Lyapunov type inequality in the frame of generalized Caputo derivatives

Fahd Jarad; Yassine Adjabi; Thabet Abdeljawad; Saed F. Mallak; Hussam Alrabaiah

doi:10.3934/dcdss.2020212

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

Lyapunov type inequality in the frame of generalized Caputo derivatives

Fahd Jarad ^1,, , Yassine Adjabi ^2, , Thabet Abdeljawad ^3,4, , Saed F. Mallak ^5, and Hussam Alrabaiah ^6,

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 July 2021 Early access 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.

Keywords: Lyapunov type inequality, generalized fractional integrals, generalized fractional derivatives, genralized Caputo fractional derivatives, generalized Mittag-Leffler function type, fractional boundary value problems, fractional eigenvalue problems, nontrivial solutions, zeros, Green's function.

Mathematics Subject Classification: Primary: 26A33; Secondary: 35A23.

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. 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

show all references

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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