The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies

Mohamed Bezzıou; Zoubir Dahmani; Rabha Ibrahim

doi:10.32323/ujma.1425363

TR EN

The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies

Abstract

Gronwall's inequalities are important in the study of differential equations and integral inequalities. Gronwall inequalities are a valuable mathematical technique with several applications. They are especially useful in differential equation analysis, stability research, and dynamic systems modeling in domains spanning from science and math to biology and economics. In this paper, we present new generalizations of Gronwall inequalities of integral versions. The proposed results involve $( \rho ,\varphi)-$Riemann-Liouville fractional integral with respect to another function. Some applications on differential equations involving $( \rho ,\varphi)-$Riemann-Liouville fractional integrals and derivatives are established.

Keywords

Fractional calculus, Fractional differential equation, Gronwall inequality

References

[1] B.N.N. Achar, J.W. Hanneken, T. Clarke, Damping characteristics of a fractional oscillator, Physica A., 339(2004), 311–319.
[2] Y. Adjabi, F. Jarad, T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457–5473.
[3] R. Almeida, A Gronwall inequality for a general Caputo fractional operator, Math. Inequal. Appl., 20 (2017), 1089–1105.
[4] J. Alzabut, T. Abdeljawad, F. Jarad, W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 101 (2019), 1–12.
[5] M. Bezziou, Z. Dahmani, A. Khameli, Some weighted inequalities of Chebyshev type via RL-approach, Mathematica, 60(83) (2018), 12–20.
[6] M. Bezziou, Z. Dahmani, M.Z. Sarikaya, New operators for fractional integration theory with some applications, J. Math. Extension, 12(1) (2018), 87-100.
[7] M. Bezziou, and Z. Dahmani, New integral operators for conformable fractional calculus with applications, J. Interdisciplinary Math., 25(4) (2022), 927-940.
[8] T. Blaszczyk, M. Ciesielski, Fractional oscillation equation: analytical solution and algorithm for its approximate computation, J. Vibration Control, 22(8) (2016), 2045–2052.
[9] K. Boukerrioua, Note on some nonlinear integral inequalities and applications to differential equations, Int. J. Diff. Eq., 456216 (2011) 1-15.
[10] D. Boucenna, A.B. Makhlouf, M.A. Hammami, On Katugampola fractional order derivatives and Darboux problem for differential equations, CUBO A Mathematical J., 22(1) (2020), 125-136.

[11] A. Carpintery, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer Verlag, Vienna-New York, 1997.
[12] D.N. Chalishajar, K. Karthikeyan, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Sci., 33 (2013), 758–772.
[13] Z. Dahmani, N. Bedjaoui, New generalized integral inequalities, J. Advan. Res. Appl. Math., 3(4) (2011), 58–66.
[14] Z. Dahmani, H.M. El Ard, Generalizations of some integral inequalities using Riemann-Liouville operator, Int. J. Open Problems Compt. Math., 4(4) (2011), 40–46.
[15] S.S. Dragomir, Some Gronwall Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2002.
[16] J.S. Duan, Z. Wang, S.Z. Fu, The zeros of the solution of the fractional oscillation equation, Fract. Calc. Appl. Anal., 17(1) (2014), 10–22.
[17] C.S. Drapaca, S.A. Sivaloganathan, Fractional model of continuum mechanics, J. Elast., 107 (2012), 107–123.
[18] S. Ferraoun, Z. Dahmani, Gronwall type inequalities: New fractional integral results with some applications on hybrid differential equations, Int. J. Nonlinear Anal. Appl., 12(2) (2021), 799–809.
[19] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12(3) ( 2009), 299–318.
[20] D.H. Jiang, C.Z. Bai, On coupled Gronwall inequalities involving a fractional integral operator with its applications, AIMS Math., 7 (2022), 7728–7741.
[21] U. Katugampola, New approach to a generalized fractional integral, Bull. Math. Anal. Appl., 6(4) (2014), 1–15.
[22] A. Kilbas, M.H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies. Vol. 204, 2006.
[23] V. Kiryakova, A brief story about the operators of generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203–220.
[24] S.Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Ineq. Appl., 549 (2013), 1–9.
[25] W.J. Liu, C.C. Li, J.W. Dong, On an open problem concerning an integral inequality, JIPAM. J. Inequal. Pure Appl. Math., 8(3) (2007), 1–5.
[26] W. Liu, Q.A. Ngo, V.N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3(2) (2009), 201–212.
[27] R.L. Magin, Fractional calculus in bioengineering , Parts 1–3. Crit. Rev. Biomed. Eng., 32(1) (2004), 1–104.
[28] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9(1996), 23–28.
[29] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7(9) (1996), 1461–1477.
[30] S. Mubeen, G.M. Habibullah, k􀀀fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2) (2012), 89–94.
[31] K. S. Nisar, G. Rahman, J. Choi, S. Mubeen, M. Arshad, Certain Gronwall-type inequalities associated with Riemann-Liouville k􀀀and Hadamard k-fractional derivatives and their applications, East Asian Math. J., 34(3) (2018), 249–263.
[32] M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal, S. Naheed, On certain fractional calculus operators and applications in mathematical physics, Phys. Scr., 95(11) (2020), 1–9.
[33] A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra, J.J. Nieto, A study on k􀀀generalized Y-Hilfer derivative operator, Vietnam J. Math., 52 (2022), 25-43.
[34] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
[35] M.Z. Sarikaya, Z. Dahmani, M.E. Kiris, F. Ahmad, (k; s)􀀀Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45(1) (2016), 77 – 89.
[36] J. Shao, F. Meng, Gronwall-Bellman type inequalities and their applications to fractional differential equations, Abst. Appl. Anal. J., Article ID 217641 (2013), 1–7.
[37] J.V.D.C. Sousa, E.C.D. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of y􀀀 Hilfer operator, Differ. Equ. Appl., 11(1) (2019), 87–106.
[38] V. Uchaikin, E. Kozhemiakina, Non-local seismo-dynamics: A Fractional Approach, Fractal Fract., 6 (2022), 513.
[39] B.J. West, M. Bologna, P. Grigolini, Physics of Fractioanl Opeartors, Springer-Verlag, Berlin, 2003.
[40] X.J. Yang, F. Gao, Y. Ju, General Fractional Derivatives with Applications in Viscoelasticity, Academic Press: Cambridge, MA, USA, 2020.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Mohamed Bezzıou
0009-0009-2911-7003
Algeria

Zoubir Dahmani
0000-0003-4659-0723
Algeria

Rabha Ibrahim ^*
0000-0001-9341-025X
Iraq

Early Pub Date

November 20, 2024

Publication Date

December 9, 2024

Submission Date

January 24, 2024

Acceptance Date

August 20, 2024

Published in Issue

Year 2024 Volume: 7 Number: 4

DOI

https://doi.org/10.32323/ujma.1425363

IZ

https://izlik.org/JA93FL46UW

APA

Bezzıou, M., Dahmani, Z., & Ibrahim, R. (2024). The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies. Universal Journal of Mathematics and Applications, 7(4), 180-191. https://doi.org/10.32323/ujma.1425363

AMA

1.Bezzıou M, Dahmani Z, Ibrahim R. The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies. Univ. J. Math. Appl. 2024;7(4):180-191. doi:10.32323/ujma.1425363

Chicago

Bezzıou, Mohamed, Zoubir Dahmani, and Rabha Ibrahim. 2024. “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator With Applications in Economic Studies”. Universal Journal of Mathematics and Applications 7 (4): 180-91. https://doi.org/10.32323/ujma.1425363.

EndNote

Bezzıou M, Dahmani Z, Ibrahim R (December 1, 2024) The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies. Universal Journal of Mathematics and Applications 7 4 180–191.

IEEE

[1]M. Bezzıou, Z. Dahmani, and R. Ibrahim, “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies”, Univ. J. Math. Appl., vol. 7, no. 4, pp. 180–191, Dec. 2024, doi: 10.32323/ujma.1425363.

ISNAD

Bezzıou, Mohamed - Dahmani, Zoubir - Ibrahim, Rabha. “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator With Applications in Economic Studies”. Universal Journal of Mathematics and Applications 7/4 (December 1, 2024): 180-191. https://doi.org/10.32323/ujma.1425363.

JAMA

1.Bezzıou M, Dahmani Z, Ibrahim R. The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies. Univ. J. Math. Appl. 2024;7:180–191.

MLA

Bezzıou, Mohamed, et al. “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator With Applications in Economic Studies”. Universal Journal of Mathematics and Applications, vol. 7, no. 4, Dec. 2024, pp. 180-91, doi:10.32323/ujma.1425363.

Vancouver

1.Mohamed Bezzıou, Zoubir Dahmani, Rabha Ibrahim. The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies. Univ. J. Math. Appl. 2024 Dec. 1;7(4):180-91. doi:10.32323/ujma.1425363