Published by De Gruyter October 6, 2022

Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay

Krishnan Kavitha and Velusamy Vijayakumar

From the journal International Journal of Nonlinear Sciences and Numerical Simulation

https://doi.org/10.1515/ijnsns-2021-0412

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Abstract

This manuscript prospects the controllability of Hilfer fractional neutral differential equations. The new results are obtained by implementing a suitable fixed point approach and the technique of measures of noncompactness and the outcomes and facts belong to fractional theory. Firstly, we focus the controllability and extend the discussion with nonlocal conditions. Finally, an interesting example is proposed to illustrate our main obtained results.

Keywords: controllability; Hilfer fractional derivative; integral solution; neutral systems; non-dense domain; nonlocal conditions

2010 Mathematics Subject Classification: 26A33; 34A08; 34K35; 34K37; 35R11; 60H10; 93E03

Corresponding author: Velusamy Vijayakumar, Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, TN, India, E-mail: vijaysarovel@gmail.com

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: Statement: There are no funders to report for this submission.
Conflict of interest statement: This work does not have any conflicts of interest.
Data availability statement: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Appendix A

Step 3: For every x ∈ B q , and t ∈ (0, a), we can prove the equicontinuity of Ω ̃ x . For ȷ ₁ = 0 and 0 < ȷ ₂ ≤ a, we get ‖ ( Ω ̃ x ) ( ȷ 2 ) − ( Ω ̃ x ) ( ȷ 1 ) ‖ ℏ → 0 , as ȷ ₂ → 0.

‖ Ω ̃ x ( ȷ 2 ) − Ω ̃ x ( ȷ 1 ) ‖ = ‖ ȷ 2 1 − ℏ K α , β ( ȷ 2 ) [ x ( 0 ) − F ( 0 , x 0 ) ] − ȷ 1 1 − ℏ K α , β ( ȷ 1 ) [ x ( 0 ) − F ( 0 , x 0 ) ] ‖ + ‖ ȷ 2 1 − ℏ F ( ȷ 2 , x ȷ 2 ) − ȷ 1 1 − ℏ F ( ȷ 1 , x ȷ 1 ) ‖ + ȷ 2 1 − ℏ lim λ → ∞ ∫ 0 ȷ 2 ( ȷ 2 − s ) β − 1 Q β ( ȷ 2 − s ) B λ G ( s , x ( s ) ) d s − ȷ 1 1 − ℏ ∫ 0 ȷ 1 ( ȷ 1 − s ) β − 1 Q β ( ȷ 1 − s ) B λ G ( s , x ( s ) ) d s + ȷ 2 1 − ℏ ∫ 0 ȷ 2 ( ȷ 2 − s ) β − 1 Q β ( ȷ 2 − s ) B λ B u ( s ) d s − ȷ 1 1 − ℏ lim λ → ∞ ∫ 0 ȷ 1 ( ȷ 1 − s ) β − 1 Q β ( ȷ 1 − s ) B λ B u ( s ) d s , ≤ 1 Γ ( α ( 1 − β ) ) ∫ 0 ȷ 2 ȷ 2 1 − ℏ ( ȷ 2 − s ) α ( 1 − β ) − 1 s β − 1 Q β ( s ) [ x ( 0 ) − F ( 0 , x 0 ) ] d s − ∫ 0 ȷ 1 ȷ 1 1 − ℏ ( ȷ 1 − s ) α ( 1 − β ) − 1 s β − 1 Q β ( s ) [ x ( 0 ) − F ( 0 , x 0 ) ] d s + ‖ ȷ 2 1 − ℏ F ( ȷ 2 , x ȷ 2 ) − ȷ 1 1 − ℏ F ( ȷ 1 , x ȷ 1 ) ‖ + ȷ 2 1 − ℏ lim λ → ∞ ∫ ȷ 1 ȷ 2 ( ȷ 2 − s ) β − 1 Q β ( ȷ 2 − s ) B λ G ( s , x ( s ) ) d s + ‖ lim λ → ∞ ∫ 0 ȷ 1 ȷ 2 1 − ℏ ( ȷ 2 − s ) β − 1 − ȷ 1 1 − ℏ ( ȷ 1 − s ) β − 1 Q β ( ȷ 2 − s ) B λ G ( s , x ( s ) ) d s ‖ + ȷ 1 1 − ℏ lim λ → ∞ ∫ 0 ȷ 1 − δ ( ȷ 1 − s ) β − 1 [ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ] B λ G ( s , x ( s ) ) d s + ȷ 1 1 − ℏ lim λ → ∞ ∫ ȷ 1 − δ ȷ 1 ( ȷ 1 − s ) β − 1 [ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ] B λ G ( s , x ( s ) ) d s + ȷ 2 1 − ℏ lim λ → ∞ ∫ ȷ 1 ȷ 2 ( ȷ 2 − s ) β − 1 Q β ( ȷ 2 − s ) B λ B u ( s ) d s + ‖ lim λ → ∞ ∫ 0 ȷ 1 ȷ 2 1 − ℏ ( ȷ 2 − s ) β − 1 − ȷ 1 1 − ℏ ( ȷ 1 − s ) β − 1 Q β ( ȷ 2 − s ) B λ B u ( s ) d s ‖ + ȷ 1 1 − ℏ lim λ → ∞ ∫ 0 ȷ 1 − δ ( ȷ 1 − s ) β − 1 [ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ] B λ B u ( s ) d s + ȷ 1 1 − ℏ lim λ → ∞ ∫ ȷ 1 − δ ȷ 1 ( ȷ 1 − s ) β − 1 [ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ] B λ B u ( s ) d s ≤ ∑ i = 1 12 T i ,

where

T 1 = M Γ ( α ( 1 − β ) ) Γ ( β ) ∫ ȷ 1 ȷ 2 ȷ 2 1 − ℏ ( ȷ 2 − s ) α ( 1 − β ) − 1 s β − 1 [ x ( 0 ) − F ( 0 , x 0 ) ] d s , T 2 = ȷ 2 1 − ℏ M Γ ( β ) ∫ 0 ȷ 1 ( ȷ 2 − s ) β − 1 − ( ȷ 1 − s ) β − 1 s β − 1 [ x ( 0 ) − F ( 0 , x 0 ) ] d s , T 3 = ȷ 2 1 − ℏ − ȷ 1 1 − ℏ M Γ ( β ) ∫ 0 ȷ 1 ( ȷ 1 − s ) β − 1 s β − 1 [ x ( 0 ) − F ( 0 , x 0 ) ] d s , T 4 = ȷ 2 1 − ℏ − ȷ 1 1 − ℏ K h ′ ( 1 + q ) , T 5 = ȷ 2 1 − ℏ M M λ Γ ( β ) ( ȷ 2 − ȷ 1 ) ( 1 + K ) ( 1 − ρ ) ( 1 + K ) ( 1 − ρ ) ‖ ϵ 1 ‖ L 1 p Ω ( q ) , T 6 = M M λ Ω ( q ) Γ ( β ) ‖ ∫ 0 ȷ 1 ȷ 2 1 − ℏ ( ȷ 2 − s ) β − 1 − ȷ 1 1 − ℏ ( ȷ 1 − s ) β − 1 ϵ 1 ( s ) d s ‖ , T 7 = ȷ 1 1 − ℏ M λ ( ȷ 1 ( 1 + K ) − δ ( 1 + K ) ) 1 − ρ ( 1 + K ) 1 − ρ sup [ 0 , ȷ 1 − δ ] ‖ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ‖ ‖ ϵ 1 ‖ L 1 p Ω ( q ) , T 8 = ȷ 1 1 − ℏ 2 M M λ Γ ( β ) δ ( 1 + K ) ( 1 − ρ ) ( 1 + K ) 1 − ρ ‖ ϵ 1 ‖ L 1 p Ω ( q ) , T 9 = ȷ 2 1 − ℏ M M λ M b Γ ( β ) ∫ ȷ 1 ȷ 2 ( ȷ 2 − s ) β − 1 u ( s ) d s , T 10 = M M λ M b Γ ( β ) ∫ 0 ȷ 1 ȷ 2 1 − ℏ ( ȷ 2 − s ) β − 1 − ȷ 1 1 − ℏ ( ȷ 1 − s ) β − 1 u ( s ) d s , T 11 = ȷ 1 1 − ℏ M λ M b ∫ 0 ȷ 1 − δ ( ȷ 1 − s ) β − 1 ‖ u ( s ) ‖ d s sup s ∈ [ 0 , ȷ 1 − δ ] ‖ Q β ( ȷ 2 − s ) − Q β ( ȷ 1 − s ) ‖ , T 12 = ȷ 1 1 − ℏ 2 M M λ M b Γ ( β ) ∫ ȷ 1 − δ ȷ 1 ( ȷ 1 − s ) β − 1 ‖ u ( s ) ‖ d s .

Applying Lebesgue dominated convergence theorem, we find that T 1 to T 12 → 0 as ȷ ₂ − ȷ ₁ → 0 and δ → 0. This proves the equicontinuous of Ω ̃ ( B q ) on I.

References

[1] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution of fractional differential equations with uncertainty,” Nonlinear Anal. Theor. Methods Appl., vol. 72, no. 6, pp. 2859–2862, 2010. https://doi.org/10.1016/j.na.2009.11.029.Search in Google Scholar

[2] C. Dineshkumar, K. S. Nisar, R. Udhayakumar, and V. Vijayakumar, “A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions,” Asian J. Control, vol. 24, no. 5, pp. 2378–94, 2022. https://doi.org/10.1002/asjc.2650.Search in Google Scholar

[3] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, A. Shukla, and K. S. Nisar, “A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order r ϵ (1, 2) with delay,” Chaos Solit. Fractals, vol. 153, p. 111565, 2021. https://doi.org/10.1016/j.chaos.2021.111565.Search in Google Scholar

[4] K. Diethelm and A. D. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity,” in Scientific Computing in Chemical Engineering II, Berlin, Heidelberg, Springer, 1999, pp. 217–224.10.1007/978-3-642-60185-9_24Search in Google Scholar

[5] K. Diethelm, The Analysis of Fractional Differential Equations, An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Berlin, Springer-Verlag, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[6] D. D. Demir, N. Bildik, and B. G. Sinir, “Application of fractional calculus in the dynamics of beams,” Bound. Value Probl., vol. 135, pp. 1687–2770, 2012.10.1186/1687-2770-2012-135Search in Google Scholar

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amesterdam, Elsevier, 2006.Search in Google Scholar

[8] V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic System, Cambridge, Cambridge Scientific Publishers, 2009.Search in Google Scholar

[9] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Anal., vol. 69, pp. 2677–2682, 2008. https://doi.org/10.1016/j.na.2007.08.042.Search in Google Scholar

[10] Y. K. Ma, K. Kavitha, W. Albalawi, A. Shukla, K. S. Nisar, and V. Vijayakumar, “An analysis on the approximate controllability of Hilfer fractional neutral differential systems in Hilbert spaces,” Alex. Eng. J., vol. 61, no. 9, pp. 7291–7302, 2022. https://doi.org/10.1016/j.aej.2021.12.067.Search in Google Scholar

[11] C. Michele, “Linear model of dissipation whose Q is almost frequency independent-II,” Geophys. J. Int., vol. 13, no. 5, pp. 529–539, 1967.10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[12] K. S. Miller and B. Ross, An Introduction To the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993.Search in Google Scholar

[13] M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, N. Sakthivel, and K. Kaliraj, “Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r ϵ (1, 2),” Optim. Control Appl. Methods, vol. 43, no. 4, pp. 996–1019, 2022. https://doi.org/10.1002/oca.2867.Search in Google Scholar

[14] M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, and H. M. Baskonus, “On the approximate controllability results for fractional integrodifferential systems of order 1 < r < 2 with sectorial operators,” J. Comput. Appl. Math., vol. 415, p. 114492, 2022. https://doi.org/10.1016/j.cam.2022.114492.Search in Google Scholar

[15] M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, and S. Rezapour, “New discussion on nonlocal controllability for fractional evolution system of order 1 < r < 2,” Adv. Differ. Equ., vol. 2021, no. 1, pp. 1–19, 2021. https://doi.org/10.1186/s13662-021-03630-3.Search in Google Scholar

[16] 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, Academic Press, 1999.Search in Google Scholar

[17] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore, World Scientific, 2014.10.1142/9069Search in Google Scholar

[18] R. Hilfer, Application of Fractional Calculus in Physics, Singapore, World Scientific, 2000.10.1142/3779Search in Google Scholar

[19] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, New York, Elsevier, 2015.10.1016/B978-0-12-804277-9.50002-XSearch in Google Scholar

[20] Y. K. Chang, A. Anguraj, and M. Mallika Arjunan, “Existence results for non-densely defined neutral impulsive conditions differential inclusions with nonlocal conditions,” J. Appl. Math. Comput., vol. 28, no. 1, pp. 79–91, 2008. https://doi.org/10.1007/s12190-008-0078-8.Search in Google Scholar

[21] K. Kavitha and V. Vijayakumar, “A discussion concerning to partial-approximate controllability of Hilfer fractional system with nonlocal conditions via approximating method,” Chaos Solit. Fractals, vol. 157, p. 111924, 2022. https://doi.org/10.1016/j.chaos.2022.111924.Search in Google Scholar

[22] K. Kavitha, V. Vijayakumar, R. Udhayakumar, N. Sakthivel, and K. S. Nisar, “A note on approximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay,” Math. Methods Appl. Sci., vol. 44, no. 6, pp. 4428–4447, 2020. https://doi.org/10.1002/mma.7040.Search in Google Scholar

[23] K. Kavitha and V. Vijayakumar, “An analysis regarding to approximate controllability for Hilfer fractional neutral evolution hemivariational inequality,” Qual. Theory Dyn. Syst., vol. 21, no. 3, pp. 1–22, 2022. https://doi.org/10.1007/s12346-022-00611-z.Search in Google Scholar

[24] A. Singh, A. Shukla, V. Vijayakumar, and R. Udhayakumar, “Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces,” Chaos Solit. Fractals, vol. 150, p. 111095, 2021. https://doi.org/10.1016/j.chaos.2021.111095.Search in Google Scholar

[25] A. Shukla, N. Sukavanam, and D. N. Pandey, “Approximate controllability of semilinear stochastic control system with nonlocal conditions,” Nonlinear Dynam. Syst. Theor., vol. 15, no. 3, pp. 321–333, 2015.Search in Google Scholar

[26] A. Shukla, V. Vijayakumar, and K. S. Nisar, “A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order r ϵ (1, 2),” Chaos Solit. Fractals, vol. 154, p. 111615, 2022. https://doi.org/10.1016/j.chaos.2021.111615.Search in Google Scholar

[27] A. Shukla, N. Sukavanam, and D. N. Pandey, “Complete controllability of semilinear stochastic systems with delay in both state and control,” Math. Rep., vol. 18, pp. 247–259, 2016.10.1093/imamci/dnw059Search in Google Scholar

[28] V. Vijayakumar, “Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille–Yosida operators,” Int. J. Control, vol. 92, no. 9, pp. 2210–2222, 2019. https://doi.org/10.1080/00207179.2018.1433331.Search in Google Scholar

[29] V. Vijayakumar, S. K. Panda, K. S. Nisar, and H. M. Baskonus, “Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay,” Numer. Methods Part. Differ. Equ., vol. 37, no. 2, pp. 1200–1221, 2020. https://doi.org/10.1002/num.22573.Search in Google Scholar

[30] V. Vijayakumar, C. Ravichandran, and R. Murugesu, “Nonlocal controllability of mixed Volterra–Fredholm type fractional semilinear integro-differential inclusions in Banach spaces,” Dyn. Continuous Discrete Impuls. Syst., vol. 20, nos. 4–5b, pp. 485–502, 2013.Search in Google Scholar

[31] V. Vijayakumar, C. Ravichandran, K. S. Nisar, and K. D. Kucche, “New discussion on approximate controllability results for fractional Sobolev type Volterra–Fredholm integro-differential systems of order 1 < r < 2,” Numer. Methods Part. Differ. Equ., pp. 1–20, 2021. https://doi.org/10.1002/num.22772.Search in Google Scholar

[32] W. K. Williams, V. Vijayakumar, R. Udhayakumar, and K. S. Nisar, “A new study on existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces,” Numer. Methods Part. Differ. Equ., vol. 37, no. 2, pp. 949–961, 2020. https://doi.org/10.1002/num.22560.Search in Google Scholar

[33] F. Xianlong and L. Xingbo, “Controllability of non-densely defined neutral functional differential systems in abstract space,” Chin. Ann. Math. Ser. B, vol. 28, pp. 243–252, 2007. https://doi.org/10.1007/s11401-005-0028-9.Search in Google Scholar

[34] H. M. Ahmed, “Controllability for Sobolev type fractional integro-differential systems in a Banach space,” Adv. Differ. Equ., vol. 2012, pp. 167, 2012.10.1186/1687-1847-2012-167Search in Google Scholar

[35] H. Gu and J. Trujillo, “Existence of mild solution for evolution equation with Hilfer fractional derivative,” Appl. Math. Comput., vol. 257, pp. 344–354, 2015. https://doi.org/10.1016/j.amc.2014.10.083.Search in Google Scholar

[36] K. Kavitha, K. S. Nisar, A. Shukla, V. Vijayakumar, and S. Rezapour, “A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems,” Adv. Differ. Equ., vol. 2021, no. 1, pp. 1–18, 2021. https://doi.org/10.1186/s13662-021-03624-1.Search in Google Scholar

[37] V. Singh, “Controllability of Hilfer fractional differential systems with non-dense domain,” Numer. Funct. Anal. Optim., vol. 40, no. 13, pp. 1572–1592, 2019. https://doi.org/10.1080/01630563.2019.1615947.Search in Google Scholar

[38] V. Vijayakumar and R. Udhayakumar, “Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay,” Chaos Solit. Fractals, vol. 139, p. 110019, 2020. https://doi.org/10.1016/j.chaos.2020.110019.Search in Google Scholar

[39] J. R. Wang and Y. R. Zhang, “Nonlocal initial value problems for differential equation with Hilfer fractional derivative,” Appl. Math. Comput., vol. 266, pp. 850–859, 2015. https://doi.org/10.1016/j.amc.2015.05.144.Search in Google Scholar

[40] H. Gu, Y. Zhou, B. Ahmad, and A. Ahmad, “Integral solutions of fractional evolution equations with non-dense domain,” Electron. J. Differ. Equ., vol. 145, pp. 1–15, 2017.Search in Google Scholar

[41] K. Jothimani, K. Kaliraj, Z. Hammouch, and C. Ravichandran, “New results on controllability in the framework of fractional integrodifferential equations with nondense domain,” Eur. Phys. J. Plus, vol. 134, no. 441, pp. 01–10, 2019. https://doi.org/10.1140/epjp/i2019-12858-8.Search in Google Scholar

[42] G. M. Mophou and G. M. N’Guerekata, “On integral solutions of some nonlocal fractional differential equations with nondense domain,” Nonlinear Anal., vol. 71, no. 10, pp. 4668–4675, 2009. https://doi.org/10.1016/j.na.2009.03.029.Search in Google Scholar

[43] G. D. Prato and E. Sinestrari, “Differential operators with non-dense domain,” Ann. Della Scuola Norm. Super. Pisa, vol. 14, pp. 85–344, 1987.Search in Google Scholar

[44] L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” J. Math. Anal. Appl., vol. 162, pp. 494–505, 1991. https://doi.org/10.1016/0022-247x(91)90164-u.Search in Google Scholar

[45] L. Byszewski and H. Akca, “On a mild solution of a semilinear functional-differential evolution nonlocal problem,” J. Appl. Math. Stochastic Anal., vol. 10, no. 3, pp. 265–271, 1997. https://doi.org/10.1155/s1048953397000336.Search in Google Scholar

[46] S. Ji, G. Li, and M. Wang, “Controllability of impulsive differential systems with nonlocal conditions,” Appl. Math. Comput., vol. 217, pp. 6981–6989, 2011. https://doi.org/10.1016/j.amc.2011.01.107.Search in Google Scholar

[47] J. Liang and H. Yang, “Controllability of fractional integro-differential evolution equations with nonlocal conditions,” Appl. Math. Comput., vol. 254, pp. 20–29, 2015. https://doi.org/10.1016/j.amc.2014.12.145.Search in Google Scholar

[48] J. R. Wang, Z. Fan, and Y. Zhou, “Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces,” J. Optim. Theor. Appl., vol. 154, no. 1, pp. 292–302, 2012. https://doi.org/10.1007/s10957-012-9999-3.Search in Google Scholar

[49] A. Pazy, Semilgroups of Linear Operators and Applications to Partial Differential Equations, New York, NY, Springer-Verlag, 1983.10.1007/978-1-4612-5561-1Search in Google Scholar

[50] R. Hilfer, “Experimental evidence for fractional time evolution in glass materials,” Chem. Phys., vol. 284, pp. 399–408, 2002. https://doi.org/10.1016/s0301-0104(02)00670-5.Search in Google Scholar

[51] Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Comput. Math. Appl., vol. 59, pp. 1063–1077, 2010.10.1016/j.camwa.2009.06.026Search in Google Scholar

[52] J. Banas and K. Goebel, “Measure of noncompactness in Banach spaces,” in Lecture Notes in Pure and Applied Mathematics, New York, Marcel Dekker, 1980.Search in Google Scholar

[53] K. Deimling, Multivalued Differential Equations, Berlin, De Gruyter, 1992.10.1515/9783110874228Search in Google Scholar

[54] H. Mönch, “Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces,” Nonlinear Anal., vol. 4, no. 5, pp. 985–999, 1980. https://doi.org/10.1016/0362-546x(80)90010-3.Search in Google Scholar

[55] L. Byszewski, “Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt, = F(x, t, u, ux),” J. Appl. Math. Stochastic Anal., vol. 3, no. 3, pp. 163–168, 1990. https://doi.org/10.1155/s1048953390000156.Search in Google Scholar

Received: 2021-11-01

Accepted: 2022-09-18

Published Online: 2022-10-06

Discussion on controllability of non-densely defined Hilfer fractional neutral differential equations with finite delay

Abstract

References

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