Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications

In 1916, Tricomi 1 introduced originally the concept of quasi-nonexpansive for real functions. Subsequently, this concept has studied for mappings in Banach and metric spaces see, e.g., 2–7 . Recently, some generalized types of quasi-nonexpansive mappings in metric and Banach spaces have appeared. For example, see Ahmed and Zeyada 8 , Qihou 9–11 and others. Unless stated to the contrary, we assume that X, d is a metric space. Let T : D ⊆ X → X be any mapping and let F T be the set of all fixed points of T . If F : X → R where R is the set of all real numbers and if c ∈ R, set Lc : {x ∈ X : F x ≤ c}. We use the symbol μ to denote the usual Kuratowski measure of noncompactness. For some properties of μ, see Zeidler 12, pages 493–495 . For a given x0 ∈ D, the Picard iteration xn is determined by: I xn T xn−1 T x0 , n ∈ N