EXISTENCE AND UNIQUENESS OF SOLUTION OF CAUCHY-TYPE PROBLEM FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS

TitleEXISTENCE AND UNIQUENESS OF SOLUTION OF CAUCHY-TYPE PROBLEM FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS
Publication TypeJournal Article
Year of Publication2018
AuthorsDHAIGUDE, DB, BHAIRAT, SANDEEPP
Secondary TitleCAUCHY-TYPE PROBLEM FOR HILFER FRACTIONAL EQUATIONS
Volume22
Issue1
Start Page121
Pagination14
Date Published01/2018
Type of Workscientific: mathematics
ISBN Number1083-2564
AMS26A33, 34A08, 34A12
Abstract

The Cauchy-type problem for a nonlinear differential equation involving Hilfer (generalized Riemann-Liouville) fractional derivative is considered. The equivalence between this Cauchy-type problem and a nonlinear Volterra integral equation in the space of weighted continuous functions is established. Using this result, existence, uniqueness and continuous dependence of solution for Cauchy-type problem are obtained by using successive approximations and the generalized Gronwall inequality.
URLhttp://www.acadsol.eu/en/articles/22/1/8.pdf
DOI10.12732/caa.v22i1.8
Refereed DesignationRefereed
Full Text

REFERENCES

[1] D.B. Dhaigude and Sandeep P. Bhairat, On existence and approximation of solution of nonlinear Hilfer fractional differential equation, Int. J. Pure. Appl. Math. (2017), To Appear.
[2] D.B. Dhaigude and Sandeep P. Bhairat, Local Existence and Uniqueness of Solution for Hilfer-Hadamard Fractional Differential Problem, Nonlinear Dyn. Syst. Theory (2017), To Appear.
[3] K.M. Furati, M.D. Kassim, N.E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. (2012), 1616-1626.
[4] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phy., 284 (2002), 399-408.
[5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[6] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Cal. Appl. Anal., 12 (2009) 299-318.
[7] A.A. Kilbas, M. Saigo, Fractional integral and derivatives of Mittag-Leffler type function, Dokl. Akad. Nauk Belarusi, 39 (1995), 22-26.
[8] A.A. Kilbas and S.A. Marzan, Cauchy Problem for Differential Equation with Caputo Derivative, Fract. Calc. Appl. Anal. (2004).
[9] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006.
[10] C. Kou, J. Liu, and Y. Ye, Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equaitions, Discrete Dyn. Nat. Soc. (2010).
[11] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis, 69 (2008), 2677-2682.
[12] J.A. Nanware, D.B. Dhaigude, Existence and Uniqueness of solution of Riemann-Liouville fractional differential equations with integral boundary conditions, Int. J. Nonlinear Sci., 14 (2012), 410-415.
[13] J.A. Nanware, D.B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, J. Nonlinear Sci. Appl., 7 (2014), 246-254.
[14] I. Podlubny, Fractional Differential Equations, New York, Academic Press, 1999.
[15] S.G. Samko, A.A. Kilbas, O.I.Marichev, Fractional Intergrals and Derivatives: Theory and Applications, Golden and Breach Science Publishers, Switzertland, 1993.
[16] R.K. Saxena and M. Saigo, Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function, Frac. Cal. Appl. Anal., 8, No. 2 (2005), 141-154.
[17] Z. Tomovski, R. Hilfer, H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler functions, Integr. Transf. Spec, F, 21,, No. 11 (2010), 797-814.
[18] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequlity and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.