Abstract
In this paper, we formulate the new set of sufficient conditions for the controllability of Hilfer fractional neutral differential systems with almost sectorial operator and infinite delay. Firstly, we show the controllability of the system using the MNC and the M\(\ddot{o}\)nch’s fixed point theorem. Finally, we extend the results to the controllability of the system with nonlocal conditions, and some applications are provided to demonstrate how the major results might be applied.
Similar content being viewed by others
References
Agarwal, R.P., Lakshmikanthan, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. TMA 72(6), 2859–2862 (2010)
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations. Springer International Publishing AG, Inclusions and Inequalities (2017)
Balachandran, K., Sakthivel, R.: Controllability of integro-differential systems in Banach spaces. Appl. Math. Comput. 118, 63–71 (2001)
Banas, J., Goebel, K.: Measure of noncompactness in Banach spaces. In: Lecture Notes in Pure and Applied Mathematics, Dekker, New York (1980)
Bedi, P., Kumar, A., Abdeljawad, T., Khan, Z.A., Khan, A.: Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators. Adv. Differ. Equ. 615, 1–15 (2020)
Byszewski, L.: Theorems about existence and uniqueness of a solutions of semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162(2), 494–505 (1991)
Byszewski, L., Akca, H.: On a mild solution of semilinear functional differential evolution nonlocal problem. J. Math. Stoch. Anal. 10(3), 265–271 (1997)
Chang, Y.K.: Controllability of impulsive differential systems with infinite delay in Banach spaces. Chaos Solit. Fractals 33, 1601–1609 (2007)
Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Shukla, A., Nisar, K.S.: A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusion of order \(r\in (1,2)\) with delay. Chaos Solit. Fractals 153, 111565 (2021)
Dineshkumar, C., Udhayakumar, R.: New results concerning to approximate controllability of Hilfer fractional neutral stochastic delay integro-differential system. Numer. Methods Partial Differ. Equ. 1–19 (2020). https://doi.org/10.1002/num.22567.
Furati, K.M., Kassim, M.D., Tatar, N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 641, 616–626 (2012)
Gu, H., Trujillo, J.J.: Existence of integral solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)
Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Jaiswal, A., Bahuguna, D.: Hilfer fractional differential equations with almost sectorial operators. Differ. Equ. Dyn. Syst., pp. 1–17 (2020). https://doi.org/10.1007/s12591-020-00514-y.
Ji, S., Li, G., Wang, M.: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217, 6981–6989 (2011)
Karthikeyan, K., Debbouche, A., Torres, D.F.M.: Analysis of Hilfer fractional integro-differential equations with almost sectorial operators. Fractal Fract. 5(1), 1–14 (2021)
Kavitha, K., Vijayakumar, V., Udhayakumar, R., Sakthivel, N., Nissar, K.S.: A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay. Math. Methods Appl. Sci. 44(6), 4428–4447 (2021)
Kavitha, K., Vijayakumar, V., Udhayakumar, R.: Results on controllability on Hilfer fractional neutral differential equations with infinite delay via measure of noncompactness. Chaos Solit. Fractals, 139, 1–9. 110035 (2020)
Khaminsou, B., Thaiprayoon, C., Sudsutad, W., Jose, S.A.: Qualitative analysis of a proportional Caputo fractional Pantograph differential equation with mixed nonlocal conditions. Nonlinear Funct. Anal. Appl. 26(1), 197–223 (2021)
Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal.: Theory Methods Appl. 69(8), 2677–2682 (2008)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
\(M\ddot{o}nch\), H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal.: Theory Methods Appl., 4(5), 985–999 (1980)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Periago, F., Straub, B.: A functional calculus for almost sectorial operators and applications to abstract evolution equations. J. Evol. Equ. 2, 41–62 (2002)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Sakthivel, R., Ganesh, R., Anthoni, S.M.: Approximate controllability of fractional nonlinear differential inclusions. Appl. Math. Comput. 225, 708–717 (2013)
Singh, V.: Controllability of Hilfer fractional differential systems with non-dense domain. Numer. Funct. Anal. Optim. 40(13), 1572–1592 (2019)
Sivasankar, S., Udhayakumar, R.: Hilfer fractional neutral stochastic Volterra integro-differential inclusions via almost sectorial operators. Mathematics 10(12), 2024 (2022). https://doi.org/10.3390/math10122074
Varun Bose, C.S., Udhayakumar, R.: A note on the existence of Hilfer fractional differential inclusions with almost sectorial operators. Math. Methods Appl. Sci., 45, 2530–2541 (2022)
Varun Bose, C. B. S., Udhayakumar, R.: Existence of mild solutions for Hilfer fractional neutral integro-differential inclusions via almost sectorial operators. Fractal Fract., 6(532), (2022). https://doi.org/10.3390/fractalfract.6090532
Vijayakumar, V., Ravichandran, C., Nisar, K.S., Kucche, K.D.: New discussion on approximate controllability results for fractional Sobolev type Volterra–Fredholm integro-differential systems of order 1 \(<\) r \(<\) 2. Numer. Methods Partial Differ. Equ. 1–19 (2021). https://doi.org/10.1002/num.22772
Wang, J.R., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154(1), 292–302 (2012)
Yang, M., Wang, Q.: Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions. Fract. Calc. Appl. Anal. 20(3), 679–705 (2017)
Yang, M., Wang, Q.: Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math. Methods Appl. Sci. 40, 1126–1138 (2017)
Zhang, L., Zhou, Y.: Fractional Cauchy problems with almost sectorial operators. Appl. Math. Comput. 257, 145–157 (2014)
Zhou, M., Li, C., Zhou, Y.: Existence of mild solutions for Hilfer fractional evolution equations with almost sectorial operators. Axioms 11(2), 1–13 (2022). https://doi.org/10.3390/axioms11040144
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)
Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Elsevier, New York (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bose, C.S.V., Udhayakumar, R. Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness. Qual. Theory Dyn. Syst. 22, 22 (2023). https://doi.org/10.1007/s12346-022-00719-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-022-00719-2
Keywords
- Hilfer fractional evolution system
- Measure of noncompactness
- Fixed point theorem
- Almost sectorial operators