[1] R.P. Agarwal and V. Lakshmikantham, Uniqueness and nonuniqueness criteria for
ordinary differential equations, Series in Real Analysis, vol. 6, World Scientific, 1993.
[2] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional
differential equation, J. Math. Anal. Appl. 311 (2), 495–505, 2005.
[3] D. Baleanu and G.M. Octavian, On the existence interval for the initial value problem
of a fractional differential equation, Hacet. J. Math. Stat. 40 (4), 2011.
[5] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional dif-
ferential equation, J. Math. Anal. Appl. 204, 609–625, 1996.
[6] J.B. Diaz and W.L. Walter, On uniqueness theorems for ordinary differential equa-
tions and for partial differential equations of hyperbolic type, Trans. Amer. Math. Soc.
96, 90–100, 1960.
[7] K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for
Caputo’s fractional calculus, Fract. Calc. Appl. Anal. 15 (2), 304–313, 2012.
[8] P. Drábek and A. Fonda, Handbook of differential equations: ordinary differential
equations, vol 3, Elsevier, North Holland, 2006.
[9] J. Dugundji and A. Granas, Fixed point theory Mathematical Monographs, 61,
Państwowe Wydawnictwo Naukowe, PWN, Warszawa, 1982.
[10] P.W. Eloe and T. Masthay, Initial value problems for Caputo fractional differential
equations, J. Fract. Calc. Appl. 9 (2), 178–195, 2018.
[11] R.A.C. Ferreira, A Nagumo-type uniqueness result for an nth order differential equa-
tion, Bull. Lond. Math. Soc. 45 (5), 930–934, 2013.
[12] A.F. Filippov, Differential equations with discontinuous right-hand sides: control sys-
tems, vol. 18, Springer Science & Business Media, 2013.
[13] N. Heymans and I. Podlubny, Physical interpretation of initial conditions for frac-
tional differential equations with Riemann-Liouville fractional derivatives, Rheologica
Acta. 45 (5), 765–771, 2006.
[14] A.A. Kilbas, H.M. Srivasta and J.J. Trujilllo, Theory and applications of fractional
differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science
Limited, 2006.
[15] V. Lakshmikantham and S. Leela, Nagumo-type uniqueness result for fractional dif-
ferential equations, Nonlinear Anal. 71 (7-8), 2886–2889, 2009.
[16] V. Lakshmikanthan and A.S. Vatsala, Basic theory of fractional differential equations,
Nonlinear Anal. 69, 2677–2682, 2008.
[18] M.D. Ortigueira, Fractional calculus for scientists and engineers, vol. 84, Springer,
Dordrecht, 2011.
[19] M.D. Ortigueira and F.J. Coito, System initial conditions vs derivative initial condi-
tions, Comput. Math. Appl. 59 (5), 1782–1789, 2010.
[20] I. Podlubny, Fractional differential equations: an introduction to fractional deriva-
tives, fractional differential equations, to methods of their solution and some of their
applications, 198, Elsevier, 1998.
[21] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives:
Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzer-
land, 1993.
[22] C. Sin and Z. Liancun, Existence and uniqueness of global solutions of Caputo-type
fractional differential equations, Fract. Calc. Appl. Anal. 19 (3), 765–774, 2016.
[23] M. Şan, Complex variable approach to the analysis of a fractional differential equation
in the real line, C. R. Math. Acad. Sci. Paris, 356 (3), 293–300, 2018.
[24] M. Şan and K.N. Soltanov, The New Existence and Uniqueness Results for Complex
Nonlinear Fractional Differential Equation, arXiv preprint arXiv:1512.04780.
[25] T. Trif, Existence of solutions to initial value problems for nonlinear fractional dif-
ferential equations on the semi-axis, Fract. Calc. Appl. Anal. 16 (3), 595–612, 2013.
[26] S. Zhang, Monotone iterative method for initial value problem involving Riemann-
Liouville fractional derivatives, Nonlinear Anal. 71, 2087–2093, 2009.
Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero
In this article, we consider an initial value problem for a nonlinear differential equation with Riemann-Liouville fractional derivative. By proposing a new approach, we prove local existence and uniqueness of the solution when the nonlinear function on the right hand side of the equation under consideration is continuous on $(0,T]\times\mathbb{R}.$
[1] R.P. Agarwal and V. Lakshmikantham, Uniqueness and nonuniqueness criteria for
ordinary differential equations, Series in Real Analysis, vol. 6, World Scientific, 1993.
[2] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional
differential equation, J. Math. Anal. Appl. 311 (2), 495–505, 2005.
[3] D. Baleanu and G.M. Octavian, On the existence interval for the initial value problem
of a fractional differential equation, Hacet. J. Math. Stat. 40 (4), 2011.
[5] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional dif-
ferential equation, J. Math. Anal. Appl. 204, 609–625, 1996.
[6] J.B. Diaz and W.L. Walter, On uniqueness theorems for ordinary differential equa-
tions and for partial differential equations of hyperbolic type, Trans. Amer. Math. Soc.
96, 90–100, 1960.
[7] K. Diethelm, The mean value theorems and a Nagumo-type uniqueness theorem for
Caputo’s fractional calculus, Fract. Calc. Appl. Anal. 15 (2), 304–313, 2012.
[8] P. Drábek and A. Fonda, Handbook of differential equations: ordinary differential
equations, vol 3, Elsevier, North Holland, 2006.
[9] J. Dugundji and A. Granas, Fixed point theory Mathematical Monographs, 61,
Państwowe Wydawnictwo Naukowe, PWN, Warszawa, 1982.
[10] P.W. Eloe and T. Masthay, Initial value problems for Caputo fractional differential
equations, J. Fract. Calc. Appl. 9 (2), 178–195, 2018.
[11] R.A.C. Ferreira, A Nagumo-type uniqueness result for an nth order differential equa-
tion, Bull. Lond. Math. Soc. 45 (5), 930–934, 2013.
[12] A.F. Filippov, Differential equations with discontinuous right-hand sides: control sys-
tems, vol. 18, Springer Science & Business Media, 2013.
[13] N. Heymans and I. Podlubny, Physical interpretation of initial conditions for frac-
tional differential equations with Riemann-Liouville fractional derivatives, Rheologica
Acta. 45 (5), 765–771, 2006.
[14] A.A. Kilbas, H.M. Srivasta and J.J. Trujilllo, Theory and applications of fractional
differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science
Limited, 2006.
[15] V. Lakshmikantham and S. Leela, Nagumo-type uniqueness result for fractional dif-
ferential equations, Nonlinear Anal. 71 (7-8), 2886–2889, 2009.
[16] V. Lakshmikanthan and A.S. Vatsala, Basic theory of fractional differential equations,
Nonlinear Anal. 69, 2677–2682, 2008.
[18] M.D. Ortigueira, Fractional calculus for scientists and engineers, vol. 84, Springer,
Dordrecht, 2011.
[19] M.D. Ortigueira and F.J. Coito, System initial conditions vs derivative initial condi-
tions, Comput. Math. Appl. 59 (5), 1782–1789, 2010.
[20] I. Podlubny, Fractional differential equations: an introduction to fractional deriva-
tives, fractional differential equations, to methods of their solution and some of their
applications, 198, Elsevier, 1998.
[21] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives:
Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzer-
land, 1993.
[22] C. Sin and Z. Liancun, Existence and uniqueness of global solutions of Caputo-type
fractional differential equations, Fract. Calc. Appl. Anal. 19 (3), 765–774, 2016.
[23] M. Şan, Complex variable approach to the analysis of a fractional differential equation
in the real line, C. R. Math. Acad. Sci. Paris, 356 (3), 293–300, 2018.
[24] M. Şan and K.N. Soltanov, The New Existence and Uniqueness Results for Complex
Nonlinear Fractional Differential Equation, arXiv preprint arXiv:1512.04780.
[25] T. Trif, Existence of solutions to initial value problems for nonlinear fractional dif-
ferential equations on the semi-axis, Fract. Calc. Appl. Anal. 16 (3), 595–612, 2013.
[26] S. Zhang, Monotone iterative method for initial value problem involving Riemann-
Liouville fractional derivatives, Nonlinear Anal. 71, 2087–2093, 2009.
Sert, U., & Şan, M. (2020). Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero. Hacettepe Journal of Mathematics and Statistics, 49(5), 1718-1725. https://doi.org/10.15672/hujms.512563
AMA
Sert U, Şan M. Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero. Hacettepe Journal of Mathematics and Statistics. October 2020;49(5):1718-1725. doi:10.15672/hujms.512563
Chicago
Sert, Uğur, and Müfit Şan. “Some Analysis on a Fractional Differential Equation With a Right-Hand Side Which Has a Discontinuity at Zero”. Hacettepe Journal of Mathematics and Statistics 49, no. 5 (October 2020): 1718-25. https://doi.org/10.15672/hujms.512563.
EndNote
Sert U, Şan M (October 1, 2020) Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero. Hacettepe Journal of Mathematics and Statistics 49 5 1718–1725.
IEEE
U. Sert and M. Şan, “Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, pp. 1718–1725, 2020, doi: 10.15672/hujms.512563.
ISNAD
Sert, Uğur - Şan, Müfit. “Some Analysis on a Fractional Differential Equation With a Right-Hand Side Which Has a Discontinuity at Zero”. Hacettepe Journal of Mathematics and Statistics 49/5 (October 2020), 1718-1725. https://doi.org/10.15672/hujms.512563.
JAMA
Sert U, Şan M. Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero. Hacettepe Journal of Mathematics and Statistics. 2020;49:1718–1725.
MLA
Sert, Uğur and Müfit Şan. “Some Analysis on a Fractional Differential Equation With a Right-Hand Side Which Has a Discontinuity at Zero”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, 2020, pp. 1718-25, doi:10.15672/hujms.512563.
Vancouver
Sert U, Şan M. Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero. Hacettepe Journal of Mathematics and Statistics. 2020;49(5):1718-25.