Comments on the paper “Weak almost-convergence theorem without opial’s condition” [B.K. Sharma et al., J. Math. Anal. Appl. 254 (2001) 636–644]

Let E be a Banach space with dual E∗. Recall that the duality map J :E→E∗ is defined by J(x) := x∗ ∈ E∗: x,x∗ = x 2 = x∗ 2 , x∈ E. It is known that E is uniformly smooth [1] if and only if J is single-valued and uniformly continuous on bounded sets in E. The normal structure coefficient [2] of E is defined as the number N(E) := inf d(C)/r(C): C convex bounded subset of E with d(C)>0 , where d(C) := sup x − y : x,y ∈ C and r(C) := inf x∈C sup y∈C x − y are the diameter of C and, respectively, the Chebyshev radius of C relative to itself. A Banach space E is said to have uniformly normal structure [2] if N(E)>1. It is known that N(H) =√2, where H is a Hilbert space.Given real numbers λ,α,β > 0, a Banach space E is said to satisfy property (U,λ,α,β) if x +y α + λ x − y α − 2β x α + y α 0, x,y∈ E. It is known that a Hilbert space satisfies (U, 1, 2, 1) and an lp (or Lp) satisfies (U,p −1, 2, 1) for 2 p ∞. Let D be a nonempty convex subset of a Banach space E and T :D→D be a mapping.