Nonlinear Analysis 71 (2009) e2833–e2838 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Convergence theorem for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Φ -hemicontractive mappings J.K. Kim a,∗ , D.R. Sahu b , Y.M. Nam a a Department of Mathematics Educations, Kyungnam University, Masan, Kyungnam, 631-701, Republic of Korea b Department of Mathematics, Banaras Hindu university, Varanasi-221005, India article info MSC: 47H09 46B20 47H10 Keywords: Asymptotically nonexpansive mapping Asymptotically pseudocontractive mapping Nearly asymptotically nonexpansive mapping Asymptotically generalized Φ-pseudocontractive mapping Asymptotically generalized Φ-hemicontractive mapping abstract In this paper, we introduce the new class of asymptotically generalized Φ-hemicontractive mappings and establish a strong convergence theorem of the iterative sequence generated by these mappings in a general Banach space. The main result of this paper is the improvement of the well-known corresponding result. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Let X be an arbitrary real normed linear space with dual space X ∗ . We denote by J the normalized duality mapping from X into 2 X ∗ defined by J (x) := {f ∗ ∈ X ∗ :〈x, f ∗ 〉=‖x‖ 2 =‖f ∗ ‖ 2 }, where 〈·, ·〉 denotes the generalized duality pairing. Definition 1.1. Let C be a nonempty subset of real normed linear space X . A mapping T : C → X is said to be (1) strongly pseudocontractive if for all x, y ∈ C , there exist constant k ∈ (0, 1) and j(x − y) ∈ J (x − y) satisfying 〈Tx − Ty, j(x − y)〉≤ k‖x − y‖ 2 , (2) φ-strongly pseudocontractive if for all x, y ∈ C , there exist strictly increasing function φ :[0, ∞) →[0, ∞) with φ(0) = 0 and j(x − y) ∈ J (x − y) satisfying 〈Tx − Ty, j(x − y)〉≤‖x − y‖ 2 − φ(‖x − y‖)‖x − y‖, (3) generalized Φ-pseudocontractive (cf. [1,2]) if for all x, y ∈ C , there exist strictly increasing function Φ :[0, ∞) →[0, ∞) with Φ(0) = 0 and j(x − y) ∈ J (x − y) satisfying 〈Tx − Ty, j(x − y)〉≤‖x − y‖ 2 − Φ(‖x − y‖), ∗ Corresponding author. Tel.: +82 55 249 2211; fax: +82 55 243 8609. E-mail addresses: jongkyuk@kyungnam.ac.kr (J.K. Kim), drsahu@bhu.ac.in (D.R. Sahu), nym4953@kyungnam.ac.kr (Y.M. Nam). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.06.091