Palestine Journal of Mathematics Vol. 8(2)(2019) , 182–190 © Palestine Polytechnic University-PPU 2019 HYBRID ITERATION METHOD FOR FIXED POINTS OF ASYMPTOTICALLY φ−DEMICONTRACTIVE MAPS IN REAL HILBERT SPACES Uko Sunday Jim Communicated by Ayman Badawi MSC 2010 Classifications: Primary 47H09, 47H10; Secondary 47J05, 65J15. Keywords and phrases: Hybrid iteration method, Asymptotically φ-demi- contractive, Fixed point, Uniformly Lips- chitzian, Hilbert spaces. We acknowledge the anonymous reviewers for their useful comments. Abstract. A strong convergence theorem of Hybrid iteration method to fixed points of asymptotically φ−demicontractive mapping is proved in real Hilbert spaces. Our results extend, generalize and complement the results of Wang [8], Osilike, Isiogugu and Nwokoro [14], and extend several others from asymptotically demicontractive to the more general class of asymp- totically φ−demicontractive maps (see for example [2, 11, 17]). 1 Introduction Let K be a nonempty subset of a real Hilbert space H. A mapping T : K −→ K is said to be asymptotically φ−demicontractive with a sequence {k n } ∞ n=1 ⊆ [1, ∞), lim n→∞ k n = 1,(see for example, [3, 4]), if F (T )= {x ∈ K : Tx = x} = ∅ and there exists an increasing continuous function φ : [0, ∞) → [0, ∞) with φ(0)= 0 such that ‖T n x − p‖ 2 ≤ k n ‖x − p‖ 2 + ‖x − T n x‖ 2 − φ(‖x − T n x‖), (1.1) ∀x ∈ K, p ∈ F (T ) and n ≥ 1. A mapping T : K −→ K is said to be asymptotically demicontractive with a sequence {a n } ∞ n=1 ⊆ [1, ∞), lim n→∞ a n = 1, if F (T ) = ∅ and ∀x ∈ K, p ∈ F (T ), ∃ a k ∈ [0, 1) ∋ ‖T n x − p‖ 2 ≤ a 2 n ‖x − p‖ 2 + k‖(I − T n )x‖ 2 , (1.2) A mapping T : K −→ K is said to be k−strictly asymptotically pseudocontractive with a sequence {k n } ∞ n=1 ⊆ [1, ∞), lim n→∞ k n = 1 if ∀x, y ∈ K, n ∈ N, ∃ a k ∈ [0, 1) ∋ ‖T n x − T n y‖ 2 ≤ k n ‖x − y‖ 2 + k‖(I − T n )x − (I − T n )y‖ 2 , (1.3) where I is the identity operator. The class of k−strictly asymptotically pseudocontractive and asymptotically demicontractive maps were first introduced in Hilbert spaces by Qihou [10]. Ob- serve that a k−strictly asymptotically pseudocontractive map with a nonempty fixed point set F (T ) is asymptotically demicontractive. An example of a k−strictly asymptotically pseudocon- tractive map is given in Osilike et al. [16]. Furthermore, T is uniformly L−Lipschitzian if there exists a constant L> 0 ∋ ‖T n x − T n y‖≤ L‖x − y‖, (1.4) ∀x, y ∈ K and n ≥ 1. The class of asymptotically φ−demicontractive maps was first introduced in arbitrary real Ba- nach spaces by Osilike and Isiogugu [13]. It is shown in [13] that the class of asymptotically demicontractive map is a proper subclass of the class of asymptotically φ−demicontractive