilinear fractional differential equations, Electron. J. Differential equations 21, 1-9 (2009) · Zbl 1179.34002 · doi:emis:journals/EJDE/Volumes/2009/21/abstr.html
[23]Mophou, G. M.; N’guérékata, G. M.: Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum 79, 315-322 (2009) · Zbl 1180.34006 · doi:10.1007/s00233-008-9117-x
[24]Zhou, Y.; Jiao, F.: Existence of mild solutions for fractional neutral evolution equations, Comput. math. Appl. 59, 1063-1077 (2010) · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026
[25]Özdemir, Necati; Karadeniz, Derya; İskender, Beyza B.: Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. lett. A 373, 221-226 (2009) · Zbl 1227.49007 · doi:10.1016/j.physleta.2008.11.019
[26]Pazy, A.: Semigroup of linear operators and applications to partial differential equations, (1983)
[27]Zeidler, E.: Nonlinear functional analysis and its application II/A, (1990)
[28]Hu, S.; Papageorgiou, N. S.: Handbook of multivalued analysis (Theory), (1997)