Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

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.