J. Appl. Math. & Informatics Vol. 31(2013), No. 3 - 4, pp. 565 - 575 Website: http://www.kcam.biz STRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS M.O. OSILIKE * , F.O. ISIOGUGU AND F.U. ATTAH Abstract. Let H be a real Hilbert space and let T : H → H be a Lipschitz pseudocontractive mapping. We introduce a modified Ishikawa iterative algorithm and prove that if F (T )= {x ∈ H : Tx = x}̸ = ∅, then our proposed iterative algorithm converges strongly to a fixed point of T . No compactness assumption is imposed on T and no further requirement is imposed on F (T ). AMS Mathematics Subject Classification : 47J25, 47H09, 65J15. Key words and phrases : Pseudocontractive maps, fixed points, Ishikawa iteration, strong convergence, Hilbert spaces. 1. Introduction Let H be a real Hilbert space with inner product ⟨., .⟩ and induced norm ||.||. Let C be a nonempty closed convex subset of H. A mapping T : C → C is said to be L-Lipschitzian if there exists L ≥ 0 such that ||Tx − Ty|| ≤ L||x − y||, ∀x, y ∈ C. (1.1) T is said to be a contraction if L ∈ [0, 1) and T is said to be nonexpansive if L = 1. T is said to be pseudocontractive in the terminology of Browder and Petryshyn [1] if ||Tx − Ty|| 2 ≤ ||x − y|| 2 + ||x − Tx − (y − Ty)|| 2 , ∀x, y ∈ C. (1.2) It is easy to verify that (1.1) is equivalent to the monotonicity condition ⟨(I − T )x − (I − T )y,x − y⟩≥ 0, ∀x, y ∈ C, (1.3) Received September 2, 2012. Revised September 24, 2012. Accepted October 11, 2012. * Corresponding author. c ⃝ 2013 Korean SIGCAM and KSCAM. 565