A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $  {r} \in (1, 2) $ with impulses

Marimuthu Mohan Raja; Velusamy Vijayakumar; Anurag Shukla; Kottakkaran Sooppy Nisar; Wedad Albalawi; Abdel-Haleem Abdel-Aty; Marimuthu Mohan Raja; Velusamy Vijayakumar; Anurag Shukla; Kottakkaran Sooppy Nisar; Wedad Albalawi; Abdel-Haleem Abdel-Aty

doi:10.3934/math.2023548

AIMS Mathematics

2023, Volume 8, Issue 5: 10802-10821. doi: 10.3934/math.2023548

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A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $ {r} \in (1, 2) $ with impulses

1.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, Tamil Nadu, India
2.
Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj-209732, India
3.
Department of Mathematics, College of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Al Kharj, 16278, Saudi Arabia
4.
School of Technology, Woxsen University- Hyderabad-502345, Telangana State, India
5.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
6.
Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha 61922, Saudi Arabia
7.
Physics Department, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt

Received: 07 January 2023 Revised: 21 February 2023 Accepted: 22 February 2023 Published: 06 March 2023
MSC : 26A33, 34A08, 35R12, 47B12, 47H08, 46E36, 93B05

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $ 1 < r < 2 $. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.
- fractional differential equations,
- controllability,
- impulsive system,
- measure of noncompactness,
- Volterra-Fredholm type,
- sectorial operators
Citation: Marimuthu Mohan Raja, Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Wedad Albalawi, Abdel-Haleem Abdel-Aty. A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order $ {r} \in (1, 2) $ with impulses[J]. AIMS Mathematics, 2023, 8(5): 10802-10821. doi: 10.3934/math.2023548

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Abstract

In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order $ 1 < r < 2 $. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.

References

[1]	W. M. Abd-Elhameed, Y. H. Youssri, Numerical solutions for Volterra-Fredholm-Hammerstein integral equations via second kind Chebyshev quadrature collocation algorithm, Adv. Math. Sci. Appl., 24 (2014), 129–141.
[2]	J. Banas, K. Goebel, Measure of noncompactness in Banach spaces, In: Lecture Notes in Pure and Applied Matyenath, Marcel Dekker, New York, 1980.
[3]	K. Deimling, Multivalued differential equations, De Gruyter, Berlin, 1992.
[4]	C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, A. Shukla, A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay, Chaos Soliton. Fract., 157 (2022), 1–17. https://doi.org/10.1016/j.chaos.2022.111916 doi: 10.1016/j.chaos.2022.111916
[5]	Z. Fu, L. Yang, Q. Xi, C. Liu, A boundary collocation method for anomalous heat conduction analysis in functionally graded materials, Comput. Math. Appl., 88 (2021), 91–109. https://doi.org/10.1016/j.camwa.2020.02.023 doi: 10.1016/j.camwa.2020.02.023
[6]	J. W. He, Y. Liang, B. Ahmad, Y. Zhou, Nonlocal fractional evolution inclusions of order $\alpha \in (1, 2)$, Mathematics, 209 (2019), 1–17. https://doi.org/10.3390/math7020209 doi: 10.3390/math7020209
[7]	S. Ji, G. Li, M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput., 217 (2011), 6981–6989. https://doi.org/10.1016/j.amc.2011.01.107 doi: 10.1016/j.amc.2011.01.107
[8]	M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter, 2001. https://doi.org/10.1515/97831108
[9]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control., 24 (2021), 1406–1415. https://doi.org/10.1002/asjc.2549 doi: 10.1002/asjc.2549
[10]	K. Kavitha, K. S. Nisar, A. Shukla, V. Vijayakumar, S. Rezapour, A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems, Adv. Differ. Equ., 2021 (2021), 467. https://doi.org/10.1186/s13662-021-03624-1 doi: 10.1186/s13662-021-03624-1
[11]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[12]	X. Liu, Z. Liu, M. Bin, The solvability and optimal controls for some fractional impulsive equations of order $1 < \alpha < 2$, Abstr. Appl. Anal., 2014 (2014), 1–9. https://doi.org/10.1155/2014/142067 doi: 10.1155/2014/142067
[13]	Y. K. Ma, M. M. Raja, K. S. Nisar, A. Shukla, V. Vijayakumar, Results on controllability for Sobolev type fractional differential equations of order $1 < r < 2$ with finite delay, AIMS Math., 7 (2022), 10215–10233. https://doi.org/10.3934/math.2022568 doi: 10.3934/math.2022568
[14]	Y. K. Ma, C. Dineshkumar, V. Vijayakumar, R. Udhayakumar, A. Shukla, K. S. Nisar, Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions, Ain Shams Eng. J., 14 (2023), 1–13. 101882. https://doi.org/10.1016/j.asej.2022.101882 doi: 10.1016/j.asej.2022.101882
[15]	Y. K. Ma, M. M. Raja, V. Vijayakumar, A. Shukla, W. Albalawi, K. S. Nisar, Existence and continuous dependence results for fractional evolution integrodifferential equations of order $r \in (1, 2)$, Alex. Eng. J., 61 (2022), 9929–9939. https://doi.org/10.1016/j.aej.2022.03.010 doi: 10.1016/j.aej.2022.03.010
[16]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
[17]	M. M. Raja, A. Shukla, J. J. Nieto, V. Vijayakumar, K. S. Nisar, A note on the existence and controllability results for fractional integrodifferential inclusions of order $r \in (1, 2]$ with impulses, Qual. Theor. Dyn. Syst., 21 (2022), 1–41. https://doi.org/10.1007/s12346-022-00681-z doi: 10.1007/s12346-022-00681-z
[18]	M. M. Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, S. Rezapour, New discussion on nonlocal controllability for fractional evolution system of order $1 < r < 2$, Adv. Differ. Equ., 2021 (2021), 481. https://doi.org/10.1186/s13662-021-03630-3 doi: 10.1186/s13662-021-03630-3
[19]	M. M. Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, N. Sakthivel, K. Kaliraj, Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order $r \in (1, 2)$, Optim. Contr. Appl. Met., 43 (2022), 996–1019. https://doi.org/10.1002/oca.2867 doi: 10.1002/oca.2867
[20]	M. M. Raja, V. Vijayakumar, Optimal control results for Sobolev-type fractional mixed Volterra-Fredholm type integrodifferential equations of order $1 < r < 2$ with sectorial operators, Optim. Contr. Appl. Met., 43 (2022), 1314–1327. https://doi.org/10.1002/oca.2892 doi: 10.1002/oca.2892
[21]	M. M. Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, H. M. Baskonus, On the approximate controllability results for fractional integrodifferential systems of order $1 < r < 2$ with sectorial operators, J. Comput. Appl. Math., 415 (2022), 1–12. https://doi.org/10.1016/j.cam.2022.114492 doi: 10.1016/j.cam.2022.114492
[22]	M. M. Raja, V. Vijayakumar, New results concerning to approximate controllability of fractional integrodifferential evolution equations of order $1 < r < 2$, Numer. Meth. Part. D. E., 38 (2022), 509–524. https://doi.org/10.1002/num.22653 doi: 10.1002/num.22653
[23]	H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal.-Real, 4 (1980), 985–999. https://doi.org/10.1016/0362-546X(80)90010-3 doi: 10.1016/0362-546X(80)90010-3
[24]	D. O'Regan, R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl., 6 (2001), 77–97. https://doi.org/10.1155/S1025583401000066 doi: 10.1155/S1025583401000066
[25]	A. E. Ofem, A. Hussain, O. Joseph, M. O. Udo, U. Ishtiaq, H. Al Sulami, et al., Solving fractional Volterra-Fredholm integro-differential equations via $A^{*}$ iteration method, Axioms*, 11 (2022), 470. https://doi.org/10.3390/axioms11090470 doi: 10.3390/axioms11090470
[26]	A. E. Ofem, U. Udofia, D. I. Igbokwe, A robust iterative approach for solving nonlinear volterra delay integro-differential equations, Ural Math. J., 7 (2021), 59–85. https://doi.org/10.15826/umj.2021.2.005 doi: 10.15826/umj.2021.2.005
[27]	G. A. Okeke, A. E. Ofem, T. Abdeljawad, M. A. Alqudah, A. Khan, A solution of a nonlinear Volterra integral equation with delay via a faster iteration method, AIMS Math., 8 (2022), 102–124. https://doi.org/10.3934/math.2023005 doi: 10.3934/math.2023005
[28]	G. A. Okeke, A. E. Ofem, A novel iterative scheme for solving delay differential equations and nonlinear integral equations in Banach spaces, Math. Method. Appl. Sci., 45 (2022), 5111–5134. https://doi.org/10.1002/mma.8095 doi: 10.1002/mma.8095
[29]	R. Patel, A. Shukla, J. J. Nieto, V. Vijayakumar, S. S. Jadon, New discussion concerning to optimal control for semilinear population dynamics system in Hilbert spaces, Nonlinear Anal.-Model., 27 (2022), 496–512. https://doi.org/10.15388/namc.2022.27.26407 doi: 10.15388/namc.2022.27.26407
[30]	R. Patel, A. Shukla, S. S. Jadon, Existence and optimal control problem for semilinear fractional order $(1, 2]$ control system, Math. Method. Appl. Sci., 2020, 1–12. https://doi.org/10.1002/mma.6662
[31]	I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to method of their solution and some of their applications, San Diego, CA: Acad. Press, 1999.
[32]	H. Qin, X. Zuo, J. Liu, L. Liu, Approximate controllability and optimal controls of fractional dynamical systems of order $1 < q < 2$ in Banach spaces, Adv. Differ. Equ., 73 (2015), 1–17. https://doi.org/10.1186/s13662-015-0399-5 doi: 10.1186/s13662-015-0399-5
[33]	C. Ravichandran, D. Baleanu, On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces, Adv. Differ. Equ., 291 (2013), 1–13. https://doi.org/10.1186/1687-1847-2013-291 doi: 10.1186/1687-1847-2013-291
[34]	L. Shu, X. B. Shu, J. Mao, Approximate controllability and existence of mild solutions for Riemann-Liouville fractional Stochastic evolution equations with nonlocal conditions of order $1 < \alpha < 2$, Fract. Calc. Appl. Anal., 22 (2019), 1086–1112. http://dx.doi.org/10.1515/fca-2019-0057 doi: 10.1515/fca-2019-0057
[35]	X. B. Shu, Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order $1 < \alpha < 2$, Comput. Math. with Appl., 64 (2012), 2100–2110. http://dx.doi.org/10.1016/j.camwa.2012.04.006 doi: 10.1016/j.camwa.2012.04.006
[36]	X. B. Shu, F. Xu, Upper and lower solution method for factional evolution equations with order $1 < \alpha < 2$, J. Korean Math. Soc., 51 (2014), 1123–1139. https://doi.org/10.4134/JKMS.2014.51.6.1123 doi: 10.4134/JKMS.2014.51.6.1123
[37]	A. Shukla, V. Vijayakumar, K. S. Nisar, A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order $r \in (1, 2)$, Chaos Soliton. Fract., 154 (2022), 1–8. https://doi.org/10.1016/j.chaos.2021.111615 doi: 10.1016/j.chaos.2021.111615
[38]	S. Sivasankaran, M. M. Arjunan, V. Vijayakumar, Existence of global solutions for second order impulsive abstract partial differential equations, Nonlinear Anal.-Theor., 74 (2011), 6747–6757. https://doi.org/10.1016/j.na.2011.06.054 doi: 10.1016/j.na.2011.06.054
[39]	Z. Tang, Z. Fu, H. Sun, X. Liu, An efficient localized collocation solver for anomalous diffusion on surfaces, Fract. Calc. Appl. Anal., 24 (2021), 865–894. https://doi.org/10.1515/fca-2021-0037 doi: 10.1515/fca-2021-0037
[40]	V. Vijayakumar, C. Ravichandran, K. S. Nisar, K. D. Kucche, New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order $1 < r < 2$, Numer. Meth. Part. D. E., 2021, 1–19. https://doi.org/10.1002/num.22772
[41]	V. Vijayakumar, K. S. Nisar, D. Chalishajar, A. Shukla, M. Malik, A. Alsaadi, et al., A note on approximate controllability of fractional semilinear integro-differential control systems via resolvent operators, Fractal Fract., 6 (2022), 1–14. https://doi.org/10.3390/fractalfract6020073 doi: 10.3390/fractalfract6020073
[42]	J. R. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynam. Part. Differ. Eq., 8 (2011), 345–361.
[43]	J. R. Wang, X. Li, W. Wei, On the natural solution of an impulsive fractional differential equations of order $q \in (1, 2)$, Commun. Nonlinear. Sci., 17 (2012), 4384–4394. https://doi.org/10.1016/j.cnsns.2012.03.011 doi: 10.1016/j.cnsns.2012.03.011
[44]	X. Wang, X. B. Shu, The existence of positive mild solutions for fractional differential evolution equations with nonlocal conditions of order $1 < \alpha < 2$, Adv. Differ. Equ., 159 (2015), 1–15. https://doi.org/10.1186/s13662-015-0461-3 doi: 10.1186/s13662-015-0461-3
[45]	Q. Xi, Z. Fu, T. Rabczuk, D. Yin, A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials, Int. J. Heat Mass Tran., 64 (2012), 2100–2110. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121778 doi: 10.1016/j.ijheatmasstransfer.2021.121778
[46]	Y. H. Youssri, R. M. Hafez, Chebyshev collocation treatment of Volterra-Fredholm integral equation with error analysis, Arab. J. Math., 9 (2020), 471–480. https://doi.org/10.1007/s40065-019-0243-y doi: 10.1007/s40065-019-0243-y
[47]	Y. Zhou, Basic theory of fractional differential equations, World Scientific, Singapore, 2014. https://doi.org/10.1142/9069
[48]	Y. Zhou, Fractional evolution equations and inclusion: Analysis and control, Elsevier, New York, 2015.
[49]	Y. Zhou, J. W. He, New results on controllability of fractional evolution systems with order $\alpha \in (1, 2)$, Evol. Equ. Control The., 10 (2021), 491–509. https://doi.org/10.3934/eect.2020077 doi: 10.3934/eect.2020077

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