ON GENERALIZATION OF PACHPATTE TYPE INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRAL

Year 2018, Volume: 8 Issue: 1, 106 - 113, 01.06.2018

F. Usta Z. Sarıkaya

Abstract

The main target addressed in this article is presenting Pachpatte type inequalities for Katugampola conformable fractional integral. In accordance with this purpose we try to use more general type of function in order to make a generalization. Thus our results cover the previous published studies for Pachpatte type inequalities.

Keywords

Pachpatte inequality, conformable fractional integral

References

• T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathe- matics, 279, 57–66, 2015.
• D. R. Anderson and D. J. Ulness, Results for conformable differential equations, preprint, 2016.
• A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Math. 13, –898, 2015.
• R. Bellman, The stability of solutions of linear differential equations, Duke Mathematical Journal, 10, 647, 1943.
• T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20, 192-296, 1919.
• M. Abu Hammad, R. Khalil, Conformable fractional heat differential equations, International Journal of Pure and Applied Mathematics, 94(2), 215–221, 2014.
• M. Abu Hammad, R. Khalil, Abel’s formula and wronskian for conformable fractional differential equations, International Journal of Differential Equations and Applications, 13(3), 2014, 177-183.
• O.S. Iyiola and E.R.Nwaeze, Some new results on the new conformable fractional calculus with ap- plication using D’Alambert approach, Progress in Fractional Differentiation and Applications, 2(2), –122, 2016.
• U. Katugampola, A new fractional derivative with classical properties, ArXiv:1410.6535v2.
• A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
• R. Khalil, M. Al horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational Apllied Mathematics, 264, 65–70, 2014.
• B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998.
• B.G. Pachpatte, On some new inequalities related to a certain inequalities in the theory of differential equations, Journal of Mathematical Analysis and Applications, 251, 736–751, 2000.
• S. G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Ap- plications, Gordonand Breach, Yverdon et alibi, 1993.
• M. Z. Sarikaya, Gronwall type inequality for conformable fractional integrals, Konuralp Journal of Mathematics, 4(2), 217–222, 2016.
• F. Usta and M.Z. Sarikaya, On generalization conformable fractional integral inequalities , 2016, preprint.
• A. Zheng, Y. Feng and W. Wang, The Hyers-Ulam stability of the conformable fractional differential equation, Mathematica Aeterna, 5(3), 485–492, 2015.