Miscellaneous properties of Sturm-Liouville problems in multiplicative calculus

Year 2023, Volume: 72 Issue: 4, 1141 - 1154, 29.12.2023

Güler Başak Öznur , Güher Gülçehre Özbey , Yelda Aygarküçükevcilioğlu , Rabia Aktaş

https://doi.org/10.31801/cfsuasmas.1272953

Abstract

The purpose of this paper is to investigate some properties of multiplicative regular and periodic Sturm-Liouville problems given in general form. We first introduce regular and periodic Sturm-Liouville (S-L) problems in multiplicative analysis by using some algebraic structures. Then, we discuss the main properties such as orthogonality of different eigenfunctions of the given problems. We show that the eigenfunctions corresponding to same eigenvalues are unique modulo a constant multiplicative factor and reality of the eigenvalues of multiplicative regular S-L problems. Finally, we present some examples to illustrate our main results.

Keywords

Multiplicative calculus, multiplicative derivative, multiplicative integral, Sturm-Liouville equation

References

• Aniszewska, D., Multiplicative Runge-Kutta method, Nonlinear Dynamics, 50 (2007), 265-272. https://doi.org/10.1007/s11071-006-9156-3.
• Bashirov, A. E., Mısırlı, E., Özyapıcı, A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1) (2008), 36-48. https://doi.org/10.1016/j.jmaa.2007.03.081.
• Bashirov, A. E., Mısırlı, E., Tandoğdu, Y., Özyapıcı, A., On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities, 26(4) (2011), 425-438. https://doi.org/10.1007/s11766-011-2767-6.
• Bashirov, A. E., Riza, M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1) (2011), 75-85.
• Campillay-Llanos, W., Guevara, F., Pinto, M., Torres, R., Differential and integral proportional calculus: how to find a primitive for f(x)=1/2πe(1/2)x2, International Journal of Mathematical Education in Science and Technology, 52(3) (2021), 463-476. https://doi.org/10.1080/0020739X.2020.1763489.
• Çakmak, A. F., Başar, F., Some new results on sequence spaces with respect to non-Newtonian calculus, Journal of Inequalities and Applications, 1 (2012), 1-7. https://doi:10.1186/1029-242X-2012-228.
• Filip, D., Piatecki, C., A non-Newtonian examination of the theory of exogenous economic growth, Mathematica Aeterna, 4(2) (2014), 101–117.
• Filip, D., Piatecki, C., An overwiew on the non-Newtonian calculus and its potential applications to economics, Applied Mathematics and Computation, 187(1) (2007), 68-78. https://hal.science/hal-00945788.
• Florack, L., Van Assen, H., Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, 42(1), (2012) 64-75. https://doi:10.1007/s10851-011-0275-1.
• Frederico, G. S. F., Odzijewicz, T., Torres, D. F. M., Noether’s theorem for non-smooth externals of variational problems with time delay, Applicable Analysis, 93 (2014), 153-170. http://dx.doi.org/10.1080/00036811.2012.762090.
• Göktas, S., A New Type of Sturm-Liouville equation in the non-Newtonian calculus, Journal of Function Spaces, (2021), 8 pages. https://doi.org/10.1155/2021/5203939.
• Grossman, M., An introduction to non-Newtonian calculus, International Journal of Mathematical Education in Science and Technology, 10(4) (1979), 525-528. https://doi.org/10.1080/0020739790100406.
• Grossman, M., Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, 1972.
• Gülsen, T., Yılmaz, E., Göktas, S., Multiplicative Dirac system, Kuwait Journal of Science, (2021). https://doi:10.48129/kjs.13411.
• Gurefe, Y., Kadak, Y., Misirli, E., Kurdi, A., A new look at the classical sequence spaces by using multiplicative calculus, University Politehnica of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics, 78(2) (2016), 9-20.
• Jost, J., Mathematical Methods in Biology and Neurobiology, Universitext, Springer, New York, 1972.
• Lemos-Paiao, A. P., Torres, C. J., Venturino, D. F. M., Optimal control of aquatic diseases: A case study of Yemen’s cholera outbreak, Journal of Optimization Theory and Applications, 185 (2020), 1008-1030. https://doi.org/10.1007/s10957-020-01668-z.
• Mora, M., Cordova-Lepe, F., Del-Valle, R., A non-Newtonian gradient for counter detection in images with multiplicative noise., Pattern Recognition Letter, 33 (2012), 1245-1256. https://doi.org/10.1016/j.patrec.2012.02.012.
• Özcan, S., Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Mathematics, 5(2) (2020), 1505-1518. https://doi.org/10.3934/math.2020103.
• Özyapıcı, A., Bilgehan, B., Finite product representation via multiplicative calculus and its applications to exponential signal processing, Numer. Algorithms, 71(2) (2016), 475-489. https://doi.org/10.1007/s11075-015-0004-8.
• Pinto, M., Torres, R., Campillay-Llanos, W., Guevara-Morales, F., Applications of proportional calculus and a non-Newtonian logistic growth model, Proyecciones, 39 (2020), 1471–1513. http://dx.doi.org/10.22199/issn.0717-6279-2020-06-0090.
• Silva, C. J., Torres., D. F. M., Two-dimensional Newton’s problem of minimal resistance, Control Cybernet, 35 (2006), 965-975. https://doi.org/10.3390/axioms10030171.
• Stanley, D., A multiplicative calculus, Primus, 9(4) (1999), 310-326. https://doi.org/10.1080/10511979908965937.
• Torres, D. F. M., On a non-Newtonian calculus of variations, Axioms, 10(3) (2021), 15 pages. https://doi.org/10.3390/axioms10030171.
• Waseem, M., Muhammad, M., Aslam Noor, F., Ahmed Shah, F., Inayat Noor, K., An efficient technique to solve nonlinear equations using multiplicative calculus, Turkish Journal of Mathematics, 42(2) (2018), 679–691. https://doi.org/10.3906/mat-1611-95.
• Yalçın, N., C¸ elik, E., Solution of multiplicative homogeneous linear differential equations constant exponentials, New Trends in Mathematical Sciences, 6(2) (2018), 58–67. http://dx.doi.org/10.20852/ntmsci.2018.270.
• Yalçın, N., Çelik, E., Multiplicative Cauchy-Euler and Legendre Differential Equation, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(3) (2019), 373 - 382. https://doi.org/10.17714/gumusfenbil.451718.
• Yalçın, N., The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials, Rendiconti del Circolo Matematico di Palermo Series 2, 70(1) (2021), 9-21. http://dx.doi.org/10.1007/s12215-019-00474-5.
• Yalçın N., Dedeturk, M., Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method, AIMS Mathematics, 6(4) (2021), 3393-3409. https://doi.org/10.3934/math.2021203.
• Yener, G., Emiroglu, İ., A q -analogue of the multiplicative calculus:q -multiplicative calculus, Discrete and Continuous Dynamical System, 8(6) (2015), 1435–1450.
• Yılmaz, E., Multiplicative Bessel equation and its spectral properties, Ricerche di Matematica, (2021). https://doi.org/10.1007/s11587-021-00674-1.
• Zettl, A., Sturm–Liouville Theory, American Mathematical Society, 2010.