We prove that a Banach space X has normal structure provided it contains a finite codimensional subspace Y such that all spreading models for Y have normal structure. We show that a Banach space X is strictly convex if the set of fixed points of any nonexpansive map defined in any convex subset C C X is convex and give a sufficient condition for uniform convexity of a space in terms of nonexpan...

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [17]. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of multivalued maps has applications in control theory, convex optimization, differential inclusions, and economics (see, e.g., [3, 8, 16, 22]). The theory of multivalu...

this paper is concerned with the best proximity pair problem in hilbert spaces. given two subsets $a$ and $b$ of a hilbert space $h$ and the set-valued maps $f:a o 2^ b$ and $g:a_0 o 2^{a_0}$, where $a_0={xin a: |x-y|=d(a,b)~~~mbox{for some}~~~ yin b}$, best proximity pair theorems provide sufficient conditions that ensure the existence of an $x_0in a$ such that $$d(g(x_0),f(x_0))=d(a,b).$$

Let E be a real q-uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively. Let {λn} be a real sequence in 0, 1 that satisfies the following condition: C1: limλn 0 and ∑ λn ∞. For δ ∈ 0, qη/dqk 1/ q−1 and σ ∈ 0, 1 , define a sequence {xn} iteratively in E by x0 ∈ E, xn 1 Tn...

Suppose C is a nonempty bounded closed convex retract of a real uniformly convex Banach space X with uniformly Gâteaux differentiable norm and P as a nonexpansive retraction of X onto C. Let T : C −→ X be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂ [1,∞), lim kn = 1, F (T ) = {x ∈ C : Tx = x}, and let u ∈ C. In this paper we study the convergence of the sequences {xn} and...

Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J {T t : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : 0,∞ → 0,∞ . Let F : F J ∩t≥0F T t / ∅ and f : K → K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ λ > 1, let G ...

Let E be a real q−uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η−strongly accretive map which is also κ− Lipschitzian, respectively. Let {λn} be a real sequence in [0, 1] satisfying some appropriate conditions. For δ ∈ (0, ( qη dqκ )q−1), define a sequence {xn} iteratively in E by x0 ∈ E, xn+1 = T n+1xn = Txn − δλn+1G(Txn), n ≥ ...

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets. In this note we generalize a famous result by Browder [3], Göhde [6] and Kirk [8], recently extended by Luc in [14], by using the ...

Let K be a closed convex subset of a Hilbert space H and T : K ⊸ K a nonexpansive multivalued map with a unique fixed point z such that {z} = T (z). It is shown that we can construct a sequence of approximating fixed points sets converging in the sense of Mosco to z.