Nonlinear Analysis 74 (2011) 4293–4299 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Solution of nonlinear integral equations of Hammerstein type C.E. Chidume a,∗ , E.U. Ofoedu b a The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy b Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria article info Article history: Received 29 April 2009 Accepted 10 February 2011 MSC: 47H06 47H09 47J05 47J25 Keywords: Accretive operators Generalized duality maps Equations of Hammerstein type Modulus of smoothness Uniformly Gâteaux differentiable norm abstract Let E be a 2-uniformly real Banach space and F , K : E → E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u + KFu = 0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction Let E be a real normed space and let S := {x ∈ E :‖x‖= 1}. E is said to have a Gâteaux differentiable norm if the limit lim t →0 + ‖x + ty‖−‖x‖ t exists for each x, y ∈ S . When this limit exists, we say that E is smooth. E is said to have a uniformly Gâteaux differentiable norm if for each y ∈ S the limit is attained uniformly for x ∈ S . Furthermore, E is said to be uniformly smooth if the limit exists uniformly for (x, y) ∈ S × S . Let E be a normed linear space with dimension greater than or equal to 2. The modulus of smoothness of E is the function ρ E :[0, ∞) →[0, ∞) defined by ρ E (t ) := sup  ‖x + y‖+‖x − y‖ 2 − 1 :‖x‖= 1, ‖y‖= t  . In terms of the modulus of smoothness, the space E is called uniformly smooth if and only if lim t →0 + ρ E (t ) t = 0. E is called q-uniformly smooth if there exists a constant c > 0 such that ρ E (t ) ≤ ct q , t > 0. L p (and ℓ p ) spaces, 1 < p < +∞ are q-uniformly smooth. In particular, L p is 2-uniformly smooth if 2 ≤ p < +∞ and p-uniformly smooth if 1 < p < 2. It is easy ∗ Corresponding author. Fax: +39 040224163. E-mail addresses: chidume@ictp.trieste.it (C.E. Chidume), euofoedu@yahoo.com (E.U. Ofoedu). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.017