Sentence examples for mapping with convex from inspiring English sources

set-valued mapping with convex and compact values.

Let E be a Banach space and y ∈ E. Let f : E → R ∪ be a proper, convex and lower semi-continuous mapping with convex domain D ( f ).

Let E be a Banach space and let f : E → R ∪ be a proper, convex and lower semicontinuous mapping with convex domain D ( f ).

The condition that is a mapping with convex values is crucial to get the desired conclusion in the previous theorem, Theorem DKP and all the results in [17].

In 1968, Browder [1] proved that every set-valued mapping with convex values and open fibers from a compact Hausdorff topological vector space to a convex space has a continuous selection.

The operator (mathbb {T}) in (2.1) is an upper semicontinuous mapping with convex and compact values (see [3, 4]), and therefore Kakutani's fixed point theorem guarantees that (mathbb {T}) has a fixed point in K.

Let T : K → C B K K ) be a multi-valued mapping with convex-values and satisfying the condition (C).

Therefore T is an SKC-type multi-valued mapping with convex-values and satisfies all conditions in Theorem 2.1.

Proof By the assumption that T : K → C B ( K ) is a multi-valued mapping with convex-values, hence Tp is a nonempty closed and convex subset of K.

(4.1) It is easy to prove that (also see [5, 21]) (T: K to P K)) is a C-type multi-valued mapping with convex-values and (0 in K) is the unique fixed point of T in K and (T 0) = {0}).

Lemma 2.5 Let K be a nonempty subset of a uniformly convex Banach space X and T : K → C B K K ) be a multi-valued mapping with convex-valued and satisfying the condition (C), then H ( T x, T y ) ≤ 2 d ( x, T x ) + ∥ x − y ∥, ∀ x, y ∈ K. Proof Let x ∈ K, since Tx is a nonempty closed and convex subset of K.