Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Existence, Uniqueness and Stability Results of Fractional Volterra-Fredholm Integro Differential Equations of $psi$-Hilfer Type

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 535--545 | DOI:10.5890/DNC.2021.09.013

Ahmed A. Hamoud$^{1}$ , Abdulrahman A. Sharif$^{2}$, Kirtiwant P. Ghadle$^{2}$

$^{1}$ Department of Mathematics, Taiz University, Taiz-380 015, Yemen

$^{2}$ Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India

Download Full Text PDF

 

Abstract

In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear $\psi$-Hilfer fractional Volterra-Fredholm integro differential equations with boundary conditions. In addition, the Ulam-Hyers stability for solutions of the given problem are also discussed. The desired results are proved by using generalized Gronwall inequality, aid of fixed point theorems due to Banach and Schauder in weighted spaces.

References

  1. [1]  Dawood, L., Hamoud, A., and Mohammed, N. (2020), { Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations}, { Journal of Mathematics and Computer Science}, {\bf21}(2), 158-163.
  2. [2]  Hamoud, A. and Ghadle, K. (2018), The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, { Probl. Anal. Issues Anal.}, {\bf7}(25), 41-58.
  3. [3]  Hamoud, A. and Ghadle, K. (2018), Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, { Tamkang J. Math.}, {\bf 49}(4), 301-315.
  4. [4]  Hamoud, A., Ghadle, K., and Atshan, S. (2019), The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, { Khayyam J. Math.}, {\bf5}(1), 21-39.
  5. [5]  Karthikeyan, K. and Trujillo, J. (2012), Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, { Commun. Nonlinear Sci. Numer. Simulat.}, {\bf17}, 4037-4043.
  6. [6]  Kilbas, A., Srivastava, H. and Trujillo, J. (2006), { Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud.}, Elsevier, Amsterdam, {\bf204}.
  7. [7]  Lakshmikantham, V. and Rao, M. (1995), { Theory of Integro-Differential Equations}, { Gordon \& Breach}, London.
  8. [8]  Miller, K. and Ross, B. (1993), { An Introduction to the Fractional Calculus and Differential Equations}, John Wiley, New York.
  9. [9]  Samko, S., Kilbas, A., and Marichev, O. (1993), { Fractional Integrals and Derivatives, Theory and Applications}, Gordon and Breach, Yverdon.
  10. [10]  Wu, J. and Liu, Y. (2009), Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, { Electronic Journal of Differential Equations}, {\bf2009}, 1-8.
  11. [11]  Wu, J. and Liu, Y. (2010), Existence and Uniqueness Results for Fractional Integro-Differential Equations with Nonlocal Conditions, { 2nd IEEE International Conference on Information and Financial Engineering}, 91-94.
  12. [12]  Ulam, S.M. (1960), { Problems in Modern Mathematics}, Chapter VI, Science Editions, Wiley, New York.
  13. [13]  Rassias, T.H.M. (1978), On the stability of the linear mapping in Banach spaces, { Proc. Amer. Math. Soc.}. 72, 297-300.
  14. [14]  Oliveira, E. and Sousa, C.J. (2018), Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math. 73(3), 111.
  15. [15]  Sousa, J.V.C., Rodrigues, F.G., and Oliveira, E.C. (2019), Stability of the fractional Volterra integro-differential equation by means of $\psi$-Hilfer operator, { Math. Methods Appl. Sci.} {\bf42}, 3033-3043.
  16. [16]  Sousa, J.V.C., Kucche, K.D., Oliveira, E.C. (2019), Stability of $\psi$-Hilfer impulsive fractional differential equations, { Appl. Math. Lett.}, {\bf88}, 73-80.
  17. [17]  Sousa, J.V.D.C. and de Oliveira, E.C. (2019), Leibniz type rule: $\psi$-Hilfer fractional operator, { Commun. Nonlinear Sci. Numer. Simulat.}, {\bf 77}, 305-311.
  18. [18]  Sousa, C.J. and de Oliveira E.C. (2018), Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, { Appl Math Lett.}, 81, 50-56.
  19. [19]  Furati, K.M., Kassim, M.D., and Tatar, N. (2012), Existence and uniqueness for a problem involving Hilfer fractional derivative, { Comput. Math. Appl.}, {\bf64}, 1616-1626.
  20. [20]  Sousa, J.V.D.C. and de Oliveira, E.C. (2018), On the $\psi$-Hilfer fractional derivative, { Commun. Nonlinear Sci. Numer. Simulat.}, {\bf60}, 72-91.
  21. [21]  Thabet, S.T.M., Ahmad, B., and Agarwal, R.P. (2019), On abstract Hilfer fractional integrodifferential equations with boundary conditions, { Arab J. Math. Sci.}
  22. [22]  Vivek, D., Elsayed, E., and Kanagarajan, K. (2018), Theory and analysis of $\psi$-fractional differential equations with boundary conditions, { Commun. Appl. Anal.}, {\bf22}, 401-414.
  23. [23]  Wang, J. and Zhang. Y. (2015), Nonlocal initial value problems for differential equations with Hilfer fractional derivative, { Appl. Math. Comput.}, {\bf266}, 850-859.
  24. [24]  Hamoud, A. and Ghadle, K. (2019), Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, { J. Appl. Comput. Mech.} {\bf 5}(1), 58-69.
  25. [25]  Hamoud, A. and Ghadle, K. (2018), Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations, { Indonesian Journal of Electrical Engineering and Computer Science}, {\bf 11}(3), 857-867.
  26. [26]  Almeida, R. (2017), A Caputo fractional derivative of a function with respect to another function, { Commun. Nonlinear Sci. Numer. Simulat.}, {\bf44}, 460-481.
  27. [27]  Vivek, D., Kanagarajan, K., and Elsayed, E. (2018), Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, { Mediterr. J. Math.}, {\bf15}, 1-15.
  28. [28]  Momani, S., Jameel, A., and Al-Azawi, S. (2007), Local and global uniqueness theorems on fractional integro-differential equations via Biharis and Gronwalls inequalities, { Soochow Journal of Mathematics}, {\bf33}(4), 619-627.
  29. [29]  Zhou, Y. (2014), { Basic Theory of Fractional Differential Equations}, { Singapore: World Scientific}.
  30. [30]  Rus, I.A. (2010), Ulam stabilities of ordinary differential equations in a Banach space, { Carpath. J. Math.}. {\bf26}, 103-107.
  31. [31]  Sousa, J.V.C. and Oliveira, E.C. (2019), A Gronwall inequality and the Cauchy-type problem bymeans of $\psi$-Hilfer operator, { Differ. Equ. Appl.}, {\bf11}, 87-106.