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Establishing the existence of Hilfer fractional pantograph equations with impulses

Yıl 2018, Cilt: 1 Sayı: 1, 36 - 42, 30.06.2018
https://doi.org/10.33401/fujma.406651

Öz

In [1], the authors established the existence of a class of fractional differential equations of a complex order. In this note, we derive some sufficient conditions for the existence of solutions to a class of Hilfer fractional pantograph equations with impulsive effect. Further, using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss various kinds of Ulam-Hyers stability.

Kaynakça

  • [1] S. Harikrishnan, RabhaW. Ibrahim, K. Kanagarajan, On y-Hilfer Fractional Differential Equation with Complex Order, Universal Journal of Mathematics and Applications(1) (2018) 33-38 Universal Journal of Mathematics and Applications
  • [2] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions, J. Contemp. Mathemat. Anal., 50, (2015), 209-219.
  • [3] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sci., 33B,(2013),1-9.
  • [4] K.M. Furati, M.D. Kassim and N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626.
  • [5] A. Granas, and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  • [6] K. Guan, Q. Wang, X. He, Oscillation of a pantograph differential equation with impulsive perturbations, Appl. Math. Comput., 219, (2012), 3147-3153.
  • [7] S. Harikrishnan, K. Kanagarajan and E. M. Elsayed, Existence and stability results for langevin equations with Hilfer fractional derivative, Res. Fixed Point Theory Appl., (2018), 10 pages.
  • [8] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, (1941), 222-224.
  • [9] R. Hilfer, Applications of Eractional Calculus in Physics, World scientific, Singapore, 1999.
  • [10] A. Iserles, On the generalized pantograph functional differential equation, European J. Appl. Math., 4, (1993), 1-38.
  • [11] R.W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstract Appl. Anal., (2012), 10 pages.
  • [12] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23(5), (2012), 9 pages.
  • [13] A. A. Kilbas, H.M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier,2006.
  • [14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989.
  • [15] X. Liu and Y. Li, Some antiperiodic boundary value problem for nonlinear fractional impulsive differential equations, Abst. Appl. Anal., (2014), 10 pages.
  • [16] Z. Luo and J. Shen, Global existence results for impusive functional differential equation, J. Math. Anal. Appl., 323(1), (2006), 644-653.
  • [17] I. Podlubny, Fractional Differential equation, Academic Press, San Diego, 1999.
  • [18] A. M. Samoilenko and N. A. Perestyuk, Impulsive differential equations, World scientific, Singapore(1995).
  • [19] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4), (2016),685-698.
  • [20] D. Vivek, K. Kanagarajan and E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15:15, (2018).
  • [21] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract., 1(1), (2017), 1-14.
  • [22] D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, 2, (2017), 105-112.
  • [23] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, (2011), 1-10.
  • [24] J. Wang, Y. Zhou and M. Feckanc, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64, (2012), 3389-3405.
  • [25] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328(2), (2007), 1075-1081.
Yıl 2018, Cilt: 1 Sayı: 1, 36 - 42, 30.06.2018
https://doi.org/10.33401/fujma.406651

Öz

Kaynakça

  • [1] S. Harikrishnan, RabhaW. Ibrahim, K. Kanagarajan, On y-Hilfer Fractional Differential Equation with Complex Order, Universal Journal of Mathematics and Applications(1) (2018) 33-38 Universal Journal of Mathematics and Applications
  • [2] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions, J. Contemp. Mathemat. Anal., 50, (2015), 209-219.
  • [3] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sci., 33B,(2013),1-9.
  • [4] K.M. Furati, M.D. Kassim and N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626.
  • [5] A. Granas, and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
  • [6] K. Guan, Q. Wang, X. He, Oscillation of a pantograph differential equation with impulsive perturbations, Appl. Math. Comput., 219, (2012), 3147-3153.
  • [7] S. Harikrishnan, K. Kanagarajan and E. M. Elsayed, Existence and stability results for langevin equations with Hilfer fractional derivative, Res. Fixed Point Theory Appl., (2018), 10 pages.
  • [8] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, (1941), 222-224.
  • [9] R. Hilfer, Applications of Eractional Calculus in Physics, World scientific, Singapore, 1999.
  • [10] A. Iserles, On the generalized pantograph functional differential equation, European J. Appl. Math., 4, (1993), 1-38.
  • [11] R.W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstract Appl. Anal., (2012), 10 pages.
  • [12] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23(5), (2012), 9 pages.
  • [13] A. A. Kilbas, H.M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier,2006.
  • [14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989.
  • [15] X. Liu and Y. Li, Some antiperiodic boundary value problem for nonlinear fractional impulsive differential equations, Abst. Appl. Anal., (2014), 10 pages.
  • [16] Z. Luo and J. Shen, Global existence results for impusive functional differential equation, J. Math. Anal. Appl., 323(1), (2006), 644-653.
  • [17] I. Podlubny, Fractional Differential equation, Academic Press, San Diego, 1999.
  • [18] A. M. Samoilenko and N. A. Perestyuk, Impulsive differential equations, World scientific, Singapore(1995).
  • [19] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4), (2016),685-698.
  • [20] D. Vivek, K. Kanagarajan and E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15:15, (2018).
  • [21] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract., 1(1), (2017), 1-14.
  • [22] D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, 2, (2017), 105-112.
  • [23] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, (2011), 1-10.
  • [24] J. Wang, Y. Zhou and M. Feckanc, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64, (2012), 3389-3405.
  • [25] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328(2), (2007), 1075-1081.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Sugumaran Harikrishnan

Rabha Ibrahim 0000-0001-9341-025X

Kuppusamy Kanagarajan

Yayımlanma Tarihi 30 Haziran 2018
Gönderilme Tarihi 16 Mart 2018
Kabul Tarihi 17 Nisan 2018
Yayımlandığı Sayı Yıl 2018 Cilt: 1 Sayı: 1

Kaynak Göster

APA Harikrishnan, S., Ibrahim, R., & Kanagarajan, K. (2018). Establishing the existence of Hilfer fractional pantograph equations with impulses. Fundamental Journal of Mathematics and Applications, 1(1), 36-42. https://doi.org/10.33401/fujma.406651
AMA Harikrishnan S, Ibrahim R, Kanagarajan K. Establishing the existence of Hilfer fractional pantograph equations with impulses. Fundam. J. Math. Appl. Haziran 2018;1(1):36-42. doi:10.33401/fujma.406651
Chicago Harikrishnan, Sugumaran, Rabha Ibrahim, ve Kuppusamy Kanagarajan. “Establishing the Existence of Hilfer Fractional Pantograph Equations With Impulses”. Fundamental Journal of Mathematics and Applications 1, sy. 1 (Haziran 2018): 36-42. https://doi.org/10.33401/fujma.406651.
EndNote Harikrishnan S, Ibrahim R, Kanagarajan K (01 Haziran 2018) Establishing the existence of Hilfer fractional pantograph equations with impulses. Fundamental Journal of Mathematics and Applications 1 1 36–42.
IEEE S. Harikrishnan, R. Ibrahim, ve K. Kanagarajan, “Establishing the existence of Hilfer fractional pantograph equations with impulses”, Fundam. J. Math. Appl., c. 1, sy. 1, ss. 36–42, 2018, doi: 10.33401/fujma.406651.
ISNAD Harikrishnan, Sugumaran vd. “Establishing the Existence of Hilfer Fractional Pantograph Equations With Impulses”. Fundamental Journal of Mathematics and Applications 1/1 (Haziran 2018), 36-42. https://doi.org/10.33401/fujma.406651.
JAMA Harikrishnan S, Ibrahim R, Kanagarajan K. Establishing the existence of Hilfer fractional pantograph equations with impulses. Fundam. J. Math. Appl. 2018;1:36–42.
MLA Harikrishnan, Sugumaran vd. “Establishing the Existence of Hilfer Fractional Pantograph Equations With Impulses”. Fundamental Journal of Mathematics and Applications, c. 1, sy. 1, 2018, ss. 36-42, doi:10.33401/fujma.406651.
Vancouver Harikrishnan S, Ibrahim R, Kanagarajan K. Establishing the existence of Hilfer fractional pantograph equations with impulses. Fundam. J. Math. Appl. 2018;1(1):36-42.

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