• Munzarin Ashfaque Ahmed Sajjan Assistant Professor, Department of Mathematics, Y & M AKIs Poona College of Arts, Science and Commerce, Camp, Pune, India
  • Amjad Salim Shaikh Assistant Professor, Department of Mathematics, Y & M AKIs Poona College of Arts, Science and Commerce, Camp, Pune, India


Riemann-Liouville, Caputo fractional derivative operator, Existence and uniqueness of solution


In this review article, we have studied existence and uniqueness of solution of some fractional differential equations using some known fixed-point theorems like Schrauder fixed point theorem, Banach fixed point theorem etc. and also listed different criteria used by authors to obtain existence and uniqueness of solutions of fractional differential equation.


I. M. Benchohra and B. A. Slimani, existence and uniqueness of solutions to impulsive fractional differential equations, electron. j. differential equations, 2009(10) (2009), 1–11.

II. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Vol. I, One Dimensional Theory. Birkhauser Basel, and Academic Press, New Y

III. K. Diethelm, The Analysis of Fractional Differential Equation, Spinger, Heidelberg, Germany, 2021.

IV. Domenico Delbosco and Luigi Rodino, Existence and Uniqueness for a Nonlinear Fractional Differential Equation, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 204, 609 625(1996).

V. Eiman, K. Shah, M. Sarwar and D. Baleanu.” Study on Krasnoselskii’s fixed point theorem for Caputo–Fabrizio fractional differential equations”. Advances in Difference Equations https://doi.org/10.1186/s13662-020-02624-x (2020).

VI. S. Fenyo and H. W. Stolle, Theorie und Praxis der linearen Integralgleichungen, Deutscher Verlag d. Wiss., Berlin, 1963.

VII. Furati, K.M.; Kassim, M.D.; Tatar, N. Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 2012, 64, 1616–1626.

VIII. R. Hilfer, Applications of fractional calculus in physics, World Scientific Singapore, 2000.

IX. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of the Fractional Differential Equations, volume 204. Elsevier, Amsterdam, 2006.

X. Lakshman Mahto, Syed Abbas,” Existence and Uniqueness of Solution of Caputo Fractional Differential Equations” (2012).

XI. V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge, 2009.

XII. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993.

XIII. Mohammed Matar, On Existence of Solution to Some Nonlinear Differential Equations of Fractional Order $2

XIV. I.Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999

XV. K. B. Oldham and J. Spanier, The Fractional Calculus Theory and applications of differentiation and integrations of arbitrary order, Academic Press, New York, 1975.

XVI. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.

XVII. A. S. Shaikh, Sontakke, B. R. “Impulsive initial value problems for a class of implicit fractional differential equations” Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 8, No. 1, 2020, pp. 141-154 DOI:10.22034/cmde.2019.9455.

XVIII. I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holland Publ., Amsterdam, 1966.

XIX. Sontakke B. R., A. S. Shaikh. And K.S.Nisar. “Existence and uniqueness of integrable solutions of fractional order initial value equations. Journal of Mathematical Modeling Vol. 6, No. 2, 2018, pp. 137-148.

XX. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.

XXI. Wen-Xue Zhou and Xu Liu, Existence of solution to a class of boundry value problem for impulsive fractional differential equation.

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How to Cite

Munzarin Ashfaque Ahmed Sajjan, & Amjad Salim Shaikh. (2023). A REVIEW ON EXISTENCE AND UNIQUENESS OF SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATION. International Education and Research Journal (IERJ), 9(5). Retrieved from https://ierj.in/journal/index.php/ierj/article/view/2702