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

On $psi$-Caputo Fractional Nonlinear Volterra-Fredholm Integro-Differential Equations

Discontinuity, Nonlinearity, and Complexity 11(1) (2022) 97--106 | DOI:10.5890/DNC.2022.03.008

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

$^{1}$ Department of Mathematics, Hodeidah University, AL-Hudaydah, Yemen

$^{2}$ Department of Mathematics, Taiz University, Taiz-380 015, Yemen

Abstract

In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear $\psi$-Caputo fractional Volterra-Fredholm integro-differential equations with boundary conditions. The desired results are proved by using Banach and Schaefer's fixed point theorems in Banach spaces. Furthermore, Ulam's type stability of the proposed problem is studied.

References

1.  [1] Kilbas, A., Srivastava, H., and Trujillo, J. (2006), Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., Elsevier, Amsterdam, 204.
2.  [2] Lakshmikantham, V. and Rao, M. (1995), Theory of Integro-Differential Equations, Gordon $\&$ Breach, London.
3.  [3] Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, {John Wiley}, New York.
4.  [4] Samko, S., Kilbas, A., and Marichev, O. (1993), Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon.
5.  [5] Sarikaya, M. and Usta, F. (2016), On comparison theorems for conformable fractional differential equations, Int. J. Anal. Appl., 12(2), 207-214.
6.  [6] Usta, F., Budak, H., and Sarikaya, M. (2018), Yang-Laplace transform method Volterra and Abel's integro-differential equations of fractional order, Int. J. Nonlinear Anal. Appl., 9(2), 203-214.
7.  [7] Usta, F. (2020), Numerical analysis of fractional Volterra integral equations via Bernstein approximation method, J. Comput. Appl. Math., 384, 1-12.
8.  [8] Bani Issa, M., Hamoud, A., and Ghadle, K. (2021), Numerical solutions of fuzzy integro-differential equations of the second kind, J. Math. Comput. Sci., 23(1), 67-74.
9.  [9] Dawood, L., Hamoud, A., and Mohammed, N. (2020), Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations, J. Math. Comput. Sci., 21(2), 158-163.
10.  [10] 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.
11.  [11] 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.
12.  [12] 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., 5(1), 21-39.
13.  [13] 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(1), 58-69.
14.  [14] Karthikeyan, K. and Trujillo, J. (2012), Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17, 4037-4043.
15.  [15] Wu, J. and Liu, Y. (2009), Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, Electron. J. Differential Equations, 2009, 1-8.
16.  [16] 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.
17.  [17] Oliveira, E. and Sousa, C.J. (2018), Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results Math., 73(3), 111.
18.  [18] 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., 42, 3033-3043.
19.  [19] Sousa, J.V.C., Kucche, K.D., and Oliveira, E.C. (2019), Stability of $\psi$-Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88, 73-80.
20.  [20] Sousa, J.V.D.C. and de Oliveira, E.C. (2019), Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simulat., 77, 305-311.
21.  [21] Sousa, C.J. and Capelas, de Oliveira E. (2018), Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81, 50-56.
22.  [22] Furati, K.M., Kassim, M.D., and Tatar, N. (2012), Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, 1616-1626.
23.  [23] Sousa, J.V.D.C. and de Oliveira, E.C. (2018), On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simulat., 60, 72-91.
24.  [24] Vivek, D., Elsayed, E., and Kanagarajan, K. (2018), Theory and analysis of $\psi$-fractional differential equations with boundary conditions, Commun. Appl. Anal., 22, 401-414.
25.  [25] Wang, J. and Zhang. Y. (2015), Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266, 850-859.
26.  [26] Hamoud, A. (2020), Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro differential equations, Adv. Theory Nonlinear Anal. Appl., 4(4), 321-331.
27.  [27] Hamoud, A., Mohammed, N., and Ghadle, K. (2020), Existence and uniqueness results for Volterra-Fredholm integro differential equations, Adv. Theory Nonlinear Anal. Appl., 4(4), 361-372.
28.  [28] Almeida, R. (2017), A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat., 44, 460-481.
29.  [29] Momani, S., Jameel, A., and Al-Azawi, S. (2007), Local and global uniqueness theorems on fractional integro-differential equations via Bihari's and Gronwall's inequalities, Soochow Journal of Mathematics, 33(4), 619-627.
30.  [30] 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., 15, 1-15.
31.  [31] Vivek, D., Elsayed, E., and Kanagarajan, K. (2020), Existence and uniqueness results for $\psi-$fractional integro-differential equations with boundary conditions, Publications De L'institut Math{ematique,} 107(121), 145-155