Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation
DOI:
https://doi.org/10.24297/jam.v12i11.3719Keywords:
Caputo fractional derivative, fractional integral, Implicit fractional differential equation, BihariandMedved inequalities.Abstract
We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.
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References
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[6] Katugampola UN, New approach to a generalized fractional integral. Appl. Math. Comput. 218(3) (2011), 860–865.
[7] Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and Application of Fractional Differential Equations, Elsevier, Amersterdam, 2006.
[8] Kucche K. D., Nieto Juan J., Venktesh Venktesh, Theory of nonlinear implicit fractional differential equations, Differ Equ Dyn Syst, DOI 10.1007/s12591-016-0297-7.
[9] Lakshmikantam V. and VatsalaA.S., Basic theory of fractional Differential Equations, Nonlinear Analysis, 69 No. 8 (2008), 2677-2682.,doi:10.1016/j.na.2007.08.042.
[10] Lakshmikantam V., Theory of Fractional functional Differential Equations, Nonlinear Analysis, 69 No. 10 (2008), 3337-3343., doi:10.1016/j.na.2007.09.025.
[11] Mainardi F., Fractional Calculus: some basic problem in continuum and stastical mechanics, in factals and fractional calculus in contiuam mechanics, (A.Carpinteri and F.Mainardi, Eds)Springer-Verlag, Wien, (1997) 291-348.
[12] Medveˇd, M., A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math.Anal.Appl. 214(1997), 349-366.
[13] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations; John Wiley: New York, NY, USA, 1993.
[14] Momani S. M., Local and global uniqueness theorem on differential equations of non-integer order via Bihari’s and Gronwall inequalities , Rev.Tec. Ing. Univ. Zulia. Vol. 23, No.1, (2000) 66-69.
[15] Nieto Juan J., AbelghaniOuahab and VenkteshVenktesh, Implicit Fractional Differential Equations via the Liouville–Caputo Derivative, Mathematics 2015, 3, 398-411; doi:10.3390/math3020398.
[16] Pachpatte B. G., Inequalities for Differential and Integral Equations, Academic Press, New York, 205, 1998.
[17] Pachpatte B. G., Integral and finite difference inequalities and applications,North Holland Mathematics Studies, 205, Elsevier Science, 2006
[18] Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999.
[19] Somko S. G., Kilbas A. A. and Marichev O. I., Fractional Integral and Derivative Theory and Application, Gordon and Breach, Yverdon, Switzerland, 1993.
[20] Haping Ye, Jianming Gao and Yongs Ding, A generalized Grownwall inequality and it’s application to fractional differential equation, Journal of Mathematical Analysis and Application. 328 No 2 (2007), 1075-1081.,doi:10.1016/j. jmaa.2006.05.061.
[21] Yong. Zhou, Existence and uniqueness of solution for system of fractionaldifferential equations, Fractional Calculus and Applied Analysis. 12, No.2,2009.
[2] Corduneanu C., Principal of Differential and Integral Equation, Allynand Bacon, Boston, 1971.
[3] Cresson J. Fractional embedding of differential operators and Lagrangian systems. J Math Phys 48(3) (2007), 34 pp.
[4] El-Borai M., Some probability densities and fundamental solutions of fractional evolutionequations, Chaos Solitons Fractals 14(2002), No. 3, 433–440.
[5] Hilfer R., Application of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[6] Katugampola UN, New approach to a generalized fractional integral. Appl. Math. Comput. 218(3) (2011), 860–865.
[7] Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and Application of Fractional Differential Equations, Elsevier, Amersterdam, 2006.
[8] Kucche K. D., Nieto Juan J., Venktesh Venktesh, Theory of nonlinear implicit fractional differential equations, Differ Equ Dyn Syst, DOI 10.1007/s12591-016-0297-7.
[9] Lakshmikantam V. and VatsalaA.S., Basic theory of fractional Differential Equations, Nonlinear Analysis, 69 No. 8 (2008), 2677-2682.,doi:10.1016/j.na.2007.08.042.
[10] Lakshmikantam V., Theory of Fractional functional Differential Equations, Nonlinear Analysis, 69 No. 10 (2008), 3337-3343., doi:10.1016/j.na.2007.09.025.
[11] Mainardi F., Fractional Calculus: some basic problem in continuum and stastical mechanics, in factals and fractional calculus in contiuam mechanics, (A.Carpinteri and F.Mainardi, Eds)Springer-Verlag, Wien, (1997) 291-348.
[12] Medveˇd, M., A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math.Anal.Appl. 214(1997), 349-366.
[13] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations; John Wiley: New York, NY, USA, 1993.
[14] Momani S. M., Local and global uniqueness theorem on differential equations of non-integer order via Bihari’s and Gronwall inequalities , Rev.Tec. Ing. Univ. Zulia. Vol. 23, No.1, (2000) 66-69.
[15] Nieto Juan J., AbelghaniOuahab and VenkteshVenktesh, Implicit Fractional Differential Equations via the Liouville–Caputo Derivative, Mathematics 2015, 3, 398-411; doi:10.3390/math3020398.
[16] Pachpatte B. G., Inequalities for Differential and Integral Equations, Academic Press, New York, 205, 1998.
[17] Pachpatte B. G., Integral and finite difference inequalities and applications,North Holland Mathematics Studies, 205, Elsevier Science, 2006
[18] Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999.
[19] Somko S. G., Kilbas A. A. and Marichev O. I., Fractional Integral and Derivative Theory and Application, Gordon and Breach, Yverdon, Switzerland, 1993.
[20] Haping Ye, Jianming Gao and Yongs Ding, A generalized Grownwall inequality and it’s application to fractional differential equation, Journal of Mathematical Analysis and Application. 328 No 2 (2007), 1075-1081.,doi:10.1016/j. jmaa.2006.05.061.
[21] Yong. Zhou, Existence and uniqueness of solution for system of fractionaldifferential equations, Fractional Calculus and Applied Analysis. 12, No.2,2009.
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Published
2016-12-30
How to Cite
Tidke, H. L. (2016). Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation. JOURNAL OF ADVANCES IN MATHEMATICS, 12(11), 6807–6811. https://doi.org/10.24297/jam.v12i11.3719
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