Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls. (English) Zbl 1228.45015

Summary: We investigate nonlocal problems for a class of fractional integrodifferential equations via fractional operators and optimal controls in Banach spaces. By using the fractional calculus, Hölder inequality, \(p\)-mean continuity and fixed point theorems, some existence results of mild solutions are obtained under the two cases of the semigroup \(T(t)\), the nonlinear terms \(f\) and \(h\), and the nonlocal item \(g\). Then, the existence conditions of optimal pairs of systems governed by a fractional integrodifferential equation with nonlocal conditions are presented. Finally, an example is given to illustrate the effectiveness of the results obtained.


45K05 Integro-partial differential equations
34A08 Fractional ordinary differential equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
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[1] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[2] Lakshmikantham, V.; Leela, S.; Devi, J.V., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002
[3] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002
[4] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[5] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, (), 217-224
[6] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. syst. signal process., 5, 81-88, (1991)
[7] Glockle, W.G.; Nonnenmacher, T.F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68, 46-53, (1995)
[8] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific Singapore · Zbl 0998.26002
[9] Mainardi, F., Fractional calculus, some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[10] 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)
[11] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., 2009, (2009), Article ID 981728, 47 pages · Zbl 1182.34103
[12] 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
[13] Ahmad, B.; Nieto, J.J., Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via leray – schauder degree theory, Topol. methods nonlinear anal., 35, 295-304, (2010) · Zbl 1245.34008
[14] Balachandran, K.; Park, J.Y., Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear anal., 71, 4471-4475, (2009) · Zbl 1213.34008
[15] Bai, Z.B., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal., 72, 916-924, (2010) · Zbl 1187.34026
[16] 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
[17] El-Borai, M.M., Semigroup and some nonlinear fractional differential equations, Appl. math. comput., 149, 823-831, (2004) · Zbl 1046.34079
[18] Henderson, J.; Ouahab, A., Fractional functional differential inclusions with finite delay, Nonlinear anal., 70, 2091-2105, (2009) · Zbl 1159.34010
[19] 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
[20] 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
[21] 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
[22] 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
[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
[24] Zhou, Yong; Jiao, Feng, Existence of mild solutions for fractional neutral evolution equations, Comput. math. appl., 59, 1063-1077, (2010) · Zbl 1189.34154
[25] Zhou, Yong; Jiao, Feng, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal., 11, 4465-4475, (2010) · Zbl 1260.34017
[26] Chang, Y.K.; Kavitha, V.; Mallika Arjunan, M., Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear anal., 71, 5551-5559, (2009) · Zbl 1179.45010
[27] Wang, JinRong; Zhou, Yong, A class of fractional evolution equations and optimal controls, Nonlinear anal., 12, 262-272, (2011) · Zbl 1214.34010
[28] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear anal., 69, 3692-3705, (2008) · Zbl 1166.34033
[29] Cuevas, C.; Lizama, C., Almost automorphic mild solutions to a class of fractional differential equations, Appl. math. lett., 21, 1315-1319, (2008) · Zbl 1192.34006
[30] 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
[31] 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
[32] 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
[33] El-Borai, M.M., On some fractional evolution equations with nonlocal conditions, Int. J. pure appl. math., 24, 405-413, (2005) · Zbl 1090.35006
[34] 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
[35] Wang, JinRong; Xiang, X.; Wei, W., Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space, Electron. J. qual. theory differ. equ., 4, 1-17, (2009) · Zbl 1178.45017
[36] Wang, JinRong; Xiang, X.; Wei, W., A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach space, J. appl. math. comput., 34, 465-484, (2010) · Zbl 1213.34092
[37] Wang, JinRong; Wei, W., A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces, Results math., 58, 379-397, (2010) · Zbl 1209.34095
[38] Wang, JinRong; Zhou, Yong, Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces, Adv. difference equ., 2011, (2011), Article ID 385324, 16pages · Zbl 1222.49006
[39] JinRong Wang, W. Wei, Yong Zhou, Fractional finite time delay evolution systems and optimal controls in infinite dimensional spaces, J. Dyn. Control Syst. (2011) (in press). · Zbl 1241.26005
[40] Wang, JinRong; Zhou, Yong; Wei, W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. nonlinear sci. numer. simul., (2011) · Zbl 1223.45007
[41] Pazy, A., Semigroup of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023
[42] Zeidler, E., Nonlinear functional analysis and its application II/A, (1990), Springer-Verlag New York
[43] Hu, S.; Papageorgiou, N.S., Handbook of multivalued analysis, theory, (1997), Kluwer Academic Publishers Dordrecht Boston, London · Zbl 0887.47001
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