Analysis of a coupled system of \(\psi\)-Caputo fractional derivatives with multipoint-multistrip integral type boundary conditions. (English) Zbl 07828224
Summary: In this paper, we investigate the existence of solutions for a new coupled system of fractional differential equations that involves \(\psi\)-Caputo fractional derivatives equipped with coupled integro multistrip-multipoint boundary conditions. The uniqueness result for the given problem is obtained by utilizing the Banach contraction principle, while the existence results are established with the help of Schaefer’s fixed point theorem under specific assumptions. We also discuss the Ulam-Hyers stability for the problem at hand. Numerical examples are constructed for the illustration of the abstract results.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47H10 | Fixed-point theorems |
| 34D10 | Perturbations of ordinary differential equations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
coupled system; integro-multipoint-multistrip boundary conditions; \(\psi\)-Caputo fractional derivative; Schaefer’s fixed point theorem; Ulam-Hyers stabilityReferences:
| [1] | Abdo, MS, Qualitative analyses of \(\psi \)-Caputo type fractional integrodifferential equations in Banach spaces, J. Adv. Appl. Comput. Math., 2022, 9, 2022 |
| [2] | Abbas, N.; Ali, M.; Shatanawi, W.; Mustafa, Z., Thermodynamic properties of Second-grade micropolar nanofluid flow past an exponential curved Riga stretching surface with Cattaneo-Christov double diffusion, Alex. Eng. J., 81, 101-117, 2023 · doi:10.1016/j.aej.2023.09.020 |
| [3] | Ahmad, M.; Jiang, J.; Zada, A.; Ali, Z.; Fu, Z.; Xu, J., Hyers-Ulam-Mittag-Leffler Stability for a system of fractional neutral differential equations, Discrete Dyn. Nat. Soc., 2020, 1-10, 2020 · Zbl 1459.34004 · doi:10.1155/2020/2786041 |
| [4] | Ahmad, B.; Kerthikeyan, P.; Buvaneswari, K., Fractional differential equations with coupled slit-strips type integral boundary conditions, AIMS Math., 4, 1596-1609, 2019 · Zbl 1486.34011 · doi:10.3934/math.2019.6.1596 |
| [5] | Ali, A.; Khalid, S.; Rahmat, G.; Ali, G.; Nisar, KS; Alshahrani, B., Controllability and Ulam-Hyers stability of fractional order linear systems with variable coefficients, Alex. Eng. J., 61, 8, 6071-6076, 2022 · doi:10.1016/j.aej.2021.11.030 |
| [6] | Ahmad, B.; Ntouyas, SK, A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips type integral boundary conditions, J. Math. Sci., 226, 175-196, 2017 · Zbl 1379.34005 · doi:10.1007/s10958-017-3528-8 |
| [7] | Agarwal, RP; Assolami, A.; Alsaedi, A.; Ahmad, B., Existence results and Ulam-Hyers stability for a fully coupled system of nonlinear sequential hilfer fractional differential equations and integro-multistrip-multipoint boundary conditions, Qual. Theory Dyn. Syst., 21, 125-158, 2022 · Zbl 1505.34006 · doi:10.1007/s12346-022-00650-6 |
| [8] | Almeida, R., A caputo fractional derivative of a function with respect to another function, Common. Nonlinear Sci. Numer. Sumer., 44, 460-481, 2014 · Zbl 1465.26005 · doi:10.1016/j.cnsns.2016.09.006 |
| [9] | Ali, SM; Shatanawi, W.; Kassim, MD; Abdo, MS; Saleh, S., Investigating a class of generalized Caputo-type fractional integro-differential equations, J. Funct. Spaces, 2022, 9, 2022 · Zbl 1485.45006 |
| [10] | Abbas, N.; Shatanawi, W.; Hasan, F.; Mustafa, Z., Thermal analysis of MHD Casson-Sutterby fluid flow over exponential stretching curved sheet, Case Stud. Therm. Eng., 52, 2023 · doi:10.1016/j.csite.2023.103760 |
| [11] | Ali Khan, HN; Zada, A.; Popa, IL; Ben Moussa, S., Impulsive coupled Langevin \(\psi \)-Caputo Fractional Problem with Slit-Strips-Generalized Type Boundary Conditions, Fractal Fact., 7, 12, 837-877, 2023 · doi:10.3390/fractalfract7120837 |
| [12] | Bastos, N.: Fractional Calculus on Time Scales (2012) |
| [13] | Bekri, Z.; Erturk, VS; Kumar, P.; Govindaraj, V., Some novel analysis of two different Caputo-type fractional-order boundary value problems, Results Nonlinear Anal., 5, 3, 299-311, 2022 · doi:10.53006/rna.1114063 |
| [14] | Carvalho, A.; Pinto, CMA, A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J. Dyn. Cont., 5, 168-186, 2017 · doi:10.1007/s40435-016-0224-3 |
| [15] | Debnath, L., A brief historical introduction to fractional calculus, Int. J. Math. Ed. Sci. Technol., 35, 487-501, 2004 · doi:10.1080/00207390410001686571 |
| [16] | Ding, Y.; Wang, Z.; Ye, H., Optimal control of a fractional-order HIV-immune system with memory, IEEE Trans. Cont. Syst. Technol., 20, 763-769, 2012 · doi:10.1109/TCST.2011.2153203 |
| [17] | Faieghi, M.; Kuntanapreeda, S.; Delavari, H., LMI-based stabilization of a class of fractional order chaotic systems, Nonlinear Dyn., 72, 301-309, 2013 · Zbl 1268.93121 · doi:10.1007/s11071-012-0714-6 |
| [18] | Govindaraj, V.; Raju, KG, Controllability of fractional dynamical systems. A functional analytic approach, MCRF, 7, 4, 537-562, 2017 · Zbl 1373.93064 · doi:10.3934/mcrf.2017020 |
| [19] | Govindaraj, V.; Raju, KG, Functional approach to observability and controllability of linear fractional dynamical systems, JDSGT, 15, 2, 111-129, 2017 · Zbl 1499.93020 · doi:10.1080/1726037X.2017.1390191 |
| [20] | Hilfer, R., Applications of Fractional Calculus in Physics, 2000, Singapore: World Scientific, Singapore · Zbl 0998.26002 · doi:10.1142/3779 |
| [21] | Javidi, M.; Ahmad, B., Dynamic analysis of time fractional order phytoplankton-touic phytoplankton-zooplankton system, Ecol. Model., 318, 8-18, 2015 · doi:10.1016/j.ecolmodel.2015.06.016 |
| [22] | Jena, RM; Chakraverty, S.; Nisar, KS, Dynamical behavior of rotavirus epidemic model with non-probabilistic uncertainty under Caputo-Fabrizio derivative, Math. Methods Appl. Sci., 46, 9, 10672-10697, 2023 · Zbl 1532.92097 · doi:10.1002/mma.9146 |
| [23] | Jadhav, CP; Dale, TB; Chinchane, VL, On Dirichlet problem of time-fractional advection-diffusion equation, J. Fract. Calc. Nonlinear Syst., 4, 2, 1-13, 2023 · doi:10.48185/jfcns.v4i2.861 |
| [24] | Kilbas, AA; Trujillo, JJ, Differential equations of fractional order, methods, results and problem, Appl. Anal., 78, 153-192, 2013 · Zbl 1031.34002 · doi:10.1080/00036810108840931 |
| [25] | Lv, Z.; Ahmad, I.; Xu, J.; Zada, A., Analysis of a hybrid coupled system Of \(\psi \)-Caputo fractional dervatives with generalized slit-strips type integral boundary conditions and impulses, Discrete Dyn. Nat. Soc., 6, 618-669, 2020 |
| [26] | Lundqvist, M., Silicon Strip Detectors for Scanned Multi-slit u-Ray Imaging, 2003, Stockholm: Kungl Tekniska Hogskolan, Stockholm |
| [27] | Mellow, T.; Karkkainen, L., On the sound fields of infinitely long strips, J. Acoust. Soc. Am., 130, 153-167, 2011 · doi:10.1121/1.3596474 |
| [28] | Morsy, A.; Nisar, KS; Ravichandran, C.; Anusha, C., Sequential fractional order neutral functional integro differential equations on time scales with Caputo fractional operator over Banach spaces, AIMS Math., 8, 3, 5934-5949, 2023 · doi:10.3934/math.2023299 |
| [29] | Nisar, KS; Anusha, C.; Ravichandran, C., A non-linear fractional neutral dynamic equations: existence and stability results on time scales, AIMS Math., 9, 1, 1911-1925, 2024 · doi:10.3934/math.2024094 |
| [30] | Nisar, KS; Munusamy, K.; Ravichandran, C., Results on existence of solutions in nonlocal partial functional integrodifferential equations with finite delay in nondense domain, Alex. Eng. J., 73, 377-384, 2023 · doi:10.1016/j.aej.2023.04.050 |
| [31] | Poovarasan, R.; Kumar, P.; Nisar, KS; Govindaraj, V., The existence, uniqueness, and stability analyses of the generalized Caputo-type fractional boundary value problems, AIMS Math., 8, 7, 1-28, 2023 |
| [32] | Poovarasan, R.; Kumar, P.; Govindaraj, V.; Murillo-Arcila, M., The existence, uniqueness, and stability results for a nonlinear coupled system using \(\psi \)-Caputo fractional derivatives, Bound. Value Probl., 75, 2023, 2023 · Zbl 1525.34029 |
| [33] | Rus, IA, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26, 103-107, 2010 · Zbl 1224.34164 |
| [34] | Rizwan, R.; Zada, A.; Wang, U., Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses, Adv. Differ. Equ., 85, 1-31, 2019 · Zbl 1459.34041 |
| [35] | Sinan, M.; Ansari, KJ; Kanwal, A.; Shah, K.; Abdeljawad, T.; Abdalla, B., Analysis of the mathematical model of cutaneous Leishmaniasis disease, Alex. Eng. J., 72, 117-134, 2023 · doi:10.1016/j.aej.2023.03.065 |
| [36] | Vijayaraj, V.; Ravichandran, C.; Nisar, KS; Valliammal, N.; Logeswari, K.; Albalawi, W.; Abdel-Aty, A., An outlook on the controllability of non-instantaneous impulsive neutral fractional nonlocal systems via Atangana-Baleanu-Caputo derivative, Arab J. Basic Appl. Sci., 30, 1, 440-451, 2023 · doi:10.1080/25765299.2023.2227493 |
| [37] | Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal., 74, 792-804, 2011 · Zbl 1214.34009 · doi:10.1016/j.na.2010.09.030 |
| [38] | Wang, J.; Zada, A.; Li, W., Ulams-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. Num. Sim., 19, 553-560, 2018 · Zbl 1401.34091 · doi:10.1515/ijnsns-2017-0245 |
| [39] | Yadav, P.; Jahan, S.; Shah, K.; Peter, OJ; Abdeljawad, T., Fractional-order modelling and analysis of diabetes mellitus: utilizing the Atangana-Baleanu Caputo (ABC) operator, Alex. Eng. J., 81, 200-209, 2023 · doi:10.1016/j.aej.2023.09.006 |
| [40] | Yan, R.; Sun, S.; Lu, H.; Zhao, Y., Existence of solutions for fractional differential equations with integral boundary condition, Adv. Differ. Equ., 2014, 25-38, 2014 · Zbl 1343.34028 · doi:10.1186/1687-1847-2014-25 |
| [41] | Zada, A.; Ali, W.; Farina, S., Ulam-Hyers stability of nonlinear differential equations with fractional integrable impulsis, Math. Methods Appl. Sci., 40, 5502-5514, 2017 · Zbl 1387.34026 · doi:10.1002/mma.4405 |
| [42] | Zada, A.; Ali, S.; Li, Y., Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ., 2017, 1-26, 2017 · Zbl 1444.34083 · doi:10.1186/s13662-017-1376-y |
| [43] | Zeidler, E., Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators, 2013, Berlin: Springer, Berlin |
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