Uniqueness and stability results for caputo fractional Volterra--Fredholm integro-differential equations

Ahmed A. Hamoud

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 3, pp. 313-325.

Résumé

In this paper, we established some new results concerning the uniqueness and Ulam's stability results of the solutions of iterative nonlinear Volterra–Fredholm integro-differential equations subject to the boundary conditions. The fractional derivatives are considered in the Caputo sense. These new results are obtained by applying the Gronwall–Bellman's inequality and the Banach contraction fixed point theorem. An illustrative example is included to demonstrate the efficiency and reliability of our results.

Détail

Keywords: Volterra–Fredholm integro-differential equation, Caputo sense, Gronwall–Bellman's inequality, Banach contraction fixed point theorem.

Bibliographie

[1] B. Ahmad, S. Sivasundaram, “Some existence results for fractional integro-differential equations with nonlinear conditions”, Communications Appl. Anal., 12 (2008), 107–112

[2] M. Ali, W. Ma, “New exact solutions of nonlinear (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation”, Advances in Mathematical Physics, 2019 (2019), 1–8

[3] M. Ali, A. Hadhoud, H. Srivastava, “Solution of fractional Volterra–Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method”, Advances in Mathematical Physics, 2019:1 (2019), 1–14 | DOI

[4] M. Bani Issa, A. Hamoud, K. Ghadle, “Numerical solutions of fuzzy integro-differential equations of the second kind”, Journal of Mathematics and Computer Science, 23 (2021), 67–74 | DOI

[5] S. Cheng, S. Talwong, V. Laohakosol, “Exact solution of iterative functional differential equation”, Computing, 76 (2006), 67–76 | DOI

[6] L. Dawood, A. Sharif, A. Hamoud, “Solving higher-order integro-differential equations by VIM and MHPM”, International Journal of Applied Mathematics, 33 (2020), 253–264 | DOI

[7] A. Hamoud, K. Ghadle, “The approximate solutions of fractional Volterra–Fredholm integro-differential equations by using analytical techniques”, Probl. Anal. Issues Anal., 7(25):1 (2018), 41–58 | DOI

[8] A. Hamoud, K. Ghadle, “Usage of the homotopy analysis method for solving fractional Volterra–Fredholm integro-differential equation of the second kind”, Tamkang J. Math., 49:4 (2018), 301–315 | DOI

[9] A. Hamoud, K. Hussain, K. Ghadle, “The reliable modified Laplace Adomian decomposition method to solve fractional Volterra–Fredholm integro differential equations”, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications $\$ Algorithms, 26 (2019), 171–184

[10] A. Hamoud, K. Ghadle, “Existence and uniqueness of the solution for Volterra–Fredholm integro-differential equations”, Journal of Siberian Federal University. Mathematics $\$ Physics, 11:6 (2018), 692–701 | DOI

[11] A. Hamoud, K. Ghadle, “Some new existence, uniqueness and convergence results for fractional Volterra–Fredholm integro-differential equations”, J. Appl. Comput. Mech., 5:1 (2019), 58–69 | DOI

[12] A. Hamoud, “Existence and uniqueness of solutions for fractional neutral Volterra–Fredholm integro-differential equations”, Advances in the Theory of Nonlinear Analysis and its Application, 4:4 (2020), 321–331 | DOI

[13] J.H. He, “Some applications of nonlinear fractional differential equations and their approximations”, Bull. Sci. Technol. Soc., 15 (1999), 86–90

[14] R. Ibrahim, S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations”, Journal of Mathematical Analysis and Applications, 334 (2007), 1–10 | DOI

[15] R. Ibrahim, “Existence of iterative Cauchy fractional differential equations”, Int. J. Math. Sci., 7 (2013), 379–384

[16] S. Kendre, V. Kharat, R. Narute, “On existence of solution for iterative integro-differential equations”, Nonlinear Anal. Differ. Equ., 3 (2015), 123–131

[17] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006

[18] V. Lakshmikantham, M. Rao, Theory of Integro-Differential Equations, Gordon $\$ Breach, London, 1995

[19] M. Lauran, “Existence results for some differential equations with deviating argument”, Filomat, 25 (2011), 21–31 | DOI

[20] W. Li, S. Cheng, “A picard theorem for iterative differential equations”, Demonstratio Math., 42 (2009), 371–380

[21] S. Muthaiah, M. Murugesan, N. Thangaraj, “Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations”, Adv. Theory Nonlinear Anal. Appl., 3 (2019), 162–173 | DOI

[22] M. Matar, “Controllability of fractional semilinear mixed Volterra–Fredholm integro differential equations with nonlocal conditions”, Int. J. Math. Anal., 4 (2010), 1105–1116

[23] S. Momani, A. Jameel, S. Al-Azawi, “Local and global uniqueness theorems on fractional integro-differential equations via Bihari's and Gronwall's inequalities”, Soochow J. Math., 33 (2007), 619–627

[24] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993

[25] R. Panda, M. Dash, “Fractional generalized splines and signal processing”, Signal Process, 86 (2006), 2340–2350

[26] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993

[27] R. Shah, A. Zada, “A fixed point approach to the stability of a nonlinear Volterra integro-differential equations with delay”, Hacet. J. Math. Stat., 47 (2018), 615–623

[28] S. Unhaley, S. Kendre, “On existence and uniqueness results for iterative fractional integro-differential equation with deviating arguments”, Applied Mathematics E-Notes, 19 (2019), 116–127

[29] J. Wu, Y. Liu, “Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces”, Electron. J. Differential Equations, 2009 (2009), 1–8