Home Mathematics Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags
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Some new uniqueness and Ulam–Hyers type stability results for nonlinear fractional neutral hybrid differential equations with time-varying lags

  • Nguyen Minh Dien
Published/Copyright: May 24, 2024
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Abstract

This paper deals with some qualitative properties of solutions to nonlinear neutral hybrid differential equations connected to ψ-Caputo fractional derivative with time-varying lags. First, we demonstrate the problem possesses a mild solution uniquely where the source function may have temporal singularities. Second, in some cases, we indicate that the problem possesses a unique mild solution under some weaker conditions than the previous one. Third, we also obtain a result on a global mild solution for the problem. Finally, the results are further enriched by studying a new type of Ulam–Hyers stability for the main equation. The main results are obtained by applying the nice inequality, first proposed and proven in this paper. Some befit examples are given to justify the applicability of the main results.

Acknowledgement

The author heartily expresses his gratitude to the handing editor and the anonymous reviewers for suggestions that improved the manuscript.

  1. Communicated by Michal Fečkan

References

[1] Arfan, M.—Mahariq, I.—Shah, K.—Abdeljawad, T.—Laouini, G.—Mohammed, P. O.: Numerical computations and theoretical investigations of a dynamical system with fractional order derivative, Alex. Eng. J. 61(3) (2022), 1982–1994.10.1016/j.aej.2021.07.014Search in Google Scholar

[2] Arfan, M.—Lashin, M. M. A.—Sunthrayuth, P.—Shah, K.—Ullah, A.—Iskakova, K.—Gorji, R. M.—Abdeljawad, T.: On nonlinear dynamics of COVID-19 disease model corresponding to nonsingular fractional order derivative, Med. Biol. Eng. Comput. 60 (2022), 3169–3185.10.1007/s11517-022-02661-6Search in Google Scholar PubMed PubMed Central

[3] Burton, T. A.: A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions, Fixed Point Theory 20(1) (2019), 107–113.10.24193/fpt-ro.2019.1.06Search in Google Scholar

[4] Camacho, E. F.—Bordons, C.: Model Predictive Control in the Process Industry, Springer-Verlag, London, 1995.Search in Google Scholar

[5] Dien, N. M.—Trong, D. D.: On the nonlinear generalized Langevin equation involving ψ-Caputo fractional derivatives, Fractals 29(6) (2021), Art. ID 2150128.10.1142/S0218348X21501280Search in Google Scholar

[6] Dien, N. M.: Generalized weakly singular Gronwall-type inequalities and their applications to fractional differential equations, Rocky Mountain J. Math. 51(2) (2021), 689–707.10.1216/rmj.2021.51.689Search in Google Scholar

[7] Dien, N. M.: Existence and continuity results for a nonlinear fractional Langevin equation with a weakly singular source, J. Integral Equations Appl. 33(3) (2021), 349–369.10.1216/jie.2021.33.349Search in Google Scholar

[8] Dien, N. M.—NANE, E.—MINH, N. D.—TRONG, D. D.: Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders, Fract. Calc. Appl. Anal. 25(3) (2022), 1166–1198.10.1007/s13540-022-00056-wSearch in Google Scholar

[9] Dien, N. M.: On mild solutions of the generalized nonlinear fractional pseudo-parabolic equation with a nonlocal condition, Fract. Calc. Appl. Anal. 25(2) (2022), 559–583.10.1007/s13540-022-00024-4Search in Google Scholar

[10] Du, F.—Lu, J. G.: Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities, Appl. Math. Comput. 375 (2020), Art. ID 125079.10.1016/j.amc.2020.125079Search in Google Scholar

[11] Erneux, T.: Applied Delay Differential Equations, Springer Sciences+Business Media, LLC, 2009.10.1007/978-0-387-74372-1_8Search in Google Scholar

[12] Erturk, V. S.—Ali, A.—Shah, K.—Kumar, P.—Abdeljawad, T.: Existence and stability results for nonlocal boundary value problems of fractional order, Bound. Value Probl. 2022 (2022), Art. No. 25.10.1186/s13661-022-01606-0Search in Google Scholar

[13] Garrappa, R.—Kaslik, E.: On initial conditions for fractional delay differential equations, Commun. Nonlinear Sci. Numer. Simul. 90 (2020), Art. ID 105359.10.1016/j.cnsns.2020.105359Search in Google Scholar

[14] Gejji, V.—Sukale, Y.—Bhalekar, S.: Solving fractional delay differential equations: a new approach, Fract. Calc. Appl. Anal. 18 (2015), 400–418.10.1515/fca-2015-0026Search in Google Scholar

[15] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.Search in Google Scholar

[16] Mahmudov, N. I.: Fractional Langevin type delay equations with two fractional derivatives, Appl. Math. Lett. 103 (2020), Art. ID 106215.10.1016/j.aml.2020.106215Search in Google Scholar

[17] Matar, M. M.: Existence of solution for fractional Neutral hybrid differential equations with finite delay, Rocky Mountain J. Math. 50(6) (2020), 2141–2148.10.1216/rmj.2020.50.2141Search in Google Scholar

[18] Radojević, D.—Lazarević, M. P.: Further results on finite-time stability of neutral nonlinear multi-term fractional order time-varying delay systems, Filomat 36(5) (2022), 1775–1787.10.2298/FIL2205775RSearch in Google Scholar

[19] Shah, K.—Abdeljawad, T.—Abdalla, B.—S Abualrub, M.: Utilizing fixed point approach to investigate piecewise equations with non-singular type derivative, AIMS Math. 7(8) (2022), 14614–14630.10.3934/math.2022804Search in Google Scholar

[20] Shah, K.—Abdeljawad, T.—Din, R. U.: To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate, Phys. A 604 (2022), Art. ID 127915.10.1016/j.physa.2022.127915Search in Google Scholar PubMed PubMed Central

[21] Shah, K.—Ahmad, I.—Nieto, J. J.—Rahman, G. U.—Abdeljawad, T..: Qualitative investigation of nonlinear fractional coupled pantograph impulsive differential equations, Qual. Theory Dyn. Syst. 21 (2022), Art. No. 131.10.1007/s12346-022-00665-zSearch in Google Scholar

[22] Tuan, H. T.—Trinh, H.: A qualitative theory of time delay nonlinear fractional-order systems, SIAM J. Control Optim. 58(3) (2020), 1491–1518.10.1137/19M1299797Search in Google Scholar

[23] Tuan, H. T.—Thai, H. D.—Garrappa, R.: An analysis of solutions to fractional neutral differential equations with delay, Commun. Nonlinear Sci. Numer. Simul. 100 (2021), Art. ID 105854.10.1016/j.cnsns.2021.105854Search in Google Scholar

[24] Wang, D.—Xiao, A.—Sun, S.: Asymptotic behavior of solutions to time fractional neutral functional differential equations, J. Comput. Appl. Math. 382 (2021), Art. ID 113086.10.1016/j.cam.2020.113086Search in Google Scholar

[25] Webb, J. R. L.: Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl. 471 (2019), 692–711.10.1016/j.jmaa.2018.11.004Search in Google Scholar

Received: 2022-11-08
Accepted: 2023-06-23
Published Online: 2024-05-24
Published in Print: 2024-04-25

© 2024 Mathematical Institute Slovak Academy of Sciences

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