# Approximate controllability of impulsive Hilfer fractional differential inclusions

Volume 10, Issue 2, pp 595--611
Publication Date: February 20, 2017 Submission Date: July 20, 2016
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### Authors

Jun Du - School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China. - Department of Applied Mathematics, Huainan Normal University, Huainan 232038, P. R. China. Wei Jiang - School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China. Azmat Ullah Khan Niazi - School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China.

### Abstract

In this paper, firstly by utilizing the theory of operators semigroup, probability density functions via impulsive conditions, we establish a new $PC_{1-\nu}$-mild solution for impulsive Hilfer fractional differential inclusions. Secondly we prove the existence of mild solutions for the impulsive Hilfer fractional differential inclusions by using fractional calculus, multi-valued analysis and the fixed-point technique. Then under some assumptions, the approximate controllability of associated system are formulated and proved. An example is provided to illustrate the application of the obtained theory

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##### ISRP Style

Jun Du, Wei Jiang, Azmat Ullah Khan Niazi, Approximate controllability of impulsive Hilfer fractional differential inclusions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 595--611

##### AMA Style

Du Jun, Jiang Wei, Niazi Azmat Ullah Khan, Approximate controllability of impulsive Hilfer fractional differential inclusions. J. Nonlinear Sci. Appl. (2017); 10(2):595--611

##### Chicago/Turabian Style

Du, Jun, Jiang, Wei, Niazi, Azmat Ullah Khan. "Approximate controllability of impulsive Hilfer fractional differential inclusions." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 595--611

### Keywords

• Approximate controllability
• impulsive system
• Hilfer fractional differential inclusions
• multivalued maps
• fixed point theorem
• semigroup theory.

•  34A08
•  34A60
•  93B05

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