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

Yıl 2024, Cilt: 7 Sayı: 4, 180 - 191, 09.12.2024

Mohamed Bezzıou , Zoubir Dahmani , Rabha Ibrahim

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

Öz

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.

Anahtar Kelimeler

Fractional calculus Fractional differential equation Gronwall inequality

Kaynakça

  • [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.

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

Yıl 2024, Cilt: 7 Sayı: 4, 180 - 191, 09.12.2024

Mohamed Bezzıou , Zoubir Dahmani , Rabha Ibrahim

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

Öz

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.

Kaynakça

  • [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.
Toplam 40 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Mohamed Bezzıou 0009-0009-2911-7003

Zoubir Dahmani 0000-0003-4659-0723

Rabha Ibrahim 0000-0001-9341-025X

Erken Görünüm Tarihi 20 Kasım 2024
Yayımlanma Tarihi 9 Aralık 2024
Gönderilme Tarihi 24 Ocak 2024
Kabul Tarihi 20 Ağustos 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 4

Kaynak Göster

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 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. Aralık 2024;7(4):180-191. doi:10.32323/ujma.1425363
Chicago Bezzıou, Mohamed, Zoubir Dahmani, ve Rabha Ibrahim. “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator With Applications in Economic Studies”. Universal Journal of Mathematics and Applications 7, sy. 4 (Aralık 2024): 180-91. https://doi.org/10.32323/ujma.1425363.
EndNote Bezzıou M, Dahmani Z, Ibrahim R (01 Aralık 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 M. Bezzıou, Z. Dahmani, ve R. Ibrahim, “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies”, Univ. J. Math. Appl., c. 7, sy. 4, ss. 180–191, 2024, doi: 10.32323/ujma.1425363.
ISNAD Bezzıou, Mohamed vd. “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 (Aralık 2024), 180-191. https://doi.org/10.32323/ujma.1425363.
JAMA 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 vd. “The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator With Applications in Economic Studies”. Universal Journal of Mathematics and Applications, c. 7, sy. 4, 2024, ss. 180-91, doi:10.32323/ujma.1425363.
Vancouver 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):180-91.
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