Existence and uniqueness results for generalized fractional integrodifferential equations with nonlocal terminal condition

Document Type : Research Paper


1 Department of Mathematics, N. B. N. Sinhgad College of Engineering, Kegaon-Solapur-413255, India.

2 Department of Mathematics, D. B. F. Dayanand College of Arts and Science, Solapur-413002, India.

3 Department of Mathematics, Shivaji University, Kolhapur-416004, India.


In this study, we give results on the existence and uniqueness of solutions for generalized fractional integrodifferential equations with a nonlocal terminal condition. We have proved the existence of solutions to the problem proposed using the Schauder fixed point theorem and the uniqueness of its solutions is proved using the Banach fixed point theorem. At the end, we discussed the examples to support our results.


  • [1] S. P. Bhairat and D. B. Dhaigude, Existence of solutions of generalized differential equation with nonlocal condi- tion, Mathematica Bohemica, 144(2) (2019), 203–220.
  • [2] L. Byszewski, Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem, J. Math. Anal. appl., 162 (1991), 497–505.
  • [3] L. Byszewski and H. Alca, Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal., 34 (1998), 65–72.
  • [4] L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40 (1990), 11–19.
  • [5] T. B. Jagtap and V. V. Kharat, On Existence of Solution to Nonlinear fractional Integrodifferential System, J. Trajectory, 22(1) (2014), 40–46.
  • [6] U. N. Katugampola, New approach to a generalized fractional integral., Appl. Math. Comput. 218 (2011), 860–865.
  • [7] U. N. Katugampola, A new approach  to generalized fractional derivatives,  Bull. Math. Anal. Appl.,  6 (2014),  1–15.
  • [8] U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differenital equations, available at https, //arxiv.org/abs/1411.5229, (2016).
  • [9] S. D. Kendre, T. B. Jagtap, and V. V. Kharat, On nonlinear fractional integrodifferential equations with nonlocal condition in Banach space, Non. Anal. Diff. Eq., 1(3) (2013), 129–141.
  • [10] S. D. Kendre, V. V. Kharat, and T. B. Jagtap, On Abstract  Nonlinear  Fractional  Integrodifferential  Equations  with Integral Boundary condition, Comm. Appl. Nonl. Anal., 22(3) (2015), 93–108.
  • [11] S. D. Kendre, V. V. Kharat, and T. B. Jagtap, On Fractional Integrodifferential Equations with Fractional Non- separated Boundary conditions, Int. Jou. Appl. Math. Sci., 13(3) (2013), 169–181.
  • [12] S. D. Kendre, V. V. Kharat and R. Narute, On existence of solution for iterative integro-differential equations, Nonl. Anal. DIffer. Equ., (3) (2015), 123–131.
  • [13] V. V. Kharat, On existance and uniqueness of Fractional Integrodifferential Equations with an Integral Fractional Boundary Condition, Malaya J. Matematik, 6(3) (2018), 485–491.
  • [14] V. V. Kharat, D. B. Dhaigude, and D. R. Hasabe, On nonlinear mixed fractional integrodifferential inclusion with four-point nonlocal Riemann-Liouville integral boundary conditions, Indian J. Pure Appl. Math., 50(4) (2019), 937–951.
  • [15] V. V. Kharat and T. B. Jagtap, Existence of iterative fractional differential equation with non local condition, The Journal of Indian Mathematical Society, 83 (2016), 97–106.
  • [16] A. A. Kilbas, H. M. Srivastava, and J. J.Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204. Elsevier, Amsterdam, 2006.
  • [17] S. Krim, S. Abbas, M. Benchohra, and E. Karapinar, Terminal value problem for implicit Katugampola fractional differential equations in b Metric spaces, J. Funct. Spaces, 2021 (2021), 7 pages.
  • [18] A. A. Nanwate and S. P. Bhairat, On Nonlocal Terminal value Problems in generalized fractional sense, Palest. J. Math., 11 (Special Issue III) (2022), 62–74.
  • [19] D. S. Oliveira and E.Capelas de Oliveira, Hilfer-Katugampola fractional derivative, 37(3) (2018), 3672–3690.
  • [20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, (1999).
  • [21] H. Sussain Shah and M. Rehman, A note on terminal value problems for fractional differential equations  on  infinite interval, Appl. Math. Lett. 52 (2016), 118–125.
  • [22] W. Shreve, Terminal value problems for second order nonlinear differential equations, SIAM J. Appl. Math., 18(4) (1970), 783–791.
  • [23] S. R. Tate, V. V. Kharat, and H. T. Dinde, A nonlocal Cauchy problem for nonlinear fractional integrodifferential equations with positive constants, J. Math. Model., 7(1) (2019), 133–151.
  • [24] J. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850–859.