Certain Gronwall type inequalities associated with Riemann-Liouville $k$- and Hadamard $k$-fractional derivatives and their applications
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We aim to establish certain Gronwall type inequalities asso- ciated with Riemann-Liouville k- and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional dif- ferential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.
Abstract
We aim to establish certain Gronwall type inequalities asso- ciated with Riemann-Liouville k- and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional dif- ferential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.
참고문헌 (23)
- R. P. Agarwal, "Generalization of a retarded Gronwall-like inequality and its applications", Appl. Math. Comput., 165, pp.599~612 (2005)
- S. Belarbi, "On some new fractional integral inequalities", J. Inequal. Pure Appl. Math., 10(3) (2009)
- P. L. Chebyshev, "Sur les expressions approximatives des integrales definies par les autres prises entre les mmes limites", Proc. Math. Soc. Charkov, 2, pp.93~98 (1882)
- C. Corduneanu, "Principles of Differential and Integral Equations, 55" (1971), Allyn and Bacon
- Z. Dahmani, "On weighted Gruss type inequalities via fractional integration", J. Adv. Res. Pure Math., 2, pp.31~38 (2010)
- D. Delbosco, "Existence and uniqueness for a nonlinear fractional differential equation", J. Math. Anal. Appl., 204, pp.609~625 (1996)[DOI]
- R. Daz, "On hypergeometric functions and k-Pochhammer symbol", Divulg. Mat., 15(2), pp.179~192 (2007)
- K. Diethelm, "Analysis of fractional differential equations", J. Math. Anal. Appl., 265, pp.229~248 (2002)[DOI]
- A. A. Kilbas, "Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204" (2006), Elsevier (North-Holland) Science Publishers
- V. S. Kiryakova, "Generalized Fractional Calculus and Applications, Pitman Res Notes Math. 301" (1994), Longman Scientific & Technical;Co-published with John Wiley
- S. Mubeen, "k-fractional integrals and application", Int. J. Contem. Math. Sci., 7, pp.89~94 (2012)
- K. S. Ntouyas, "On Polya-Szego and Chebyshev types inequalities involving the Riemann-Liouville fractional integral operators", J. Math. Inequal., 10(2), pp.491~504 (2016)
- B. G. Pachpatte, "Inequalities for Differential and Integral Equations" (1998), Academic Press
- B. G. Pachpatte, "On some generalizations of Bellman's lemma", J. Math. Anal. Appl., 5, pp.141~150 (1975)
- I. Podlubny, "Fractional Differential Equations. An Introduction to Fractional Deriva-tives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications" (1999), Academic Press
- "A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives"[URL]
- M. Z. Sarikaya, "(k; s)-Riemann-Liouville fractional integral and applications", Hacet. J. Math. Stat., 45(1), pp.77~89 (2016)
- E. Set, "On generalized Gruss type inequalities for k-fractional integrals", Appl. Math. Comput., 269, pp.29~34 (2015)
- H. M. Srivastava, "Zeta and q-Zeta Functions and Associated Series and Integrals" (2012), Elsevier Science Publishers
- M. Tomar, "Certain inequalities associated with Hadamard k-fractional integral operators", J. Inequal. Appl., 2016 (2016)
- A. Wiman, "Uber den fundamentalsatz in der theorie der funktionen <TEX>$E_{\alpha}(x)$<", TEX>, 1905, pp.29, 191~201
- A. Wiman, "Uber die Nullstellen der funktionen <TEX>$E_{\alpha}(x)$<", TEX>, 1905, pp.29, 217~234
- H. Ye, "A generalized Gronwall inequality and its application to a fractional differential equation", J. Math. Anal. Appl., 328, pp.1075~1081 (2007)[DOI]
