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**Theory of fractional hybrid differential equations.**
*(English)*
Zbl 1228.45017

Summary: We develop the theory of fractional hybrid differential equations involving Riemann-Liouville differential operators of order \(0<q<1\). An existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.

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\textit{Y. Zhao} et al., Comput. Math. Appl. 62, No. 3, 1312--1324 (2011; Zbl 1228.45017)

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### References:

[1] | Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equation, (1993), John Wiley New York · Zbl 0789.26002 |

[2] | Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 |

[3] | Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999), Academic Press New York, London, Toronto |

[4] | Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integral and derivative. theory and applications, (1993), Gordon and Breach Switzerland · Zbl 0818.26003 |

[5] | Li, Q.; Sun, S., On the existence of positive solutions for initial value problem to a class of fractional differential equation, (), 886-889 |

[6] | Li, Q.; Sun, S.; Zhang, M.; Zhao, Y., On the existence and uniqueness of solutions for initial value problem of fractional differential equations, J. univ. jinan, 24, 312-315, (2010) |

[7] | Q. Li, S. Sun, Z. Han, Y. Zhao, On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations, in: 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, 2010, pp. 452-457. |

[8] | Zhang, M.; Sun, S.; Zhao, Y.; Yang, D., Existence of positive solutions for boundary value problems of fractional differential equations, J. univ. jinan, 24, 205-208, (2010) |

[9] | Zhao, Y.; Sun, S., On the existence of positive solutions for boundary value problems of nonlinear fractional differential equations, (), 682-685 |

[10] | Y. Zhao, S. Sun, Z. Han, M. Zhang, Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations, in: 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, Qingdao, 2010, pp. 480-485. |

[11] | Zhao, Y.; Sun, S.; Han, Z.; Li, Q., The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. nonlinear sci. numer. simul., 16, 4, 2086-2097, (2011) · Zbl 1221.34068 |

[12] | Zhao, Y.; Sun, S.; Han, Z.; Li, Q., Positive solutions to boundary value problems of nonlinear fractional differential equations, Abst. appl. anal., 2011, 1-16, (2011) |

[13] | Zhao, Y.; Sun, S.; Han, Z.; Zhang, M., Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. math. comput., 217, 16, 6950-6958, (2011) · Zbl 1227.34011 |

[14] | Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal. TMA, 71, 7-8, 2724-2733, (2009) · Zbl 1175.34082 |

[15] | Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. TMA, 71, 7-8, 3249-3256, (2009) · Zbl 1177.34084 |

[16] | Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. RWA, 11, 4465-4475, (2010) · Zbl 1260.34017 |

[17] | Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear anal. RWA, 12, 262-272, (2011) · Zbl 1214.34010 |

[18] | Agarwal, R.P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Comput. math. appl., 59, 3, 1095-1100, (2010) · Zbl 1189.34152 |

[19] | Li, C.F.; Luo, X.N.; Zhou, Y., Existence of positive solutions of boundary value problem for fractional differential equations, Comput. math. appl., 59, 3, 1363-1375, (2010) · Zbl 1189.34014 |

[20] | Diethelm, K., The analysis of fractional differential equations, (2010), Springer-Verlag Berlin |

[21] | Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal. TMA, 69, 2677-2682, (2008) · Zbl 1161.34001 |

[22] | Lakshmikantham, V.; Vatsala, A.S., Theory of fractional differential inequalities and applications, Commun. appl. anal., 11, 395-402, (2007) · Zbl 1159.34006 |

[23] | Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear anal. TMA, 69, 3337-3343, (2008) · Zbl 1162.34344 |

[24] | Lakshmikantham, V.; Devi, J.V., Theory of fractional differential equations in Banach space, Eur. J. pure appl. math., 1, 38-45, (2008) · Zbl 1146.34042 |

[25] | Kilbas, A.A.; Srivastava, H.H.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003 |

[26] | Caputo, M., Linear models of dissipation whose Q is almost independent, II, Geophy. J. roy. astronom., 13, 529-539, (1967) |

[27] | Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003 |

[28] | Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comput., 154, 621-640, (2004) · Zbl 1060.65070 |

[29] | Dhage, B.C., On \(\alpha\)-condensing mappings in Banach algebras, Math. student, 63, 146-152, (1994) · Zbl 0882.47033 |

[30] | Dhage, B.C.; Lakshmikantham, V., Basic results on hybrid differential equations, Nonlinear anal. hybrid, 4, 414-424, (2010) · Zbl 1206.34020 |

[31] | Dhage, B.C., A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear funct. anal. appl., 8, 563-575, (2004) · Zbl 1067.47070 |

[32] | Dhage, B.C., Fixed point theorems in ordered Banach algebras and applications, Panamer. math. J., 9, 4, 93-102, (1999) · Zbl 0964.47026 |

[33] | Lakshmikantham, V.; Leela, S., Differential and integral inequalities, (1969), Academic Press New York · Zbl 0177.12403 |

[34] | Dhage, B.C., On a fixed point theorem in Banach algebras with applications, Appl. math. lett., 18, 273-280, (2005) · Zbl 1092.47045 |

[35] | Heikkilä, S.; Lakshmikantham, V., Monotone iterative technique for nonlinear discontinues differential equations, (1994), Marcel Dekker Inc. New York · Zbl 0882.34063 |

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