Solvability of an infinite system of $ n $-th order differential equations in a paranormed sequence space and its numerical solution

Hojjatollah Amiri Kayvanloo; Mohammad Mehrabinezhad; Mahnaz Khanehgir; Reza Allahyari; Mohammad Mursaleen; Hojjatollah Amiri Kayvanloo; Mohammad Mehrabinezhad; Mahnaz Khanehgir; Reza Allahyari; Mohammad Mursaleen

doi:10.3934/math.2025755

AIMS Mathematics

2025, Volume 10, Issue 7: 16804-16821. doi: 10.3934/math.2025755

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Research article

Solvability of an infinite system of $ n $-th order differential equations in a paranormed sequence space and its numerical solution

1.
Department of Mathematics, Ma.C., Islamic Azad University, Mashhad, Iran
2.
Department of Mathematical Sciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamilnadu, India
3.
Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan
4.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received: 27 December 2024 Revised: 06 July 2025 Accepted: 17 July 2025 Published: 25 July 2025
MSC : 47H09, 47H10, 34A12, 11B68

In this paper, we introduce a ball measure of noncompactness in the Banach sequence space $ bv_q $ $ (1\leq q < \infty) $ containing the space $ l_q $. By applying the technique of measures of noncompactness and Meir–Keeler condensing mappings, we investigate the existence of solutions of an infinite system of differential equations of order $ n $ with boundary conditions. Some examples are provided to support our main results. A numerical spectral method based on Bernoulli polynomials is applied to find the approximate solution of an example.
- Bernoulli polynomials,
- infinite system,
- measure of noncompactness,
- Meir–Keeler condensing mapping,
- sequence spaces
Citation: Hojjatollah Amiri Kayvanloo, Mohammad Mehrabinezhad, Mahnaz Khanehgir, Reza Allahyari, Mohammad Mursaleen. Solvability of an infinite system of $ n $-th order differential equations in a paranormed sequence space and its numerical solution[J]. AIMS Mathematics, 2025, 10(7): 16804-16821. doi: 10.3934/math.2025755

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Abstract

In this paper, we introduce a ball measure of noncompactness in the Banach sequence space $ bv_q $ $ (1\leq q < \infty) $ containing the space $ l_q $. By applying the technique of measures of noncompactness and Meir–Keeler condensing mappings, we investigate the existence of solutions of an infinite system of differential equations of order $ n $ with boundary conditions. Some examples are provided to support our main results. A numerical spectral method based on Bernoulli polynomials is applied to find the approximate solution of an example.

References

[1]	W. M. Abd-Elhameed, H. M. Ahmed, Spectral solutions for the time-fractional heat differential equation through a novel unified sequence of Chebyshev polynomials, AIMS Math., 9 (2024), 2137–2166. http://dx.doi.org/10.3934/math.2024107 doi: 10.3934/math.2024107
[2]	A. Aghajani, J. Banaś, Y. Jalilian, Existence of solution for a class of nonlinear Volterra singular integral equation, Comput. Math. Appl., 62 (2011), 1215–1227. http://dx.doi.org/10.1016/j.camwa.2011.03.049 doi: 10.1016/j.camwa.2011.03.049
[3]	A. Aghajani, M. Mursaleen, A. S. Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci., 35B (2015), 552–566. https://doi.org/10.1016/S0252-9602(15)30003-5 doi: 10.1016/S0252-9602(15)30003-5
[4]	A. Aghajani, E. Pourhadi, Application of measure of noncompactness to $l_1$-solvability of infinite systems of second order differential equations, B. Belg. Math. Soc.-Sim., 22 (2015), 105–118. https://doi.org/10.36045/bbms/1426856862 doi: 10.36045/bbms/1426856862
[5]	A. Alotaibi, M. Mursaleen, B. A. S. Alamri, Solvability of second order linear differential equations in the sequence space $n(\phi)$, Adv. Differ. Equ., 2018 (2018), 377. https://doi.org/10.1186/s13662-018-1810-9 doi: 10.1186/s13662-018-1810-9
[6]	H. A. Kayvanloo, M. Khanehgir, R. Allahyari, A family of measures of noncompactness in the Hölder space $ C^{n, \gamma}(\mathbb{R_+})$ and its application to some fractional differential equations and numerical methods, J. Comput. Appl. Math., 363 (2020), 256–272. https://doi.org/10.1016/j.cam.2019.06.012 doi: 10.1016/j.cam.2019.06.012
[7]	H. A. Kayvanloo, M. Khanehgir, M. Mursaleen, Solution of infinite system of third-order three-point nonhomogeneous boundary value problem in weighted sequence space $bv_p^{\omega}$, Rend. Circ. Mat. Palerm. Ser., 73 (2023), 1329–1342. https://doi.org/10.1007/s12215-023-00986-1 doi: 10.1007/s12215-023-00986-1
[8]	H. A. Kayvanloo, H. Mehravaran, M. Mursaleen, R. Allahyari, A. Allahyari, Solvability of infinite systems of Caputo-Hadamard fractional differential equations in the triple sequence space $c^3(\Delta)$, J. Pseudo-Differ. Oper., 15 (2024), 26. https://doi.org/10.1007/s11868-024-00601-6 doi: 10.1007/s11868-024-00601-6
[9]	H. A. Kayvanloo, M. Mursaleen, M. Mehrabinezhad, F. P. Najafabadi, Solvability of some fractional differential equations in the Hölder space $\mathcal{H}_{\gamma}(\mathbb{R_+})$ and their numerical treatment via measures of noncompactness, Math. Sci., 17 (2022), 1–11. https://doi.org/10.1007/s40096-022-00458-0 doi: 10.1007/s40096-022-00458-0
[10]	S. H. Amora, A. Traiki, Meir-Keeler condensing operators and applications, Filomat, 35 (2021), 2175–2188. https://doi.org/10.2298/FIL2107175H doi: 10.2298/FIL2107175H
[11]	S. Banach, Sur les op$\acute{e}$rations dans les ensembles abstraits et leur application aux $\acute{e}$quations integrales, Fund. Math., 3 (1922), 133–181. Available from: http://eudml.org/doc/213289.
[12]	J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics, Marcel Dekker, New York, 60 (1980).
[13]	J. Banaś, M. Mursaleen, S. M. H. Rizvi, Existence of solutions to a boundary-value problem for an infinite system of differential equations, Electron. J. Differ. Eq., 2017 (2017), 1–12. Available from: http://ejde.math.txstate.edu.
[14]	F. Basar, Summability theory and its appliactions, Bentham Science Publishers, 2011. https://doi.org/10.2174/97816080545231120101
[15]	F. Basar, B. Altay, On the space of sequences of $p$-bounded variation and related matrix mappings, Ukr. Math. J., 55 (2003), 136–147. https://doi.org/10.1023/A:1025080820961 doi: 10.1023/A:1025080820961
[16]	M. Belhadj, A. B. Amar, M. Boumaiza, Some ﬁxed point theorems for Meir-Keeler condensing operators and application to asystem of integral equations, B. Belg. Math. Soc.-Sim., 26 (2019), 223–239. https://doi.org/10.36045/bbms/1561687563 doi: 10.36045/bbms/1561687563
[17]	D. Bugajewska, P. Kasprzak, Nonlinear composition operators in $bv_p$-spaces, Banach J. Math. Anal., 19 (2025), 47. https://doi.org/10.1007/s43037-025-00429-2 doi: 10.1007/s43037-025-00429-2
[18]	E. Doha, R. Hafez, M. Abdelkawy, S. Ezz-Eldien, T. Taha, M. Zaky, et al., Composite Bernoulli-Laguerre collocation method for a class of hyperbolic telegraph-type equations, Rom. Rep. Phys., 69 (2017), 1–21.
[19]	D. G. Duffy, Green's function with applications, Chapman and Hall/CRC Press, London, 2001.
[20]	M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79. https://doi.org/10.1112/jlms/s1-37.1.74 doi: 10.1112/jlms/s1-37.1.74
[21]	M. Ghasemi, M. Khanehgir, R. Allahyari, H. A. Kayvanloo, Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means, AIMS Math., 7 (2022), 2680–2694. https://doi.org/10.3934/math.2022151 doi: 10.3934/math.2022151
[22]	R. Hafez, E. Doha, A. Bhrawy, D. Baleanu, Numerical solutions of two-dimensional mixed Volterra-Fredholm integral equations via Bernoulli collocation method, Rom. J. Phys., 62 (2017), 1–11. Available from: http://hdl.handle.net/20.500.12416/2486.
[23]	M. Heydari, Z. Avazzadeh, New formulation of the orthonormal Bernoulli polynoimals for solving the variable-order time fractional couled Boussinesq-Burger's equations, Eng. Comput., 37 (2020), 3509–3517. https://doi.org/10.1007/s00366-020-01007-w doi: 10.1007/s00366-020-01007-w
[24]	M. Imaninezhad, M. Miri, The dual space of the sequence space $bv_p$ $1\leq p < \infty$, Acta Math. Univ. Comeniance, 1 (2010), 143–149.
[25]	A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat, 17 (2003), 59–78. https://doi.org/10.2298/FIL0317059J doi: 10.2298/FIL0317059J
[26]	K. Kuratowski, Sur les espaces complets, Fund. Math., 15 (1930), 301–309. Available from: http://eudml.org/doc/212357.
[27]	K. Kuratowski, Topology: Volume I, Academic Press, Warsaw, 1966. https://doi.org/10.1017/S0008439500030022
[28]	B. Lalrinhlua, A. M. Saeed, A. H. Ganie, R. Tiwari, S. Das, F. Mofarreh, et al., Study of vibrations in smart materials semiconductor under differential imperfect contact mechanism and nanoscale effect with electromechanical coupling effect, Acta Mech., 236 (2025), 2383–2403. https://doi.org/10.1007/s00707-025-04279-9 doi: 10.1007/s00707-025-04279-9
[29]	J. R. Loh, C. Phang, Numerical solution of Fredholm fractional integro-differential equation with right-sided Caputo's derivative using Bernoulli polynomials operational matrix of fractional derivative, Mediterr. J. Math., 16 (2019), 1–25. https://doi.org/10.1007/s00009-019-1300-7 doi: 10.1007/s00009-019-1300-7
[30]	Y. K. Ma, M. M. Raja, V. Vijayakumar, A. Shukla, W. Albalawi, K. S. Nisar, Existence and continuous dependence results for fractional evolution integrodifferential equations of order $r\in(1, 2)$, Alex. Eng. J., 61 (2022), 9929–9939. https://doi.org/10.1016/j.aej.2022.03.010 doi: 10.1016/j.aej.2022.03.010
[31]	M. Mehrabinezhad, N. Eftekhari, Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems, J. Differ. Eq. Appl., 27 (2021), 1329–1354. https://doi.org/10.1080/10236198.2021.1983551 doi: 10.1080/10236198.2021.1983551
[32]	A. Meir, E. A. Keeler, A Theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326–329. https://doi.org/10.1016/0022-247X(69)90031-6 doi: 10.1016/0022-247X(69)90031-6
[33]	S. A. Mohiuddine, A. Das, A. Alotaibi, Existence of solutions for infinite system of nonlinear $q$-fractional boundary value problem in Banach spaces, Filomat, 37 (2023), 10171–10180. https://doi.org/10.1007/978-981-96-2476-8-3 doi: 10.1007/978-981-96-2476-8-3
[34]	M. Mursaleen, Some geometric properties of a sequence space related to $l_p$, Bull. Aust. Math. Soc., 67 (2003), 343–347. https://doi.org/10.1017/S0004972700033803 doi: 10.1017/S0004972700033803
[35]	M. Mursaleen, B. Bilalov, S. M. H. Rizvi, Applications of measures of noncompactness to infinite system of fractional differential equations, Filomat, 31 (2017), 3421–3432. https://doi.org/10.2298/FIL1711421M doi: 10.2298/FIL1711421M
[36]	M. Mursaleen, S. A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $l_p$ spaces, Nonlinear Anal., 75 (2012), 2111–2115. https://doi.org/10.1016/j.na.2011.10.011 doi: 10.1016/j.na.2011.10.011
[37]	M. Mursaleen, V. Rakouevic, A survey on measures of noncompactness with some applications in infinite systems of differential equations, Aequationes Math., 96 (2022), 489–514. https://doi.org/10.1007/s00010-021-00848-0 doi: 10.1007/s00010-021-00848-0
[38]	M. Mursaleen, S. M. H. Rizvi, Solvability of infinite system of second order differential equations in $c_0$ and $l_1$ by Meir-Keeler condensing operator, Proc. Am. Math. Soc., 144 (2016), 4279–4289. https://doi.org/10.1090/proc/13048 doi: 10.1090/proc/13048
[39]	M. Mursaleen, E. Savas, Solvability of an infinite system of fractional differential equations with $p$-Laplacian operator in a new tempered sequence space, J. Pseudo-Differ. Oper., 14 (2023). https://doi.org/10.1007/s11868-023-00552-4 doi: 10.1007/s11868-023-00552-4
[40]	C. Phang, Y. T. Toh, F. S. M. Nasrudin, An operational matrix method based on poly-Bernoulli polynomials for solving fractional delay differential equations, Computation, 8 (2020), 82. https://doi.org/10.3390/computation8030082 doi: 10.3390/computation8030082
[41]	M. M. Raja, V. Vijayakumar, K. C. Veluvolu, An analysis on approximate controllability results for impulsive fractional differential equations of order $1 < r < 2$ with infinite delay using sequence method, Math. Method. Appl. Sci., 47 (2024), 336–351. https://doi.org/10.1002/mma.9657 doi: 10.1002/mma.9657
[42]	M. M. Raja, V. Vijayakumar, K. C. Veluvolu, Improved order in Hilfer fractional differential systems: Solvability and optimal control problem for hemivariational inequalities, Chaos Soliton. Fract., 188 (2024), 115558. https://doi.org/10.1016/j.chaos.2024.115558 doi: 10.1016/j.chaos.2024.115558
[43]	W. L. C. Sargent, Some sequence spaces related to the $l^p$ spaces, J. London Math. Soc., 35 (1960), 161–171. https://doi.org/10.1112/jlms/s1-35.2.161 doi: 10.1112/jlms/s1-35.2.161
[44]	Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., 62 (2024), 1323–1356. https://doi.org/10.1007/s10910-024-01592-x doi: 10.1007/s10910-024-01592-x
[45]	Y. Shi, X. Yang, The pointwise error estimate of a new energy-preserving nonlinear difference method for supergeneralized viscous Burgers' equation, Comput. Appl. Math., 44 (2025), 256–276. https://doi.org/10.1007/s40314-025-03222-x doi: 10.1007/s40314-025-03222-x
[46]	F. Tennie, L. Magri, Solving nonlinear differential equations on quantum computers: A Fokker-Planck approach, arXiv preprint, 2024. https://doi.org/10.48550/arXiv.2401.13500
[47]	A. Vakeel, A. Khan, A new type of difference sequence spaces, Appl. Sci., 12 (2010), 102–108. Available from: http://eudml.org/doc/225421.
[48]	Y. Yang, H. Li, Neural ordinary differential equations for robust parameter estimation in dynamic systems with physical priors, Appl. Soft Comput., 169 (2025). https://doi.org/10.1016/j.asoc.2024.112649
[49]	B. Zhang, Y. Tang, X. Zhang, A new method for solving variable coefficients fractional differential equations based on a hybrid of bernoulli polynomials and block pulse functions, Math. Method. Appl. Sci., 46 (2023), 8054–8073. https://doi.org/10.1002/mma.7352 doi: 10.1002/mma.7352
[50]	J. Zhang, X. Yang, S. Wang, The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations, AIMS Math., 33 (2025), 3305–3327. https://doi.org/10.3934/era.2025146 doi: 10.3934/era.2025146

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