Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Existence and Stability Results for Boundary Value Problem for Differential Equation with ψ-Hilfer Fractional Derivative

Journal of Environmental Accounting and Management 8(2) (2019) 251--259 | DOI:10.5890/JAND.2019.06.008

S. Harikrishnan, K. Kanagarajan, D. Vivek

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641020, India

Download Full Text PDF



In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equations with ψ-Hilfer fractional derivative. The arguments are based upon Schaefer’s fixed point theorem, Banach contraction principle and ulam type stability.


  1. [1]  Hilfer, R. (1999), Applications of fractional Calculus in Physics, World scientific, Singapore.
  2. [2]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Amsterdam: Elsevier.
  3. [3]  Podlubny, I. (1999), Fractional differential equations, Academic Press, San Diego.
  4. [4]  Agarwal, R.P., Benchohra, M., Hamani, S., and Pinelas, S. (2011), Boundary value problem for differential equations involving Riemann-Liouville fractional derivative on the half line, Dynamics of Continuous, Discrete and Impulsive Systems, 18, 235-244.
  5. [5]  Balachandran, K., Kiruthika, S., Rivero, M., and Trujillo, J.J. (2012), Existence of Solutions for Fractional Delay Integrodifferential Equations, Journal of Applied Nonlinear Dynamics, 1(4), 309-319.
  6. [6]  Furati, K.M., Kassim, M.D., and Tatar, N.E. (2012), Existence and uniqueness for a problem involving hilfer fractional derivative, Computer and Mathematics with Application, 64, 1616-1626.
  7. [7]  Joice, N.R. and Balachandran, K. (2016), Controllability of Nonlinear Fractional Delay Integrodifferential Systems, Discontinuity, Nonlinearity, and Complexity, 5(1), 59-73.
  8. [8]  Nanware, J.A. and Dhaigude, D.B. (2014), Boundary value problems for differential equations of non-integer order involving Caputo fractional derivative, Advanced Studies in Contemporary Mathematics, 3, 369-376.
  9. [9]  Wang, J., Lv, L., and Zhou, Y. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 63, 1-10.
  10. [10]  Sousa, Va.D.C.J. and de Oliveira, E.C., On the ψ-Hilfer fractional derivative, arXiv:1704.08186.
  11. [11]  Abbas, M.I. (2015), Ulam stability of fractional impulsive differential equations with riemann-liouville integral boundary conditions, Journal of Contemporary Mathematical Analysis, 50, 209-219.
  12. [12]  Chen, F. and Zhou, Y. (2013), Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem, Discrete Dynamics in Nature and Society, 7pages.
  13. [13]  Vivek, D., Kanagarajan, K., and Harikrishnan, S., Dynamics and stability of Hilfer-Hadamard type fractional pantograph equations with boundary conditions, Journal of Nonlinear Analysis and Application, (Accepted article-ID-jnaa-00387).
  14. [14]  Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.