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**Existence results for class of fractional order boundary value problems with integrable impulses.**
*(English)*
Zbl 1397.34135

Summary: In this article, we study existence and uniqueness results for fractional functional boundary value problems with integrable impulses. The Boyd and Wang, Schaefer’s and Burton and Kirk fixed point theorems on a complex Banach space are applied to obtain the main out comes. Also, we present examples to illustrate the claimed results.

### MSC:

34K37 | Functional-differential equations with fractional derivatives |

34K45 | Functional-differential equations with impulses |

34K10 | Boundary value problems for functional-differential equations |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

fractional order differential equation; periodic boundary value problems; contractions; impulsive conditions; existence; uniqueness
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\textit{V. Gupta} and \textit{J. Dabas}, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 25, No. 4, 267--285 (2018; Zbl 1397.34135)

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### References:

[1] | [1] N.Giribabu, J. Vasundhara Devi, G.V.S.R.Deekshitulu, The method of upper and lower solutions for initial value problem of Caputo fractional differential equations with variable moments of impulse, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 24, 2017, pp 41-54. · Zbl 1362.34013 |

[2] | [2] M. Benchohra, S. Bouriah, Existence and stability results for neutral functional differential equations of fractional order with delay, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23, 2016, pp 295-307. · Zbl 1408.34051 |

[3] | [3] J. Pedjeu, S. Sathananthan, Fundamental existence and uniqueness of stocastic integro-differential equations with markovian switching, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23, 2016, pp 153- 162. · Zbl 1341.60058 |

[4] | [4] L. Wang, Existence and uniqueness of solutions for abstract two point baoundary value problems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23, 2016, pp 97-111. · Zbl 1345.34114 |

[5] | [5] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley and Sons: New York, USA, 1993. · Zbl 0789.26002 |

[6] | [6] S.G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Yverdon, 1993. · Zbl 0818.26003 |

[7] | [7] B. Ed. Ross, The Fractional Calculus and Its Application, Lecture notes in mathematics, Springer-Verlag: Berlin, Germany, 1975. |

[8] | [8] Y. Tian, Z. Bai, Impulsive Boundary Value Problem for Differential Equations with Fractional Order, Differential Equations and Dynamical Systems, Volume 21, Issue 3, 2013, pp 253-260. · Zbl 1273.34015 |

[9] | [9] E. Hernandez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141, 2013, pp 1641-1649. · Zbl 1266.34101 |

[10] | [10] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219, 2013, pp 6743-6749. · Zbl 1293.34019 |

[11] | [11] M. Pierri, H. R. Henriquez, A. Prokopczyk, Global Solutions for Abstract Differential Equations with Non-Instantaneous Impulses, Mediterr. J. Math., 2015, pp 1-24. · Zbl 1353.34071 |

[12] | [12] A. Anguraj, S. Kanjanadevi, Existence results for fractional non- instantaneuos impulsive integro-differential equations with nonlocal conditions, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23, 2016, pp 429-445. · Zbl 1353.35301 |

[13] | [13] V. Colao, L. Muglia, H. Xu, Existence of solutions for a second-order differential equation with non-instantaneous impulses and delay, Annali di Matematica, 2015, pp 1-20. · Zbl 1344.34084 |

[14] | [14] M. Feckan, J. Wang, Y. Zhou, Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses, Nonauton. Dyn. Syst., 1, 2014, pp 93-101. · Zbl 1311.34094 |

[15] | [15] A. Sood, S. K. Srivastava, On Stability of Differential Systems with Noninstantaneous Impulses, Hindawi Publishing Corporation Mathematical Problems in Engineering, Article ID 691687, 2015, pp 1-5. · Zbl 1394.34110 |

[16] | [16] X. Yu, Existence and B-Ulam-Hyers stability for a class of fractional differential equations with non-instantaneous impulses, Advances in Difference Equations, 104, 2015. · Zbl 1348.34030 |

[17] | [17] A. Anguraj, M. C. Ranjini, M. Rivero, J. J. Trujillo, Existence Results for Fractional Neutral Functional Differential Equations with Random Impulses, Mathematics, 3, 2015, pp 16-28. Existence Results for Class of FBVP with Integrable Impulses285 · Zbl 1315.34082 |

[18] | [18] J. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J Appl Math Comput, 46, 2014, pp 321-334. · Zbl 1296.34036 |

[19] | [19] G. R. Gautam, J. Dabas, Existence result of fractional functional integro-differential equation with not instantaneous impulse, Int. J. Adv. Appl. Math. and Mech., 1(3), 2014, pp 11-21. · Zbl 1359.34087 |

[20] | [20] A. Chadha, D. N. Pandey, Periodic BVP for a class of nonlinear differential equation with a deviated argument and integrable impulses, CUBO A Mathematical Journal, Vol. 17, No-01, 2015, pp 11-27. · Zbl 1330.34113 |

[21] | [21] Z. Lin, J. Wang, W. Wei, Fractional differential equation models with pulses and criterion for pest management, Applied Mathematics and Computation, 257, 2015, pp 398-408. · Zbl 1338.91106 |

[22] | [22] D. Foukrach, T. Moussaoui, S. K. Ntouyas, Existence and uniqueness results for a class of BVPs for nonlinear fractional differential equations, Georgian Math. J., 22, 2015, pp 45-55. · Zbl 1317.34009 |

[23] | [23] D. R. Smart, Fixed Point Theorems, vol. 66. Cambridge University Press, Cambridge 1980. · Zbl 0427.47036 |

[24] | [24] F. Li, H. Wang, Solvability of boundary value problems for impulsive fractional differential equations in Banach spaces, Advances in Difference Equations, 202, 2014. · Zbl 1417.34016 |

[25] | [25] T. A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr., 189, 1998, pp 23-31. · Zbl 0896.47042 |

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