×

Lyapunov type inequality in the frame of generalized Caputo derivatives. (English) Zbl 1493.34025

In this article, the authors study Lyapunov type inequalities for certain classes of fractional boundary value problems involving generalized Caputo fractional derivatives and subject to two point boundary conditions. As applications they use the obtained Lyapunov-type inequalities to obtain intervals where certain generalized Mittag-Leffler functions have no real zeros.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
33E12 Mittag-Leffler functions and generalizations
34B09 Boundary eigenvalue problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

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. · Zbl 1368.26003
[2] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017 (2017), Paper No. 313, 11 pp. · Zbl 1444.26003
[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. · Zbl 1373.39015
[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. · Zbl 1444.34045
[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. · Zbl 1246.39005
[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.
[7] T. Abdeljawad; 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, 3355-3368 (2017)
[8] A. Atangana; D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20, 763-769 (2016)
[9] A. Atangana; J. F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos Solitons Fractals, 102, 285-294 (2017) · Zbl 1374.34296
[10] D. Çakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput., 216, 368-373 (2010) · Zbl 1189.26040
[11] M. Caputo; M. Fabrizio, A new definition of fractional derivative without singular kerne, Prog. Frac. Diff. Appl., 1, 73-85 (2015)
[12] S. Clark; D. Hinton, A Liapunov inequality for linear Hamiltonian systems, Math. Inequal. Appl., 1, 201-209 (1998) · Zbl 0909.34033
[13] B. Cuahutenango-Barro; M. A. Taneco-Hernández; J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115, 283-299 (2018) · Zbl 1416.35289
[14] K. Diethelm, The Analysis of Fractional Differential Equations, , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
[15] R. A. C. Ferreira, A Lyapunov-type inequality for a fractional initial value problem, Fract. Calc. Appl. Anal., 16, 978-984 (2013) · Zbl 1312.34013
[16] R. A. C. Ferreira, Lyapunov-type inequalities for some sequential fractional boundary value problems, Adv. Dyn. Syst. Appl., 11, 33-43 (2016)
[17] R. A. C. Ferreira, On a Lyapunov-type inequality and the zeros of a certain Mittag-Leffler function, J. Math. Anal. Appl., 412, 1058-1063 (2014) · Zbl 1323.34006
[18] F. Jarad; T. Abdeljawad; D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10, 2607-2619 (2017) · Zbl 1412.26006
[19] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Difference Equ., 2012, (2012), 142, 8 pp. · Zbl 1346.26002
[20] F. Jarad; T. Abdeljawad; Z. Hammouch, On a class of ordinary differential equations in the frame of Atagana-Baleanu fractional derivative, Chaos Solitons Fractals, 117, 16-20 (2018) · Zbl 1442.34016
[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. · Zbl 1318.34007
[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.
[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.
[24] J. F. Gómez-Aguilar; H. Yépez-Martínez; R. F. Escobar-Jiménez; C. M. Astorga-Zaragoza; J. Reyes-Reyes, Analytical and numerical solutions of electrical circuits described by fractional derivatives, Appl. Math. Model., 40, 9079-9094 (2016) · Zbl 1480.94053
[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. · Zbl 1309.33001
[26] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218, 860-865 (2011) · Zbl 1231.26008
[27] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6, 1-15 (2014) · Zbl 1317.26008
[28] A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38, 1191-1204 (2001) · Zbl 1018.26003
[29] A. A. Kilbas; 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, 22-26 (1995)
[30] A. A. Kilbas; M. Saigo, On solutions of integral equations of Abel-Volterra type, Differential Integral Equations, 8, 993-1011 (1995) · Zbl 0823.45002
[31] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, , Elsevier, Amsterdam, 2006. · Zbl 1092.45003
[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. Ma; C. Ma; J. Wang, A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11, 135-141 (2017) · Zbl 1361.34009
[34] G. M. Mittag-Leffler, Sur la nouvelle fonction \(E_{\alpha }\left(z\right) \), C. R. Acad. Sci. Paris, 137, 554-558 (1903) · JFM 34.0435.01
[35] N. Parhi; S. Panigrahi, A Lyapunov-type integral inequality for higher order differential equations, Math. Slovaca, 52, 31-46 (2002) · Zbl 1019.34039
[36] J. P. Pinasco, Lyapunov-Type Inequalities, Springer Briefs in Mathematics, Springer, New York, 2013.
[37] T. R. Prabhakar, A singular integral equation with a generalised Mittag-Leffler function in the kernel, Yokohama Math. J., 19, 7-15 (1971) · Zbl 0221.45003
[38] J. Rongand and C. Bai, Lyapunov-type inequality for afractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015, (2015), 82, 10 pp. · Zbl 1343.34021
[39] X. Yang, On Lyapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput., 134, 307-317 (2003) · Zbl 1030.34019
[40] X. Yang; K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215, 3884-3890 (2010) · Zbl 1202.34021
[41] H. Ye; J. Gao; Y. Ding, A generalized Lyapunov inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 1075-1081 (2007) · Zbl 1120.26003
[42] H. Yépez-Martínez; 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, 247-260 (2019) · Zbl 1402.26005
[43] H. Yépez-Martínez; J. F. Gómez-Aguilar; I. O. Sosa; J. M. Reyes; J. Torres-Jiménez, The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mexicana Fís., 62, 310-316 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.