On Riemann-Liouville Operators of type (α, k, P) and Related Function Spaces
DOI:
https://doi.org/10.19136/jobs.a12n34.6686Keywords:
Fractional Integral, Fractional Derivative, k-Riemann-Liouville operators, (α, k, P)-Riemann-Liouville operators, function spaces, path choices, arc-length functionsAbstract
In this paper, a first study of the integrals and differentials of type (α, k, P) was carried out. A generalization of the k-fractional operators proposed in previous works for one-variable functions was obtained, alongside with the trajectory fractional recently proposed. Relevant associated function spaces were proposed and studied, as well as basic properties concerning this generalization. Moreover, it was proven that several known fractional derivatives and integrals are particular cases of our proposal and finally, we complemented with fundamental theorems of calculus for these operators.
References
[1] E. G. Bazhlekova, “Fractional evolution equations in banach spaces”, Ph.D. dissertation, Technische Universiteit Eindhoven, 2001. doi: 10.6100/IR549476 [Online]. Available: https: //research.tue.nl/en/publications/fractional-evolution-equations-in-banachspaces( dc185d63-bb34-4122-a3ea-34033f791577).html
[2] Y. Luchko and M. Yamamoto, “The general fractional derivative and related fractional differential equations”, Mathematics, vol. 8, no. 12, p. 2115, Nov. 26, 2020, issn: 2227-7390. doi: 10.3390/math8122115 [Online]. Available: https://www.mdpi.com/2227-7390/8/12/2115
[3] V. E. Tarasov, “General fractional calculus in multi-dimensional space: Riesz form”, Mathematics, vol. 11, no. 7, p. 1651, Mar. 29, 2023, issn: 2227-7390. doi: 10.3390/math11071651 [Online]. Available: https://www.mdpi.com/2227-7390/11/7/1651
[4] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach Science Publishers, 1993.
[5] A. Kochubei and Y. Luchko, Eds., Handbook of Fractional Calculus with Applications, red. by J. Tenreiro-Machado, vol. 1, 8 vols., Berlin/Boston: De Gruyter, 2019, isbn: 978-3-11-057162-2.
[6] M. Aziz-Alaoui, “Study of a leslie–gower-type tritrophic population model”, Chaos, Solitons & Fractals, vol. 14, no. 8, pp. 1275–1293, Nov. 2002, issn: 09600779. doi: 10.1016/S0960-0779(02)00079- 6 [Online]. Available: https://linkinghub.elsevier.com/retrieve/pii/S0960077902000796
[7] H.-L. Li, L. Zhang, C. Hu, Y.-L. Jiang, and Z. Teng, “Dynamical analysis of a fractionalorder predator-prey model incorporating a prey refuge”, Journal of Applied Mathematics and Computing, vol. 54, no. 1, pp. 435–449, Jun. 2017, issn: 1598-5865, 1865-2085. doi: 10.1007/
s12190-016-1017-8 [Online]. Available: http://link.springer.com/10.1007/s12190-016-1017-8
[8] J. E. Escalante-Mart´ınez, P. Toledo, and J. A. Zavaleta-Viveros, “Fractional derivatives and hopf bifurcation in ecological interactions among plants, pollinators, and herbivores”, in Advances in Fractional Calculus, J. J. Rosales Garc´ıa, J. E. Escalante-Mart´ınez, F. A. God´ınez Rojano, J. C. Mendiola Fuentes, and L. Torres, Eds., vol. 21, Series Title: Modeling and Optimization in Science and Technologies, Cham: Springer Nature Switzerland, 2025, pp. 49– 60, isbn: 978-3-031-83988-7. doi: 10.1007/978- 3- 031- 83989- 4_3 [Online]. Available: https://link.springer.com/10.1007/978-3-031-83989-4_3
[9] J. F. Gomez-Aguilar, J. J. Rosales-Garcıa, J. J. Bernal-Alvarado, T. Cordova-Fraga, and R. Guzman-Cabrera, “Fractional mechanical oscillators”, Rev. Mex. Fis., 2012.
[10] J. F. Gomez-Aguilar, T. Cordova-Fraga, J. E. Escalante-Martınez, C. Calderon-Ramon, and R. F. Escobar-Jimenez, “Electrical circuits described by a fractional derivative with regular kernel”, Rev. Mex. Fis., 2016.
[11] S. Alizadeh, D. Baleanu, and S. Rezapour, “Analyzing transient response of the parallel RCL circuit by using the caputo–fabrizio fractional derivative”, Advances in Difference Equations, vol. 2020, no. 1, p. 55, Dec. 2020, issn: 1687-1847. doi: 10 . 1186 / s13662 - 020 - 2527 -
0 [Online]. Available: https : / / advancesindifferenceequations . springeropen . com / articles/10.1186/s13662-020-2527-0
[12] S. Majee, T. K. Kar, S. Jana, D. K. Das, and J. J. Nieto, “Complex dynamics and fractionalorder optimal control of an epidemic model with saturated treatment and incidence”, International Journal of Bifurcation and Chaos, vol. 33, no. 16, p. 2 350 192, Dec. 30, 2023,
issn: 0218-1274, 1793-6551. doi: 10.1142/S0218127423501924 [Online]. Available: https: //www.worldscientific.com/doi/10.1142/S0218127423501924
[13] S. Soulaimani and A. Kaddar, “Analysis and optimal control of a fractional order SEIR epidemic model with general incidence and vaccination”, IEEE Access, vol. 11, pp. 81 995–82 002, 2023, issn: 2169-3536. doi: 10 . 1109 / ACCESS . 2023 . 3300456 [Online]. Available: https://ieeexplore.ieee.org/document/10198259/
[14] S. Mangal, O. Misra, and J. Dhar, “Modeling infectious respiratory diseases considering fear effect and latent period”, Results in Control and Optimization, vol. 16, p. 100 439, Sep. 2024, issn: 26667207. doi: 10 . 1016 / j . rico . 2024 . 100439 [Online]. Available: https :
//linkinghub.elsevier.com/retrieve/pii/S2666720724000699
[15] S. Borjon-Espejel, J. Escalante-Martinez, and P. Padilla-Longoria, “Newtonian gravity and MOND: A fractional fourier approach”, Indian Journal of Physics, vol. 96, no. 12, pp. 3405–3411, Oct. 2022, issn: 0973-1458, 0974-9845. doi: 10.1007/s12648-022-02296-1 [Online].
Available: https://link.springer.com/10.1007/s12648-022-02296-1
[16] J. Mendiola-Fuentes, E. Guerrero-Ruiz, and J. Rosales-Garc´ıa, “Multivariate mittag-lefflersolution for a forced fractional-order harmonic oscillator”, Mathematics, vol. 12, no. 10, p. 1502, May 11, 2024, issn: 2227-7390. doi: 10.3390/math12101502 [Online]. Available:
https://www.mdpi.com/2227-7390/12/10/1502
[17] M. Li, “Analytic theory of seven classes of fractional vibrations based on elementary functions: A tutorial review”, Symmetry, vol. 16, no. 9, p. 1202, Sep. 12, 2024, issn: 2073-8994. doi: 10.3390/sym16091202 [Online]. Available: https://www.mdpi.com/2073-8994/16/9/1202
[18] Y. Luchko, “A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis”, Mathematical Modelling of Natural Phenomena, vol. 11, no. 3, A. Nepomnyashchy and V. Volpert, Eds., pp. 1–17, 2016, issn: 0973-5348, 1760-6101. doi: 10.1051/
mmnp/201611301 [Online]. Available: http://www.mmnp- journal.org/10.1051/mmnp/201611301
[19] S. Fomin, V. Chugunov, and T. Hashida, “Mathematical modeling of anomalous diffusion in porous media”, Fractional Differential Calculus, no. 1, pp. 1–28, 2011, issn: 1847-9677. doi: 10.7153/fdc-01-01 [Online]. Available: https://fdc.ele-math.com/01-01
[20] H. Sun, W. Chen, C. Li, and Y. Chen, “Fractional differential models for anomalous diffusion”, Physica A: Statistical Mechanics and its Applications, vol. 389, no. 14, pp. 2719–2724, Jul. 2010, issn: 03784371. doi: 10.1016/j.physa.2010.02.030 [Online]. Available: https:
//linkinghub.elsevier.com/retrieve/pii/S0378437110001573
[21] V. E. Tarasov and S. S. Tarasova, “Fractional derivatives and integrals: What are they needed for?”, Mathematics, vol. 8, no. 2, p. 164, Jan. 25, 2020, issn: 2227-7390. doi: 10.3390/ math8020164 [Online]. Available: https://www.mdpi.com/2227-7390/8/2/164
[22] S. T. Mohyud-Din, A. Ali, and B. Bin-Mohsin, “On biological population model of fractional order”, International Journal of Biomathematics, vol. 09, no. 5, p. 1 650 070, Sep. 2016, issn: 1793-5245, 1793-7159. doi: 10.1142/S1793524516500704 [Online]. Available: https://www. worldscientific.com/doi/abs/10.1142/S1793524516500704
[23] X. Zhou, Y. Mao, C. Zhang, and Q. He, “A comprehensive performance improvement control method by fractional order control”, IEEE Photonics Journal, vol. 10, no. 5, pp. 1–11, Oct.
2018, issn: 1943-0655, 1943-0647. doi: 10.1109/JPHOT.2018.2865032 [Online]. Available: https://ieeexplore.ieee.org/document/8434365/
[24] V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (Nonlinear physical science). Beijing: Higher Education Press, 2010, 504 pp., isbn: 978-7-04-029473-6.
[25] M. Al-Refai and Y. Luchko, “The general fractional integrals and derivatives on a finite interval”, Mathematics, vol. 11, no. 4, p. 1031, Feb. 17, 2023, issn: 2227-7390. doi: 10.3390/ math11041031 [Online]. Available: https://www.mdpi.com/2227-7390/11/4/1031
[26] J. Cheng and W. Dai, “Fractional vector taylor and cauchy mean value formulas”, Journal of Fractional Calculus and Applications, vol. 11, no. 2, pp. 130–147, Jul. 2020, issn: 2090-5858.
[27] R. C. Balc´azar-Araiza and M. Navarro, “On riemann-liouville operators for functions of one and several variables”, Journal of Basic Sciences, vol. 11, no. 30, pp. 1–15, 2025, issn: 2448-4997.
[28] R. C. Balc´azar-Araiza, “A fractional fundamental theorem of inmersions”, Master Thesis, Universidad Aut´onoma de Yucat´an, M´exico, 2025, 139 pp.
[29] S. Mubeen and G. M. Habibullah, “K -fractional integrals and application”, International Journal of Contemporary Mathematical Sciences, vol. 7, no. 2, pp. 89–94, 2012.
[30] G. A. Dorrego, “An alternative definition for the k-riemann-liouville fractional derivative”, Applied Mathematical Sciences, vol. 9, pp. 481–491, 2015, issn: 13147552. doi: 10.12988/ams.2015.411893 [Online]. Available: http://www.m-hikari.com/ams/ams-2015/ams-9- 12-2015/411893.html
[31] R. D´ıaz and E. Pariguan, “On hypergeometric functions and pochhammer k-symbol”, Divulgaciones Matem´aticas, vol. 15, no. 2, pp. 179–192, 2007.
[32] L. G. Romero and L. L. Luque, “K-weyl fractional derivative, integral and integral transform”, International Journal of Contemporary Mathematical Sciences, vol. 8, pp. 263–270, 2013, issn: 1314-7544. doi: 10.12988/ijcms.2013.13028 [Online]. Available: http://www.mhikari.
com/ijcms/ijcms-2013/5-8-2013/13028.html
[33] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory And Applications of Fractional Differential Equations (North-Holland Mathematical Studies 204), red. by Jan van Mill. Elsevier, 2006.
[34] V. Guillemin and A. Pollack, Differential topology. Englewood Cliffs, N.J: Prentice-Hall, 1974, 222 pp., isbn: 978-0-13-212605-2.
[35] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Oct. 22, 2001. doi: 10.48550/arXiv.math/0110241 arXiv: math/0110241. [Online]. Available: http://arxiv.org/abs/math/0110241
[36] H. Royden and P. Fitzpatrick, Real Analysis, 4th ed. China Machine Press, 2010.
[37] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Lecture Notes in Mathematics). Berlin, Heidelberg: Springer Berlin Heidelberg, 2010, vol. 2004, isbn: 978-3-642-14573-5. doi: 10.1007/978-3-642-14574-2 [Online]. Available: https://link.springer.com/10.1007/ 978-3-642-14574-2
[38] H. D. Junghenn, Principles of analysis: measure, integration, functional analysis, and applications. Boca Raton: CRC Press, Taylor & Francis Group, 2018, 520 pp., isbn: 978-1-4987-7328-7.
[39] T. M. Apostol, Calculus II: C´alculo con funciones de varias variables y ´algebra lineal, con aplicaciones a las ecuaciones diferenciales y las probabilidades, 2nd ed., trans. by Jos´e Pla Carrera, 2 vols. Espa˜na: Editorial Revert´e, 1976, vol. 2, isbn: 978-84-291-5004-9.
[40] T. M. Apostol, Calculus I: C´alculo con funciones de una variable, con una introducci´on al ´ Algebra Lineal, trans. by Francisco V´elez Cantarell, 2 vols. M´exico: Revert´e Ediciones, 1984, vol. 1, isbn: 978-968-6708-10-3.
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