Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

Yıl 2019, , 53 - 63, 30.06.2019

Jagan Mohan Jonnalagadda , Dipak Kumar Satpathi Debananda Basua

https://doi.org/10.31197/atnaa.471245

Öz

In this article, we establish Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with well-posed fractional boundary conditions. To illustrate the applicability of established results, we estimate lower bounds for eigenvalues of the corresponding eigenvalue problems and deduce criteria for the nonexistence of real zeros of certain Mittag-Leffler functions.

Anahtar Kelimeler

Riemann-Liouville type fractional derivative, boundary value problem, Green's function, Lyapunov-type inequality

Kaynakça

• Reference 1 Z. Bai and H. L\"{u}, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495 - 505.
• Reference 2 R.C. Brown and D.B. Hinton, Lyapunov Inequalities and Their Applications, In: Survey on Classical Inequalities (Ed. T.M. Rassias), Math. Appl. 517, Kluwer Acad. Publ., Dordrecht - London, 1 - 25, 2000.
• Reference 3 S. Dhar, Q. Kong and M. McCabe, Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations, 2016 (2016), No. 43, 1 - 16.
• Reference 4 P.W. Eloe, J.W. Lyons and J.T. Neugebauer, An ordering on Green's functions for a family of two-point boundary value problems for fractional differential equations, Communications in Applied Analysis 19 (2015), 453 - 462.
• Reference 5 R.A.C. Ferreira, A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal., 16 (2013), No. 4, 978 - 984.
• Reference 6 C.A. Hollon and J.T. Neugebauer, Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, 2015, 615 - 620.
• Reference 7 Jagan Mohan Jonnalagadda and Debananda Basua, Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems, Novi Sad journal o Mathematics, to appear.
• Reference 8 A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• Reference 9 A. Liapounoff, Probl\`{e}me g\'{e}n\'{e}ral de la stabilit\'{e} du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., (2) (1907), No. 9, 203 - 474.
• Reference 10 S.K. Ntouyas, B. Ahmad and T.P. Horikis, Recent developments of Lyapunov-type inequalities for fractional differential equations, arXiv:1804.10760v1 [math.CA] 28 Apr 2018.
• Reference 11 B.G. Pachpatte, On Lyapunov type inequalities for certain higher order differential equations, J. Math. Anal. Appl., 195 (1995), No. 2, 527 - 536.
• Reference 12 J.P. Pinasco, Lyapunov-type Inequalities with Applications to Eigenvalue Problems, Springer Briefs in Mathematics, Springer, New York, 2013.
• Reference 13 I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• Reference 14 J. Rong and C.Z. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Difference Equ., 2015 (2015), 10 Pages.
• Reference 15 A. Tiryaki, Recent developments of Lyapunov-type inequalities, Adv. Dyn. Syst. Appl., 5 (2010), No. 2, 231 - 248.
• Reference 16 Y. Wang, S. Liang and C. Xia, A Lyapunov-type inequality for a fractional differential equation under Sturm-Liouville boundary conditions, Math. Inequal. Appl., 20 (2017), No. 1, 139 - 148.
• Reference 17 X. Yang, Y. Kim and K. Lo, Lyapunov-type inequality for a class of even-order linear differential equations, Appl. Math. Comput., 245 (2014), 145 - 151.
• Reference 18 X. Yang, Y. Kim and K. Lo, Lyapunov-type inequalities for a class of higher-order linear differential equations, Appl. Math. Lett., 34 (2014), 86 - 89.