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Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness

Year 2020, Volume: 3 Issue: 2, 85 - 99, 30.06.2020

Senthil Raja Duraisamy Ponnusamy Sundararajan Kulandhaivel Karthikeyan

Abstract

We consider the controllability problem for a class of fractional impulsive evolution systems of mixed type in an infinite dimensional Banach space. The existence of mild solutions and controllability results are discussed by a new estimation technique of the measure of noncompactness and a fixed point theorem with respect to a convex-power condensing operator. However, the main results do not need any restrictive conditions on estimated parameters of the measure of noncompactness. Since we do not assume that the semigroup is compact and other conditions are more general, the outcomes we obtain here improve and generalize many known controllability results. An example is also given to demonstrate the applications of our main results.

Keywords

Controllability fixed point theorem Banach space fractional evolution system

References

  • [1] Z. Yan: Controllability of fractional order partial neutral functional integrodifferential inclusions with infinite delay. J. Franklin Inst. 348, 2156-2173 (2011)
  • [2] K. Balachandran, JY. Park: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst. 3, 363-367 (2009)
  • [3] Y.K. Chang, D.N. Chalishajar: Controllability of mixed Volterra-Fredholm type integro-diferential inclusions in Banach spaces. J. Franklin Inst. 345, 499-507 (2008)
  • [4] D.N. Chalishajar: Controllability of mixed Volterra-Fredholm type integrodifferential systems in Banach space. J. Franklin Inst. 344, 12-21 (2007)
  • [5] E.M. Hernández, D. O'Regan: Controllability of Volterra-Fredholm type systems in Banach spaces. J. Franklin Inst. 346, 95-101 (2009)
  • [6] F. Wang, Z. Liu, J. Li: Complete controllability of fractional neutral differential systems in abstract space. Abstr. Appl. Anal. 2013, Article ID 529025 (2013)
  • [7] M. Feˇ ckan, J. Wang, Y. Zhou: Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory Appl. 156, 79-95 (2013)
  • [8] S. Ji, G. Li, M. Wang: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217, 6981-6989 (2011)
  • [9] J.A. Machado, C. Ravichandran, M. Rivero, J.J. Trujillo: Controllability results for impulsive mixed type functional integrodifferential evolution equations with nonlocal conditions. Fixed Point Theory Appl. 2013, Article ID 66 (2013)
  • [10] L. Zhu, Q. Huang, G. Li: Abstract semilinear evolution equations with convex power condensing operators. J. Funct. Spaces Appl. 2013, Article ID 473876 (2013)
  • [11] X. Xue: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal. 70, 2593-2601 (2009) [12] B. Ahmad, K. Malar, K. Karthikeyan: A study of nonlocal problems of impulsive integrodifferential equations with measure of noncompactness. Adv. Di?er. Equ. 2013, Article ID 205 (2013)
  • [13] D.N. Chalishajar,K. Karthikeyan: Existence and uniqueness results for boundary value problems of higher order frac- tional integro-differential equations involving gronwall's inequality in banach spaces.Acta Mathematica Scientia, 33B(3):1- 16(2013)
  • [14] K. Karthikeyan,J.J. Trujillo: Existence and Uniqueness results for fractional integrodifferential equations with boundary value conditions, Communication in Nonlinear Science and Numerical Simulation, 17 4037-4043(2012)
  • [15] K. Karthikeyan, P. Sundararajan and D. Senthil Raja: Existence of solutions for impulsive second order abstract functional neutral differential equation with nonlocal conditions and state dependent-delay, Research and Reports on Mathematics, Volume 2, Issue 1(2018)
  • [16] D. Chalishajar, D. Senthil Raja, K. Karthikeyan, P. Sundararajan: Existence Results for Nonautonomous Impulsive Fractional Evolution Equations, Results in Nonlinear Analysis, No. 3, 133-147(2018)
  • [17] J. Wang, Y. Zhou: Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. 17, 4346-4355 (2012)
  • [18] P. Chen, Y. Li: Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness. Abstr. Appl. Anal. 2013, Article ID 784816 (2013)
  • [19] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego (1999)
  • [20] K.S. Miller, B. Ross: An Introduction to Fractional Calculus and Differential Equations. Wiley, New York (1993)
  • [20] [21] K. Deimling: Nonlinear Functional Analysis. Springer, New York (1985)
  • [21] J. Banás, L. Goebel: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)
  • [22] H.P. Heinz: On the behaviour of measure of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal., Theory Methods Appl. 7(12), 1351-1371 (1983)
  • [23] J. Sun, X. Zhang: The fixed point theorem of convex-power condensing evolution equations. Acta Math. Sin. 48(3), 439-446 (2005)

Year 2020, Volume: 3 Issue: 2, 85 - 99, 30.06.2020

Senthil Raja Duraisamy Ponnusamy Sundararajan Kulandhaivel Karthikeyan

Abstract

References

  • [1] Z. Yan: Controllability of fractional order partial neutral functional integrodifferential inclusions with infinite delay. J. Franklin Inst. 348, 2156-2173 (2011)
  • [2] K. Balachandran, JY. Park: Controllability of fractional integrodifferential systems in Banach spaces. Nonlinear Anal. Hybrid Syst. 3, 363-367 (2009)
  • [3] Y.K. Chang, D.N. Chalishajar: Controllability of mixed Volterra-Fredholm type integro-diferential inclusions in Banach spaces. J. Franklin Inst. 345, 499-507 (2008)
  • [4] D.N. Chalishajar: Controllability of mixed Volterra-Fredholm type integrodifferential systems in Banach space. J. Franklin Inst. 344, 12-21 (2007)
  • [5] E.M. Hernández, D. O'Regan: Controllability of Volterra-Fredholm type systems in Banach spaces. J. Franklin Inst. 346, 95-101 (2009)
  • [6] F. Wang, Z. Liu, J. Li: Complete controllability of fractional neutral differential systems in abstract space. Abstr. Appl. Anal. 2013, Article ID 529025 (2013)
  • [7] M. Feˇ ckan, J. Wang, Y. Zhou: Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory Appl. 156, 79-95 (2013)
  • [8] S. Ji, G. Li, M. Wang: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217, 6981-6989 (2011)
  • [9] J.A. Machado, C. Ravichandran, M. Rivero, J.J. Trujillo: Controllability results for impulsive mixed type functional integrodifferential evolution equations with nonlocal conditions. Fixed Point Theory Appl. 2013, Article ID 66 (2013)
  • [10] L. Zhu, Q. Huang, G. Li: Abstract semilinear evolution equations with convex power condensing operators. J. Funct. Spaces Appl. 2013, Article ID 473876 (2013)
  • [11] X. Xue: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal. 70, 2593-2601 (2009) [12] B. Ahmad, K. Malar, K. Karthikeyan: A study of nonlocal problems of impulsive integrodifferential equations with measure of noncompactness. Adv. Di?er. Equ. 2013, Article ID 205 (2013)
  • [13] D.N. Chalishajar,K. Karthikeyan: Existence and uniqueness results for boundary value problems of higher order frac- tional integro-differential equations involving gronwall's inequality in banach spaces.Acta Mathematica Scientia, 33B(3):1- 16(2013)
  • [14] K. Karthikeyan,J.J. Trujillo: Existence and Uniqueness results for fractional integrodifferential equations with boundary value conditions, Communication in Nonlinear Science and Numerical Simulation, 17 4037-4043(2012)
  • [15] K. Karthikeyan, P. Sundararajan and D. Senthil Raja: Existence of solutions for impulsive second order abstract functional neutral differential equation with nonlocal conditions and state dependent-delay, Research and Reports on Mathematics, Volume 2, Issue 1(2018)
  • [16] D. Chalishajar, D. Senthil Raja, K. Karthikeyan, P. Sundararajan: Existence Results for Nonautonomous Impulsive Fractional Evolution Equations, Results in Nonlinear Analysis, No. 3, 133-147(2018)
  • [17] J. Wang, Y. Zhou: Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. 17, 4346-4355 (2012)
  • [18] P. Chen, Y. Li: Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness. Abstr. Appl. Anal. 2013, Article ID 784816 (2013)
  • [19] I. Podlubny: Fractional Differential Equations. Academic Press, San Diego (1999)
  • [20] K.S. Miller, B. Ross: An Introduction to Fractional Calculus and Differential Equations. Wiley, New York (1993)
  • [20] [21] K. Deimling: Nonlinear Functional Analysis. Springer, New York (1985)
  • [21] J. Banás, L. Goebel: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)
  • [22] H.P. Heinz: On the behaviour of measure of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal., Theory Methods Appl. 7(12), 1351-1371 (1983)
  • [23] J. Sun, X. Zhang: The fixed point theorem of convex-power condensing evolution equations. Acta Math. Sin. 48(3), 439-446 (2005)
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Senthil Raja Duraisamy 0000-0001-7129-8180

Ponnusamy Sundararajan This is me

Kulandhaivel Karthikeyan This is me

Publication Date June 30, 2020
Published in Issue Year 2020 Volume: 3 Issue: 2

Cite

APA Duraisamy, S. R., Sundararajan, P., & Karthikeyan, K. (2020). Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness. Results in Nonlinear Analysis, 3(2), 85-99.
AMA Duraisamy SR, Sundararajan P, Karthikeyan K. Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness. RNA. June 2020;3(2):85-99.
Chicago Duraisamy, Senthil Raja, Ponnusamy Sundararajan, and Kulandhaivel Karthikeyan. “Controllability Problem for Fractional Impulsive Integrodifferential Evolution Systems of Mixed Type With the Measure of Noncompactness”. Results in Nonlinear Analysis 3, no. 2 (June 2020): 85-99.
EndNote Duraisamy SR, Sundararajan P, Karthikeyan K (June 1, 2020) Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness. Results in Nonlinear Analysis 3 2 85–99.
IEEE S. R. Duraisamy, P. Sundararajan, and K. Karthikeyan, “Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness”, RNA, vol. 3, no. 2, pp. 85–99, 2020.
ISNAD Duraisamy, Senthil Raja et al. “Controllability Problem for Fractional Impulsive Integrodifferential Evolution Systems of Mixed Type With the Measure of Noncompactness”. Results in Nonlinear Analysis 3/2 (June 2020), 85-99.
JAMA Duraisamy SR, Sundararajan P, Karthikeyan K. Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness. RNA. 2020;3:85–99.
MLA Duraisamy, Senthil Raja et al. “Controllability Problem for Fractional Impulsive Integrodifferential Evolution Systems of Mixed Type With the Measure of Noncompactness”. Results in Nonlinear Analysis, vol. 3, no. 2, 2020, pp. 85-99.
Vancouver Duraisamy SR, Sundararajan P, Karthikeyan K. Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness. RNA. 2020;3(2):85-99.