Abstract :
Let E be a real normed linear space, K be a nonempty subset of E and T :K →E be a uniformly
continuous generalized Φ-hemi-contractive mapping, i.e., F(T ) := {x ∈ K: T x = x} = Φ, and there
exist x∗ ∈ F(T ) and a strictly increasing function Φ : [0,∞) → [0,∞), Φ(0) = 0 such that for all
x ∈ K, there exists j (x − x∗) ∈ J(x − x∗) such that
T x − x∗, j (x −x∗) x − x∗ 2 − Φ x −x∗ .
(a) If y∗ ∈ K is a fixed point of T , then y∗ = x∗ and so T has at most one fixed point in K.
(b) Suppose there exists x0 ∈ K, such that the sequence {xn} defined by
xn+1 = anxn +bnT xn +cnun, ∀n 0,
is contained in K, where {an}, {bn} and {cn} are real sequences satisfying the following conditions:
(i) an + bn + cn = 1;
(ii) ∞n=0(bn +cn)=∞;
(iii) ∞n=0(bn +cn)2 <∞;
(iv) ∞n=0 cn <∞; and {un} is a bounded sequence in E.
