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  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.

Citation: Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[11]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85. 

[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.

[13]

B. Cuahutenango-BarroM. 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.

[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.

[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.

[16]

R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43. 

[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.

[18]

F. JaradT. 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.

[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.

[20]

F. JaradT. 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.

[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.

[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.

[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.

[24]

J. F. Gómez-AguilarH. Yépez-MartínezR. F. Escobar-JiménezC. 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.

[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.

[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.

[27]

U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15. 

[28]

A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. 

[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. 

[30]

A. A. Kilbas and M. Saigo, On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011. 

[31]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006.

[32]

A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949.

[33]

Q. MaC. 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.

[34]

G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha }\left(z\right) $, C. R. Acad. Sci. Paris, 137 (1903), 554-558. 

[35]

N. Parhi and S. Panigrahi, A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46. 

[36]

J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8523-0.

[37] I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999. 
[38]

T. R. Prabhakar, A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15. 

[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.

[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.

[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.

[42]

H. YeJ. Gao and Y. Ding, A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. 

[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.

[44]

H. Yépez-MartínezJ. F. Gómez-AguilarI. O. SosaJ. 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. 

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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[11]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1 (2015), 73-85. 

[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.

[13]

B. Cuahutenango-BarroM. 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.

[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.

[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.

[16]

R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11 (2016), 33-43. 

[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.

[18]

F. JaradT. 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.

[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.

[20]

F. JaradT. 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.

[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.

[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.

[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.

[24]

J. F. Gómez-AguilarH. Yépez-MartínezR. F. Escobar-JiménezC. 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.

[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.

[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.

[27]

U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15. 

[28]

A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. 

[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. 

[30]

A. A. Kilbas and M. Saigo, On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8 (1995), 993-1011. 

[31]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006.

[32]

A. M. Liapunov, Problème général de la stabilitie du mouvement, Ann. of Math. Stud., 17, Princeton Univ. Press, Princeton, N. J., 1949.

[33]

Q. MaC. 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.

[34]

G. M. Mittag-Leffler, Sur la nouvelle fonction $E_{\alpha }\left(z\right) $, C. R. Acad. Sci. Paris, 137 (1903), 554-558. 

[35]

N. Parhi and S. Panigrahi, A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52 (2002), 31-46. 

[36]

J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8523-0.

[37] I. Podlubny, Fractional Differential Equations,, Academic Press, an Diego, California, 1999. 
[38]

T. R. Prabhakar, A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15. 

[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.

[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.

[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.

[42]

H. YeJ. Gao and Y. Ding, A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. 

[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.

[44]

H. Yépez-MartínezJ. F. Gómez-AguilarI. O. SosaJ. 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. 

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