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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

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)

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

Khalid Hilal Lahcen Ibnelazyz Karim Guida , Said Melliani

Abstract

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)
There are 14 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Khalid Hilal This is me

Lahcen Ibnelazyz This is me

Karim Guida

Said Melliani 0000-0002-5150-1185

Publication Date July 31, 2018
Submission Date May 15, 2018
Acceptance Date August 5, 2018
Published in Issue Year 2018 Volume: 1 Issue: 2

Cite

APA Hilal, K., Ibnelazyz, L., Guida, K., Melliani, S. (2018). EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION. Journal of Universal Mathematics, 1(2), 130-147.
AMA Hilal K, Ibnelazyz L, Guida K, Melliani S. EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION. JUM. July 2018;1(2):130-147.
Chicago Hilal, Khalid, Lahcen Ibnelazyz, Karim Guida, and Said Melliani. “EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION”. Journal of Universal Mathematics 1, no. 2 (July 2018): 130-47.
EndNote Hilal K, Ibnelazyz L, Guida K, Melliani S (July 1, 2018) EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION. Journal of Universal Mathematics 1 2 130–147.
IEEE K. Hilal, L. Ibnelazyz, K. Guida, and S. Melliani, “EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION”, JUM, vol. 1, no. 2, pp. 130–147, 2018.
ISNAD Hilal, Khalid et al. “EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION”. Journal of Universal Mathematics 1/2 (July 2018), 130-147.
JAMA Hilal K, Ibnelazyz L, Guida K, Melliani S. EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION. JUM. 2018;1:130–147.
MLA Hilal, Khalid et al. “EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION”. Journal of Universal Mathematics, vol. 1, no. 2, 2018, pp. 130-47.
Vancouver Hilal K, Ibnelazyz L, Guida K, Melliani S. EXISTENCE OF MILD SOLUTIONS FOR AN IMPULSIVE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NON-LOCAL CONDITION. JUM. 2018;1(2):130-47.
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