The Existence of Mild Solutions and Approximate Controllability for Nonlinear Fractional Neutral Evolution Systems

1Agarwal, R.P., Leiva, H., Riera, L. and Lalvay, S. (2022). Existence of solutions for impulsive neutral semilinear evolution equations with nonlocal conditions, Discontinuity, Nonlinearity, and Complexity 11(2): 1–18.Search in Google Scholar

2Almarri, B. and Elshenhab, A.M. (2022). Controllability of fractional stochastic delay systems driven by the Rosenblatt process, Fractal and Fractional 6(11): 664.Search in Google Scholar

3Babiarz, A., Klamka, J. and Niezabitowski, M. (2016). Schauder’s fixed point theorem in approximate controllability problems, International Journal of Applied Mathematics and Computer Sciences 26(2): 263–275, DOI: 10.1515/amcs-2016-0018.Search in Google Scholar

4Bárcenas, D., Leiva, H. and Sívoli, Z. (2005). A broad class of evolution equations are approximately controllable but never exactly controllable, IAM Journal of Mathematical Control and Information 22(3): 310–320.Search in Google Scholar

5Chang, J. and Liu, H. (2009). Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm, Nonlinear Analysis: Theory, Methods & Applications 71(9): 3759–3768.Search in Google Scholar

6Dhayal, R., Malik, M. and Abbas, S. (2019). Approximate and trajectory controllability of fractional neutral differential equation, Advances in Operator Theory 4(4): 802–820.Search in Google Scholar

7Diethelm, K. and Ford, N.J. (2004). Multi-order fractional differential equations and their numerical solution, Applied Mathematics and Computation 154(3): 621–640.Search in Google Scholar

8Diethelm, K. and Freed, A.D. (1999). On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in F. Keil et al. (Eds), Scientific Computing in Chemical Engineering II—Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, pp. 217–224.Search in Google Scholar

9Du, J., Cui, D., Sun, Y. and Xu, J. (2020). Approximate controllability for a kind of fractional neutral differential equations with damping, Mathematical Problems in Engineering 2020: 9, Article ID: 7592818.Search in Google Scholar

10Ech-chaffani, Z., Aberqi, A. and Karite, T. (n.d.). Controllability and optimal control for fractional neutral evolution systems with Caputo derivative, Boletim da Sociedade Paranaense de Matemática, (accepted for publication).Search in Google Scholar

11Ech-chaffani, Z., Aberqi, A., Karite, T. and Torres, D.F.M. (2022). Minimum energy problem in the sense of Caputo of fractional neutral evolution systems in Banach spaces, Axioms 11(8): 379.Search in Google Scholar

12Heymans, N. and Podlubny, I. (2006). Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheologica Acta 45: 765–771.Search in Google Scholar

13Hu, L., Ren, Y. and Sakthivel, R. (2009). Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum 79: 507–514.Search in Google Scholar

14Kilbas, A.A., Srivastava, H.M. and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam.Search in Google Scholar

15Kiryakova, V. (1994). Generalized Fractional Calculus and Applications, Vol. 301, Longman-Wiley, New York.Search in Google Scholar

16Leiva, H. and Sundar, P. (2017). Existence of solutions for a class of semilinear evolutions equations with impulses and delays communicated by Toka Diagana, Journal of Nonlinear Evolution Equations and Applications 2017(7): 95–108.Search in Google Scholar

17Li, K., Peng, J. and Gao, J. (2013). Controllability of nonlocal fractional differential systems of order 1 <q < 2 in Banach spaces, Reports on Mathematical Physics 71(1): 33–43.Search in Google Scholar

18Li, X., Liu, X. and Tang, M. (2021). Approximate controllability of fractional evolution inclusions with damping, Chaos, Solitons & Fractals 148: 111073.Search in Google Scholar

19Li, Y. (2015). Regularity of mild solutions for fractional abstract Cauchy problem with order 1 < q < 2, Zeitschrift für angewandte Mathematik und Physik 66: 3283–3298.Search in Google Scholar

20Li, Y., Sun, H. and Feng, Z. (2016). Fractional abstract Cauchy problem with order 1 < q < 2, Dynamics of PDE 13(2): 155–177.Search in Google Scholar

21Liang, Y. (2022). Existence and approximate controllability of mild solutions for fractional evolution systems of Sobolev-type, Fractal and Fractional 6(2): 56.Search in Google Scholar

22Liu, Z. and Li, X. (2015). Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives, SIAM Journal on Control and Optimization 53(4): 1920–1933.Search in Google Scholar

23Mabel Lizzy, R. and Balachandran, K. (2018). Boundary controllability of nonlinear stochastic fractional systems in Hilbert spaces, International Journal of Applied Mathematics and Computer Science 28(1): 123–133, DOI: 10.2478/amcs-2018-0009.Search in Google Scholar

24Mahmudov, N.I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604–1622.Search in Google Scholar

25Mainardi, F. (1997). Fractional calculus: Some basic problems in continuum and statistical mechanics, in A. Carpinteri and F. Mainardi (Eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna, pp. 291–348.Search in Google Scholar

26Mingyuan, S., Chunhai, K. and Shaojun, G. (2016). Approximate controllability of fractional neutral evolution equations with nonlocal conditions, Journal of Shanghai Normal University (Natural Sciences) 45(3): 253–264.Search in Google Scholar

27Mokkedem, F.Z. and Fu, X.L. (2014). Approximate controllability of semi-linear neutral integrodifferential systems with finite delay, Applied Mathematics and Computation 242: 202–215.Search in Google Scholar

28Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York.Search in Google Scholar

29Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.Search in Google Scholar

30Sakthivel, R., Mahmudov, N.I. and Nieto, J.J. (2012). Controllability for a class of fractional-order neutral evolution control systems, Applied Mathematics and Computation 218(20): 10334–10340.Search in Google Scholar

31Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon.Search in Google Scholar

32Triggiani, R. (1977). A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM Journal on Control and Optimization 15(3): 407–411.Search in Google Scholar

33Vijayakumar, V., Udhayakumar, R. and Kavitha, K. (2021). On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evolution Equations and Control Theory 10(2): 271–296.Search in Google Scholar

34Xi, X.X., Hou, M., Zhou, X.F. and Wen, Y. (2022a). Approximate controllability of fractional neutral evolution systems of hyperbolic type, Evolution Equations and Control Theory 11(4): 1037–1069.Search in Google Scholar

35Xi, X.X., Zhou, X.F. and Wen, Y. (2022b). Approximate controllability of fractional neutral evolution systems of hyperbolic type, Evolution Equations and Control Theory 11(4): 1037–1069.Search in Google Scholar

36Yan, Z. (2012). Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA Journal of Mathematical Control and Information 30(4): 443–462.Search in Google Scholar

37Ye, H.P., Gao, J.M. and Ding, Y.S. (2007). A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications 328(2): 1075–1081.Search in Google Scholar

38Zhang, S. (2006). Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electronic Journal of Differential Equations 2006(36): 1–12.Search in Google Scholar

39Zhao, D. and Liu, Y. (2022). New discussion on approximate controllability for semilinear fractional evolution systems with finite delay effects in Banach spaces via differentiable resolvent operators, Fractal and Fractional 6(8): 424.Search in Google Scholar

40Zhou, Y. and He, J.W. (2021). New results on controllability of fractional evolution systems with order 1 < q < 2, Evolution Equations and Control Theory 10(3): 491–509.Search in Google Scholar

41Zhou, Y. and Jiao, F. (2010a). Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications 59(3): 1063–1077.Search in Google Scholar

42Zhou, Y. and Jiao, F. (2010b). Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis: Real World Applications 11(5): 4465–4475.Search in Google Scholar

43Zhu, B., Han, B.Y. and Yu, W.G. (2020). Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay, Acta Mathematicae Applicatae Sinica 36(4): 870–878.Search in Google Scholar