Stud. Univ. Babe¸ s-Bolyai Math. 61(2016), No. 3, 331–341 On some numerical iterative methods for Fredholm integral equations with deviating arguments Sanda Micula Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversary Abstract. In this paper we develop iterative methods for nonlinear Fredholm inte- gral equations of the second kind with deviating arguments, by applying Mann’s iterative algorithm. This proves the existence and the uniqueness of the solution and gives a better error estimate than the classical Banach Fixed Point Theo- rem. The iterates are then approximated using a suitable quadrature formula. The paper concludes with numerical examples. Mathematics Subject Classification (2010): 45B05, 47H10, 47N20, 65R20. Keywords: Fredholm integral equations, deviating arguments, numerical approx- imations, Altman’s algorithm, Mann’s iterative algorithm. 1. Preliminaries Integral equations arise in many fields of mathematics, engineering, physics, etc., as they provide a strong tool for modeling various applications, phenomena and pro- cesses occurring in actuarial sciences, statistical study of dynamic living population, elasticity theory, diffraction problems, quantum mechanics, etc. In addition, a large class of initial and boundary value problems can be reformulated as integral equa- tions. Thus, many researchers aim to find efficient and rapidly convergent algorithms for the numerical solution of Fredholm integral equations (see e.g. [2], [10], [11], [9]). In this paper, we consider a Fredholm integral equation of the type x(t) = b  a K ( t, s, x(s),x(g(s)) ) ds + f (t), t ∈ [a, b], (1.1) where K ∈ C ( [a, b] × [a, b] × R 2 ) , f ∈ C [a, b] and g ∈ C ([a, b], [a, b]). Other assumptions will be made on K, g and f later on.