ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

Existence, Uniqueness and Stability Results for Nonlocal Fractional Nonlinear Volterra-Fredholm Integro Differential Equations

Discontinuity, Nonlinearity, and Complexity 11(2) (2022) 343--352 | DOI:10.5890/DNC.2022.06.013

Ahmed A. Hamoud$^{1}$, Nedal M. Mohammed$^{2}$, Kirtiwant P. Ghadle$^{3}$

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

$^{2}$ Department of Computer Science \& IT, Taiz University, Taiz, Yemen

$^{3}$ Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India

Abstract

In this paper, we prove the existence and uniqueness of solutions for a class of nonlinear fractional Volterra-Fredholm integro differential equations with nonlocal conditions. In addition, the Ulam-Hyers and Ulam-Hyers-Rassias stability for solutions of the given problem are also discussed. The desired results are proved by using Pachpatte's integral inequality, aid of fixed point theorems due to Banach and Schaefer's fixed point theorems.

Acknowledgments

The authors are grateful to the editor and the referees for the careful reading of the paper.

References

1.  [1] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 204.
2.  [2] Lakshmikantham, V. and Rao, M. (1995), Theory of Integro-Differential Equations, Gordon \& Breach, London.
3.  [3] Ulam, S.M. (1960), Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York.
4.  [4] Zhou, Y. (2014), Basic Theory of Fractional Differential Equations, World Scientific, Singapore.
5.  [5] Balachandran, K. and Trujillo, J. (2010), The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal. Theory Meth. Applic., 72, 4587-4593.
6.  [6] Benchohra, M. and Bouriahi, S. (2015), Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure Appl. Anal., 1, 22-37.
7.  [7] Hamoud, A. and Ghadle, K. (2018), The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., 7(25), 41-58.
8.  [8] Hamoud, A. and Ghadle, K. (2018), Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, J. Math. Model., 6(1), 91-104.
9.  [9] Kendre, S., Jagtap, T., and Kharat, V. (2013), On nonlinear fractional integrodifferential equations with non local condition in Banach spaces, Nonl. Analysis and Differential Equations, 1(3), 129-141.
10.  [10] Dawood, L., Hamoud, A., and Mohammed, N. (2020), Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations, J. Math. Computer Sci., 21(2), 158-163.
11.  [11] Hamoud, A. and Ghadle, K. (2019), Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comput. Mech., 5, 58-69.
12.  [12] 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., 49(4), 301-315.
13.  [13] Rassias, TH.M. (1978), On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300.
14.  [14] Rassias, TH.M. (2000), On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62, 23-130.
15.  [15] Rus, I.A. (2010), Gronwall lemma approach to the Hyers-Ulam-Rassias stability of an integral equation, Springer Optim. Appl., 147-152.
16.  [16] Sousa, C.J. and Oliveira, E. (2018), Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl Math Lett., 81, 50-56.
17.  [17] Sousa, C.J. and Oliveira, E. (2018), On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\Psi$-Hilfer operator, J. Fixed Point Theory Appl., 20(3), 1-21.
18.  [18] Oliveira, E. and Sousa, C.J. (2018), Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math. 73(3), 1-11.
19.  [19] Baleanu, D., Rezapour, S., and Mohammadi, H. (2013), Some existence results on nonlinear fractional differential equations, Phil. Trans. R Soc. A, 1-7.
20.  [20] Devi, J. and Sreedhar, C. (2016), Generalized monotone iterative method for Caputo fractional integro-differential equation, European Journal of Pure and Applied Mathematics, 9(4), 346-359.
21.  [21] Wang, J. and Zhou, L. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, 1-10.
22.  [22] Dong, L., Hoa, N., and Vu, H. (2020), Existence and Ulam stability for random fractional integro-differential equation, Afr. Mat., 1-12.