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 delay nonlinear integrodifferential controlled systems in Banach spaces. (English) Zbl 1223.45007
This work deals with the fractional delay nonlinear integrodifferential controlled system $$\cases\text{}^C\!D_t^qx(t)+Ax(t)=f\left(t,x_t,\displaystyle\int_0^tg(t,s,x_s)ds\right)+B(t)u(t),\,\,\,0<t\le T,\\ x(t)=\varphi(t),\,\,\,-r\le t\le 0,\endcases.\tag1$$ where $\text{}^C\!D_t^q$ denotes the Caputo fractional derivative of order $q\in (0,1)$, $-A:D(A)\to X$ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators $\{S(t),\,\,t\ge 0\}$ on a separable reflexive Banach space $X$, $f$ is $X$-value function and $g$ is $X_{\alpha}$-value function. Here $X_{\alpha}=D(A^{\alpha})$ is a Banach space with the norm $\|x\|_{\alpha}=\|A^{\alpha}x\|$ for $x\in X_{\alpha}$, $u$ takes values from another separable reflexive Banach space $Y$, $B$ is a linear operator from $Y$ into $X$, and $x_t:[-r,0]\to X_{\alpha},\,\,t\ge 0$ represents the history of the state from time $t-r$ up to the present time $t$, defined by $x_t=\{x(t+s),\,\,\,s\in [-r,0]\}$. The authors prove the existence and uniqueness of $\alpha$-mild solutions for $(1)$, and the continuous dependence result of these solutions. The Lagrange problem of system $(1)$ is also formulated and an existence result of optimal controls is presented. To illustrate the obtained results, an example is finally addressed.

45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
49J21Optimal control problems involving relations other than differential equations
93C30Control systems governed by other functional relations
45G10Nonsingular nonlinear integral equations
Full Text: DOI
[1] 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
[2] Agarwal, R. P.; Belmekki, M.; Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv diff equ, 47 (2009) · Zbl 1182.34103 · doi:10.1155/2009/981728
[3] Agarwal, R. P.; Zhou, Yong; He, Yunyun: Existence of fractional neutral functional differential equations, Comp math appl 59, 1095-1100 (2010) · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010
[4] Amann, H.: Invariant sets and existence for semilinear parabolic and elliptic systems, J math anal appl 65, 432-469 (1978) · Zbl 0387.35038 · doi:10.1016/0022-247X(78)90192-0
[5] Balder, E.: Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear anal 11, 1399-1404 (1987) · Zbl 0638.49004 · doi:10.1016/0362-546X(87)90092-7
[6] 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
[7] 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
[8] Belmekki, M.; Benchohra, M.: Existence results for fractional order semilinear functional differential with nondense domain, Nonlinear anal 72, 925-932 (2010) · Zbl 1179.26018 · doi:10.1016/j.na.2009.07.034
[9] Chang, Y. K.; Kavitha, V.; Arjunan, M. M.: 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
[10] El-Borai, M. M.: Some probability densities and fundamental solutions of fractional evolution equations, Chaos soliton fract 14, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[11] El-Borai, M. M.: The fundamental solutions for fractional evolution equations of parabolic type, J appl math stoch anal 3, 197-211 (2004) · Zbl 1081.34053 · doi:10.1155/S1048953304311020
[12] 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
[13] Hu, L.; Ren, Y.; 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
[14] Hernández, E.; O’regan, D.; Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear anal 73, 3462-3471 (2010) · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035
[15] 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
[16] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993) · Zbl 0789.26002
[17] 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
[18] Lakshmikantham, V.; Leela, S.; Devi, J. V.: Theory of fractional dynamic systems, (2009) · Zbl 1188.37002
[19] Mophou, G. M.; N’guérékata, G. M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl math comput 216, 61-69 (2010) · Zbl 1191.34098 · doi:10.1016/j.amc.2009.12.062
[20] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[21] Pazy, A.: Semigroup of linear operators and applications to partial differential equations, (1983) · Zbl 0516.47023
[22] Ren, Y.; Qin, Y.; Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay, Integral equ oper theory 67, 33-49 (2010) · Zbl 1198.45009 · doi:10.1007/s00020-010-1767-x
[23] Xiang, X.; Kuang, H.: Delay systems and optimal controls, Acta math appl sin 16, 27-35 (2000) · Zbl 1005.49017 · doi:10.1007/BF02670961
[24] Wang, Jinrong; Zhou, Yong: Time optimal control problem of a class of fractional distributed systems, Int J dyn diff eq 3, 363-382 (2010) · Zbl 1245.49010
[25] Wang, Jinrong; Zhou, Yong: A class of fractional evolution equations and optimal controls, Nonlinear anal 12, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[26] Wang, Jinrong; Zhou, Yong: Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces, Adv diff equ 2011, 1-16 (2011) · Zbl 1222.49006 · doi:10.1155/2011/385324
[27] Ye, Q.; Li, Z.: Introductory to reaction-diffusion equations, (1999)
[28] Zhou, Yong: Existence and uniqueness of fractional functional differential equations with unbounded delay, Int J dyn diff eq 1, 239-244 (2008) · Zbl 1175.34081 · doi:10.1504/IJDSDE.2008.022988
[29] Zhou, Yong; Jiao, Feng; Li, Jing: Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal 71, 3249-3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[30] Zhou, Yong; Jiao, Feng: Existence of extremal solutions for discontinuous fractional functional differential equations, Int J dyn diff eq 2, 237-252 (2009) · Zbl 1188.34108 · doi:10.1504/IJDSDE.2009.031104
[31] Zhou, Yong; Jiao, Feng: 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
[32] Zhou, Yong; Jiao, Feng: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal 11, 4465-4475 (2010) · Zbl 1260.34017