EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION

Year 2018, Volume: 1 Issue: 2, 130 - 147, 31.07.2018

Khalid Hilal Lahcen Ibnelazyz Karim Guida , Said Melliani

https://izlik.org/JA72PB39MB

Abstract

In this paper we are interested in studying the existence of solutions for a controlled impulsive fractional evolution equations. We use several tools such as fractional calculus, xed point theorems and the theory of semi-group. We rst give some preliminaries and notations, the second part of the work we provide an existence result for our problem and in the nal section, we give some examples to show the importance of our results.

Keywords

Mild solution , fractional differential equation

References

• A. Angaraj, K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with non-local conditions, Nonlinear Anal. 70(2009), 2717-2721.
• A. Anguraj, M. Lathamaheshwari, Existence results for fractional differential equations with infinite delay and interval impulsive conditions, Malaya J. Mat 2 (1) (2014) 16-23.
• X. Fu, X. Liu and B. Lu, On a new class of impulsive fractional evolution equations, Advances in difference equations (2015) 2015:227.
• H. James, Liu, Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Systems 6 (1999), 77-85.
• Hernandez, E., O'Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641-1649 (2013)
• Kilbas, AA, Srivastava, HM, Trujillo, JJ:Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amesterdam (2006)
• V. Lakshmikantham, DD. Bainov, P.S. Simeonov, Theory of impulsive differential equations, World scientific, Singapore, 1989.
• Shengda Liu, JinRong Wang, Optimal Controls of Systems Governed by Semilinear Fractional Differential Equations with Not Instantaneous Impulses, J. Optim. Theory Appl. DOI 10.1007/s10957-017-1122-3.
• J. J. Nieto, D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal, Real World Appl. 10 (2009), 680-690.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
• W. Wei, X. Xiang, Y. Peng:Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55, 141-156 (2006).
• Y. Zhou, F. Jiao: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal, Real World Appl. 11, 4465-4475 (2010)
• Peter L.Falb, Infinite Dimensional Control Problems:On the Closure of the Set of Attainable States for Linear Systems, Mathematical Analysis and Application 9, 12-22 (1964).
• Y. Zhou, F. Jiao:Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063-1077 (2010)