Abstract :
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.
