HYBRID-TYPE ITERATION SCHEME FOR APPROXIMATING FIXED POINTS OF LIPSCHITZ α-HEMICONTRACTIVE MAPPINGS

We propose a two-step iteration Scheme of hybrid-type for α-hemicontractive mappings and establish some weak and strong convergence theorems of the scheme to the fixed points of the mappings in real Hilbert spaces. Our results extend and generalize the results of Wang [18], compliments the results of Osilike and Onah [17] and other numerous results currently existing in literature.

In 2007, Wang [18] obtained weak and strong convergence of (1.12) to the fixed point of as follows: Let be a Hilbert space, ∶ → a nonexpansive mapping. If for any given 0 ∈ , { } is defined by where { } and { } ⊂ [0,1) satisfy the following conditions (I) ≤ ≤ for some , ∈ (0,1) Motivated by the work of Wang [18], Igbokwe and Jim [9] and the results of Osilike and Onah [17], it is our purpose in this paper to extend the results of Wang [18] and other related results currently in literature from nonexpansive mapping to the more general -hemicontractive 5 I. K. AGWU AND D. I. IGBOKWE mapping. Our results are much more general and also more applicable than the results of Wang [18] because the strong monotonicity condition imposed on is not required in our results. .

PRELIMINARIES.
For the sake of convenience, we restate the following concepts and results.
Let be a real Banach space. A mapping with domain ( ) in E is said to be demiclosed at 0, if for any sequence ⊂ , ⇀ ∈ ( ) and ‖ − ‖ → 0, then = . A Furthermore, for any ∈ and z ∈ K, z = if and only if 〈 − , − 〉 ≥ 0, ∀ ∈ .
. ln a real Hilbert space , the following inequalities hold: , then lim →∞ exists.
A Banach space is reflexive if and only if every (normed) bounded sequence in has a subsequence which converges weakly to an element of .
Theorem . . Let be a real Hilbert space and be a nonempty closed convex subset of .