American Institute of Mathematical Sciences

July  2021, 14(7): 2335-2355. doi: 10.3934/dcdss.2020212

Lyapunov type inequality in the frame of generalized Caputo derivatives

 1 Department of Mathematics, Çankaya University 06790, Ankara, Turkey 2 Department of Mathematics, Faculty of Sciences, University of M'hamed Bougara, Boumerdes, 35000, Algeria 3 Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia 4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 5 Department of Applied mathematics, Palestine Technical University-Kadoorie, Palestine 6 College of Engineering, Al Ain University of Science and Technology, Al Ain, UAE, College of Science, Tafila Technical University, Tafila, Jordan

* Corresponding author

Received  April 2019 Revised  October 2020 Published  May 2021

Fund Project: The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17

In this paper, we establish the Lyapunov-type inequality for boundary value problems involving generalized Caputo fractional derivatives that unite the Caputo and Caputo-Hadamrad fractional derivatives. An application about the zeros of generalized types of Mittag-Leffler functions is given.

Citation: Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2335-2355. doi: 10.3934/dcdss.2020212
References:
 [1] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5.  Google Scholar [2] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0.  Google Scholar [3] T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.  Google Scholar [4] T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Difference Equ., 2017 (2017), Paper No. 321, 10 pp. doi: 10.1186/s13662-017-1383-z.  Google Scholar [5] T. Abdeljawad, B. Benli and D. Baleanu, A generalized $q$-Mittag-Leffler function by $q$-Captuo fractional linear equations, Abstr. Appl. Anal., 2012 (2012), Article ID 546062, 11 pp. doi: 10.1155/2012/546062.  Google Scholar [6] T. Abdeljawad, F. Jarad, S. F. Mallak and J. Alzabut, Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag-Leffler functions, Eur. Phys. J. Plus, 134 (2019), 247. doi: 10.1140/epjp/i2019-12772-1.  Google Scholar [7] T. Abdeljawad and F. Madjidi, A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order $2\leq \alpha < 5/2$, Eur. Phys. J. Spec. Top., 226 (2017), 3355-3368.   Google Scholar [8] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769.   Google Scholar [9] A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294.  doi: 10.1016/j.chaos.2017.03.022.  Google Scholar [10] D. Çakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373.  doi: 10.1016/j.amc.2010.01.010.  Google Scholar [11] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85.   Google Scholar [12] S. Clark and D. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequal. Appl., 1 (1998), 201-209.  doi: 10.7153/mia-01-18.  Google Scholar [13] B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar [14] K. Diethelm, The Analysis of Fractional Differential Equations, , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar [15] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional initial value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.  doi: 10.2478/s13540-013-0060-5.  Google Scholar [16] R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43.   Google Scholar [17] R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063.  doi: 10.1016/j.jmaa.2013.11.025.  Google Scholar [18] F. Jarad, T. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar [19] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012, (2012), 142, 8 pp. doi: 10.1186/1687-1847-2012-142.  Google Scholar [20] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atagana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar [21] M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary value problems equation with fractional initial conditions, Electron. J. Differential Equations, 2015 (2015), 11 pp.  Google Scholar [22] J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. 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Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar [27] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar [28] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.   Google Scholar [29] A. A. Kilbas and M. Sa${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over i} }}$go, Fractional integrals and derivatives of Mittag-Leffler type function (Russian), Dokl. Akad. Nauk Belarusi, 39 (1995), 22-26.   Google Scholar [30] A. A. Kilbas and M. Saigo, On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011.   Google Scholar [31] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006.  Google Scholar [32] A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949. Google Scholar [33] Q. Ma, C. Ma and J. Wang, A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11 (2017), 135-141.  doi: 10.7153/jmi-11-13.  Google Scholar [34] G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha }\left(z\right)$, C. R. Acad. Sci. Paris, 137 (1903), 554-558.   Google Scholar [35] N. Parhi and S. Panigrahi, A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.   Google Scholar [36] J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8523-0.  Google Scholar [37] I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999.   Google Scholar [38] T. R. Prabhakar, A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.   Google Scholar [39] J. Rongand and C. Bai, Lyapunov-type inequality for afractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015, (2015), 82, 10 pp. doi: 10.1186/s13662-015-0430-x.  Google Scholar [40] X. Yang, On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317.  doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar [41] X. Yang and K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890.  doi: 10.1016/j.amc.2009.11.032.  Google Scholar [42] H. Ye, J. Gao and Y. Ding, A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.   Google Scholar [43] H. Yépez-Martínez and J. F. Gómez-Aguilar, A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J. Comput. Appl. Math., 346 (2019), 247-260.  doi: 10.1016/j.cam.2018.07.023.  Google Scholar [44] H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.   Google Scholar

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References:
 [1] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5.  Google Scholar [2] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0.  Google Scholar [3] T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.  Google Scholar [4] T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Difference Equ., 2017 (2017), Paper No. 321, 10 pp. doi: 10.1186/s13662-017-1383-z.  Google Scholar [5] T. Abdeljawad, B. Benli and D. Baleanu, A generalized $q$-Mittag-Leffler function by $q$-Captuo fractional linear equations, Abstr. Appl. Anal., 2012 (2012), Article ID 546062, 11 pp. doi: 10.1155/2012/546062.  Google Scholar [6] T. Abdeljawad, F. Jarad, S. F. Mallak and J. Alzabut, Lyapunov type inequalities via fractional proportional derivatives and application on the free zero disc of Kilbas-Saigo generalized Mittag-Leffler functions, Eur. Phys. J. Plus, 134 (2019), 247. doi: 10.1140/epjp/i2019-12772-1.  Google Scholar [7] T. Abdeljawad and F. Madjidi, A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order $2\leq \alpha < 5/2$, Eur. Phys. J. Spec. Top., 226 (2017), 3355-3368.   Google Scholar [8] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769.   Google Scholar [9] A. Atangana and J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102 (2017), 285-294.  doi: 10.1016/j.chaos.2017.03.022.  Google Scholar [10] D. Çakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216 (2010), 368-373.  doi: 10.1016/j.amc.2010.01.010.  Google Scholar [11] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85.   Google Scholar [12] S. Clark and D. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequal. Appl., 1 (1998), 201-209.  doi: 10.7153/mia-01-18.  Google Scholar [13] B. Cuahutenango-Barro, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar [14] K. Diethelm, The Analysis of Fractional Differential Equations, , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar [15] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional initial value problem, Fract. Calc. Appl. Anal., 16 (2013), 978-984.  doi: 10.2478/s13540-013-0060-5.  Google Scholar [16] R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43.   Google Scholar [17] R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412 (2014), 1058-1063.  doi: 10.1016/j.jmaa.2013.11.025.  Google Scholar [18] F. Jarad, T. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar [19] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012, (2012), 142, 8 pp. doi: 10.1186/1687-1847-2012-142.  Google Scholar [20] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atagana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar [21] M. Jleli and B. Samet, Lyapunov-type inequalities for fractional boundary value problems equation with fractional initial conditions, Electron. J. Differential Equations, 2015 (2015), 11 pp.  Google Scholar [22] J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus, 132 (2017), 13. doi: 10.1140/epjp/i2017-11293-3.  Google Scholar [23] J. F. Gómez-Aguilar, A. Atangana and V. F. Morales-Delgado, Electrical circuits RC, LC and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. theor. Appl., 45 (2017), 1514–1533. doi: 10.1002/cta.2348.  Google Scholar [24] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza and J. Reyes-Reyes, Analytical and numerical solutions of electrical circuits described by fractional derivatives, Appl. Math. Model., 40 (2016), 9079-9094.  doi: 10.1016/j.apm.2016.05.041.  Google Scholar [25] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, Springer Heidelberg New York Dordrecht London, 2014. doi: 10.1007/978-3-662-43930-2.  Google Scholar [26] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar [27] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.   Google Scholar [28] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.   Google Scholar [29] A. A. Kilbas and M. Sa${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over i} }}$go, Fractional integrals and derivatives of Mittag-Leffler type function (Russian), Dokl. Akad. Nauk Belarusi, 39 (1995), 22-26.   Google Scholar [30] A. A. Kilbas and M. Saigo, On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011.   Google Scholar [31] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006.  Google Scholar [32] A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949. Google Scholar [33] Q. Ma, C. Ma and J. Wang, A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11 (2017), 135-141.  doi: 10.7153/jmi-11-13.  Google Scholar [34] G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha }\left(z\right)$, C. R. Acad. Sci. Paris, 137 (1903), 554-558.   Google Scholar [35] N. Parhi and S. Panigrahi, A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46.   Google Scholar [36] J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8523-0.  Google Scholar [37] I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999.   Google Scholar [38] T. R. Prabhakar, A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.   Google Scholar [39] J. Rongand and C. Bai, Lyapunov-type inequality for afractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015, (2015), 82, 10 pp. doi: 10.1186/s13662-015-0430-x.  Google Scholar [40] X. Yang, On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134 (2003), 307-317.  doi: 10.1016/S0096-3003(01)00285-5.  Google Scholar [41] X. Yang and K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890.  doi: 10.1016/j.amc.2009.11.032.  Google Scholar [42] H. Ye, J. Gao and Y. Ding, A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.   Google Scholar [43] H. Yépez-Martínez and J. F. Gómez-Aguilar, A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM), J. Comput. Appl. Math., 346 (2019), 247-260.  doi: 10.1016/j.cam.2018.07.023.  Google Scholar [44] H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62 (2016), 310-316.   Google Scholar
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