An existence and uniqueness result for linear fractional impulsive boundary value problems as an application of Lyapunov type inequality

Year 2018, Volume: 47 Issue: 2, 287 - 297, 01.04.2018

Zeynep Kayar

Abstract

A new and different approach to the investigation of the existence and uniqueness of solution of nonhomogenous impulsive boundary value problems involving the Caputo fractional derivative of order  $\alpha$ ($1<\alpha\leq 2$) is brought by using Lyapunov type inequality. To express and to analyze the unique solution, Green's function and its bounds are established, respectively. As far as we know, this approach based on the link between fractional boundary value problems and Lyapunov type inequality, has not been revealed even in the absence of impulse effect. Besides, the novel Lyapunov type inequality generalizes the related ones in the literature.

Keywords

Linear impulsive fractional boundary value problems, Green's function, Lyapunov type inequality, Disconjugacy

References

• Abdeljawad, T. and Baleanu, D. and Jarad, F. and Mustafa, O. G. and Trujillo, J. J. A Fite type result for sequential fractional differential equations, Dynam. Systems Appl. 19 (2), 383394, 2011.
• Ahmad, B. and Nieto, J. J. Anti-periodic fractional boundary value problems, Comput. Math. Appl. 62, (3) 11501156, 2011.
• Ahmad, B. and Nieto, J. J. Existence of solutions for impulsive anti-periodic boundary value problems of fractional order, Taiwanese J. Math. 15 (3), 981993, 2011.
• Ahmad, B. and Nieto, J. J. and Johnatan, P. Some boundary value problems of fractional differential equations and inclusions, Comput. Math. Appl. 62 (3), 12381250, 2011.
• Ahmad, B. and Nieto, J. J. Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative, Fract. Calc. Appl. Anal. 15, (3), 451462, 2012.
• Ahmad, B. and Sivasundaram, S. Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (3), 251258, 2009.
• Akhmet, M. Principles of discontinuous dynamical systems (Springer, New York, 2010).
• Banov, D. D. and Simeonov, P. S. Systems with impulse eect, Ellis Horwood Series: Mathematics and its Applications (Ellis Horwood Ltd., Chichester, 1989).
• Borg, G. On a Liapouno criterion of stability, Amer. J. Math. 71 6770, 1949.
• Cheng, S. S. Lyapunov inequalities for differential and difference equations, Fasc. Math. 304 (23), 2541, 1991.
• Coppel, W. A. Disconjugacy, Lecture Notes in Mathematics, 220 (Springer-Verlag, Berlin- New York, 1971).
• Eloe, P. and Neugebauer, J. T. Conjugate points for fractional differential equations, Fract. Calc. Appl. Anal. 17 (3), 855871, 2014.
• Ferreira, R. A. C. A Lyapunov-type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal. 16 (4), 978984, 2013.
• Ferreira, R. A. C. On a Lyapunov-type inequality and the zeros of a certain Mittag-Leer function, J. Math. Anal. Appl. 412 (2), 10581063, 2014.
• Guseinov, G. S. and Zafer, A. Stability criterion for second order linear impulsive differential equations with periodic coecients, Math. Nachr. 281 (9), 12731282, 2008.
• Hartman, P. Disconjugate nth order differential equations and principal solutions, Bull. Amer. Math. Soc. 74 125129, 1968.
• Hartman, P. Ordinary differential equations, Classics in Applied Mathematics 38 (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002).
• Jleli, M. and Ragoub L. and Samet B.A Lyapunov-type inequality for a fractional differential equation under a Robin boundary condition, J. Funct. Spaces 2015 Art. ID 468536, 5 pp, 2015.
• Jleli, M. and Samet, B. Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions, Math. Inequal. Appl. 18 (2), 443451, 2015.