Nonlinear Functional Analysis and Applications Vol. 23, No. 4 (2018), pp. 797-805 ISSN: 1229-1595(print), 2466-0973(online) http://nfaa.kyungnam.ac.kr/journal-nfaa Copyright c  2018 Kyungnam University Press KUP ress EXISTENCE AND UNIQUENESS RESULTS FOR NONLINEAR VOLTERRA-FREDHOLM INTEGRO DIFFERENTIAL EQUATIONS Ahmed A. Hamoud 1 , M.SH. Bani Issa 2 and Kirtiwant P. Ghadle 1 1 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004 India. e-mail: drahmedselwi985@gmail.com 2 Department of Mathematics, P.E.T. Research Foundation Mandya, University of Mysore, Mysore-570 401 India. e-mail: moh.smarh@yahoo.com Abstract. In this article, variational iteration technique is successfully applies to find the approximate solutions of nonlinear Volterra-Fredholm integro-differential equations. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Moreover, we prove the existence and uniqueness results. Finally, the examples are included to demonstrate the validity and applicability of the proposed technique. 1. Introduction In this paper, we consider nonlinear Volterra-Fredholm integro-differential equation of the form: k  j =0 ξ j (x)u (j ) (x)= f (x)+ λ 1  x a K 1 (x,t)G 1 (u(t))dt (1.1) 0 Received May 4, 2018. Revised October 17, 2018. 0 2010 Mathematics Subject Classification: 35A15, 34A12, 45J05. 0 Keywords: Variational iteration method, Volterra-Fredholm integro-differential equa- tion, existence and uniqueness results, approximate solution. 0 Corresponding author: A. A. Hamoud(drahmedselwi985@gmail.com)