Research Article A Note on Asymptotic Contractions

for all x, y ∈ X and all natural numbers n, where φn : [0,∞)→ [0,∞) and limn→∞φn = φ, uniformly on any bounded interval [0,b]. Suppose that φ is upper semicontinuous and that φ(t) < t for all t > 0. Furthermore, suppose that there exists a positive integer n∗ such that φn∗ is upper semicontinuous and φn∗(0)= 0. If there exists x0 ∈ X which has a bounded orbit O(x0)= {x0,Tx0,T2x0, . . .}, then T has a unique fixed point x∗ ∈ X and limn→∞Tx = x∗ for all x ∈ X .