Existence and Uniqueness Results of Hadamard Fractional Volterra-Fredholm Integro-Differential Equations

Year 2021, Volume: 9 Issue: 1, 143 - 147, 28.04.2021

Ahmed Hamoud , Abdulrahman Sharif Aishwary K. Ghadle Kirtiwant Ghadle

Abstract

In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear Hadamard fractional Volterra-Fredholm integro-differential equations with initial conditions. The desired results are proved by using Arzela-Ascoli theorem aid of fixed point theorems due to Banach and Krasnoselskii in Banach spaces. 

Keywords

Volterra-Fredholm integro differential equation, Hadamard fractional derivative, Fixed point method

References

• [1] S. Abbas, M. Benchohra, J. Lazreg and Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability, Chaos, Solitons & Fractals, Vol:102, (2017), 47-71.
• [2] S. Abbas, M. Benchohra, J. Lazreg and Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Advances in Difference Equations, Vol:180, (2017), 1-14.
• [3] B. Ahmad, S. Ntouyas and J. Tariboon, A study of mixed Hadamard and Riemann-Liouville fractional integro-differential inclusions via endpoint theory, Applied Mathematics Letters, Vol:52, (2016), 9-14.
• [4] P. Butzer, A. Kilbas and J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. Vol:269, (2002), 387-400.
• [5] B. Ahmad and S. Ntouyas, Initial-value problems for hybrid Hadamard fractional differential equations, Electronic Journal of Differential Equations, Vol:2014, No. 161 (2014), 1-8.
• [6] L. Dawood, A. Hamoud and N. Mohammed, Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations, Journal of Mathematics and Computer Science, Vol:21, No.2 (2020), 158-163.
• [7] J. Hadamard, Essai sur letude` des fonctions donnees` par leur developpement` de Taylor, J. Mat. Pure Appl. Ser. Vol:4, No.8 (1892), 101-186.
• [8] 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, Vol:4, No.4 (2020), 321-331.
• [9] A. Hamoud, N. Mohammed and K. Ghadle, Existence and uniqueness results for Volterra-Fredholm integro-differential equations, Advances in the Theory of Nonlinear Analysis and its Application, Vol:4, No.4 (2020), 361-372.
• [10] A. Hamoud and K. Ghadle, The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., Vol:7 (25), No.1 (2018), 41-58.
• [11] A. Hamoud and K. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, J. Math. Model., Vol:6, No. 1 (2018), 91-104.
• [12] A. Hamoud and K. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, Tamkang J. Math. Vol:49, No.4 (2018), 301-315.
• [13] A. Hamoud and K. Ghadle, Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian J. Math. Vol:60, No.3 (2018), 375-395.
• [14] A. Hamoud, K. Ghadle and S. Atshan, The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, Khayyam J. Math. Vol:5, No.1 (2019), 21-39.
• [15] A. Hamoud and K. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comput. Mech. Vol:5, No.1 (2019), 58-69.
• [16] K. Karthikeyan and J. Trujillo, Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., Vol:17, (2012), 4037-4043.
• [17] A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. Vol:38, (2001), 1191-1204.
• [18] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., Elsevier, Amsterdam, Vol:204, 2006.
• [19] V. Lakshmikantham and M. Rao, Theory of Integro-Differential Equations, Gordon & Breach, London, 1995.
• [20] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• [21] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [22] J. Wang, Y. Zhou and M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topological Methods in Nonlinear Analysis, Vol:41, No.1 (2013), 113-133.
• [23] J. Wu and Y. Liu, Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, Electronic Journal of Differential Equations, Vol:2009, (2009), 1-8.
• [24] J. Wu and Y. Liu, Existence and uniqueness results for fractional integro-differential equations with nonlocal conditions, 2nd IEEE International Conference on Information and Financial Engineering, (2010), 91-94.
• [25] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 2014.