×

On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions. (English) Zbl 1331.34017

Summary: We study the existence of solutions for an impulsive hybrid system of multi-orders Caputo-Hadamard fractional differential equations equipped with nonlinear integral boundary conditions. We make use of fixed point theorems due to Krasnoselskii-Zabreiko, Sadovski and O’Regan to establish our main results. Examples illustrating the main results are presented.

MSC:

34A08 Fractional ordinary differential equations
34A37 Ordinary differential equations with impulses
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science BV: Elsevier Science BV Amsterdam) · Zbl 1092.45003
[2] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[3] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[4] Jarad, F.; Abdeljawad, T.; Baleanu, D., Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2012:142, 8 (2012) · Zbl 1346.26002
[5] Ahmad, B.; Ntouyas, S. K., Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal., 15, 362-382 (2012) · Zbl 1279.34003
[6] Ahmad, B.; Ntouyas, S. K.; Alsaedi, A., An existence result for fractional differential inclusions with nonlinear integral boundary conditions, J. Inequal. Appl., 2013:296, 9 (2013) · Zbl 1291.34005
[7] Nyamoradi, N.; Baleanu, D.; Agarwal, R. P., On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval, Adv. Math. Phys. (2013), Art. ID 823961. p. 9 · Zbl 1272.34009
[8] Ahmad, B.; Ntouyas, S. K., Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions, Electron. J. Qual. Theory Differ. Equ., 20, 1-19 (2013) · Zbl 1340.34055
[10] Alsaedi, A.; Ntouyas, S. K.; Agarwal, R. P.; Ahmad, B., On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Adv. Difference Equ., 2015:33, 12 (2015) · Zbl 1350.34003
[11] Yukunthorn, W.; Ntouyas, S. K.; Tariboon, J., Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions, Adv. Difference Equ., 2014:315 (2014) · Zbl 1417.34027
[12] Tariboon, J.; Ntouyas, S. K.; Sudsutad, W., Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations, Bound. Value Probl., 2014:253, 16 (2014) · Zbl 1307.34017
[13] Tariboon, J.; Ntouyas, S. K.; Thaiprayoon, C., Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014, 15 (2014), Article ID 372749 · Zbl 1311.34020
[14] Alsaedi, A.; Ntouyas, S. K.; Ahmad, B., New existence results for fractional integro-differential equations with nonlocal integral boundary conditions, Abstr. Appl. Anal., 11 (2014), Article ID 205452
[15] Bhrawy, A. H.; Abdelkawy, M. A., A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys., 294, 462-483 (2015) · Zbl 1349.65503
[16] Bhrawy, A. H.; Zaky, M. A., Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dynam., 80, 101-116 (2015) · Zbl 1345.65060
[17] Hadamard, J., Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Math. Pures Appl. Ser., 8, 101-186 (1892) · JFM 24.0359.01
[18] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269, 387-400 (2002) · Zbl 1027.26004
[19] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl., 269, 1-27 (2002) · Zbl 0995.26007
[20] Butzer, P. L.; Kilbas, A. A.; Trujillo, J. J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270, 1-15 (2002) · Zbl 1022.26011
[21] Kilbas, A. A., Hadamard-type fractional calculus, J. Korean Math. Soc., 38, 1191-1204 (2001) · Zbl 1018.26003
[22] Kilbas, A. A.; Trujillo, J. J., Hadamard-type integrals as G-transforms, Integral Transforms Spec. Funct., 14, 413-427 (2003) · Zbl 1043.26004
[23] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[24] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[25] Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions (2006), Hindawi Publishing: Hindawi Publishing New York · Zbl 1130.34003
[26] (Krogh, B.; Lynch, N., Hybrid Systems: Computation and Control. Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 1790 (2000), Springer Verlag: Springer Verlag New York) · Zbl 0934.00029
[27] (Vaandrager, F.; Van Schuppen, J., Hybrid Systems: Computation and Control. Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 1569 (1999), Springer Verlag: Springer Verlag New York) · Zbl 0911.00057
[28] Egbunonu, P.; Guay, M., Identification of switched linear systems using subspace and integer programming techniques, Nonlinear Anal. Hybrid Syst., 1, 577-592 (2007) · Zbl 1131.93316
[29] Engell, S.; Frehse, G.; Schnieder, E., (Modelling, Analysis and Design of Hybrid Systems. Modelling, Analysis and Design of Hybrid Systems, Lecture Notes In Control and Information Sciences (2002), Springer Verlag: Springer Verlag Heidelberg) · Zbl 0991.00028
[30] Altafini, C.; Speranzon, A.; Johansson, K. H., Hybrid control of a truck and trailer vehicle, (Tomlin, C. J.; Greenstreet, M. R., Hybrid Systems: Computation and Control. Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2289 (2002), Springer Verlag: Springer Verlag New York) · Zbl 1044.93533
[31] Balluchi, A.; Benvenutti, L.; Di Benedetto, M.; Pinello, C.; Sangiovanni-Vincentelli, A., Automotive engine control and hybrid systems: Challenges and opportunities, Proc. IEEE, 7, 888-912 (2000)
[32] Balluchi, A.; Soueres, P.; Bicchi, A., Hybrid feedback control for path tracking by a bounded-curvature vehicle, (Di Benedetto, M.; Sangiovanni-Vincentelli, A. L., Hybrid Systems: Computation and Control. Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2034 (2001), Springer Verlag: Springer Verlag New York) · Zbl 1026.93036
[33] Engell, S.; Kowalewski, S.; Schultz, C.; Strusberg, O., Continuous-discrete interactions in chemical process plants, Proc. IEEE, 7, 1050-1068 (2000)
[35] Lygeros, J.; Godbole, D. N.; Sastry, S., A verified hybrid controller for automated vehicles, IEEE Trans. Automat. Control, 43, 522-539 (1998) · Zbl 0904.90057
[36] Varaiya, P., Smart cars on smart roads: Problems of control, IEEE Trans. Automat. Control, 38, 195-207 (1993)
[37] Pepyne, D. L.; Cassandras, C. G., Optimal control of hybrid systems in manufacturing, Proc. IEEE, 7, 1108-1123 (2000)
[38] 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
[39] Zhang, X.; Huang, X.; Liu, Z., The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybrid Syst., 4, 775-781 (2010) · Zbl 1207.34101
[40] Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal., 74, 792-804 (2011) · Zbl 1214.34009
[41] Ahmad, B.; Wang, G., A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations, Comput. Math. Appl., 62, 1341-1349 (2011) · Zbl 1228.34012
[42] Wang, J. R.; Zhou, Y.; Fec˘kan, M., On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64, 3008-3020 (2012) · Zbl 1268.34032
[43] Fec˘kan, M.; Zhou, Y.; Wang, J. R., On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17, 3050-3060 (2012) · Zbl 1252.35277
[44] Guo, T. L., Nonlinear impulsive fractional differential equations in Banach spaces, Topol. Methods Nonlinear Anal., 42, 221-232 (2013) · Zbl 1375.34006
[45] Tariboon, J.; Ntouyas, S. K.; Agarwal, P., New concepts of fractional quantum calculus and applications to impulsive fractional \(q\)-difference equations, Adv. Differential Equations, 2015, 18 (2015) · Zbl 1346.39012
[46] Stamov, G. T.; Stamova, I. M., Impulsive fractional functional differential systems and Lyapunov method for the existence of almost periodic solutions, Rep. Math. Phys., 75, 73-84 (2015) · Zbl 1322.34088
[47] Wang, J.; Zhou, Y.; Fec˘kan, M., Nonlinear implusive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64, 3389-3405 (2012) · Zbl 1268.34033
[48] Wang, J.; Ibrahim, A. G.; Fec˘kan, M., Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput., 257, 103-118 (2015) · Zbl 1338.34027
[49] Wang, J.; Zhang, Y., On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39, 85-90 (2015) · Zbl 1319.34017
[50] Wang, G.; Liu, S.; Baleanu, D.; Zhang, L., A new impulsive multi-orders fractional differential equation involving multipoint fractional integral boundary conditions, Abstr. Appl. Anal. (2014), Article ID 932747, 10 pages · Zbl 1474.34056
[51] Krasnoselskii, M. A.; Zabreiko, P. P., Geometrical Methods of Nonlinear Analysis (1984), Springer-verlag: Springer-verlag New York
[52] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer: Springer New York · Zbl 1025.47002
[53] Zeidler, E., Nonlinear Functional Analysis and Its Application: Fixed Point-Theorems, Vol. 1 (1986), Springer: Springer New York
[54] Sadovskii, B. N., On a fixed point principle, Funct. Anal. Appl., 1, 74-76 (1967) · Zbl 0165.49102
[55] O’Regan, D., Fixed-point theory for the sum of two operators, Appl. Math. Lett., 9, 1-8 (1996) · Zbl 0858.34049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.