The Continuity of Linear and Sublinear Correspondences Defined on Cones (original) (raw)
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Journal of Applied Analysis, 2019
Let K be a closed convex cone in a real Banach space, H : K → cc ( K ) {H\colon K\to\operatorname{cc}(K)} a continuous sublinear correspondence with nonempty, convex and compact values in K, and let f : ℝ → ℝ {f\colon\mathbb{R}\to\mathbb{R}} be defined by f ( t ) = ∑ n = 0 ∞ a n t n {f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}} , where t ∈ ℝ {t\in\mathbb{R}} , a n ≥ 0 {a_{n}\geq 0} , n ∈ ℕ {n\in\mathbb{N}} . We show that the correspondence F t ( x ) : = ∑ n = 0 ∞ a n t n H n ( x ) , ( x ∈ K ) {F^{t}(x)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}a_{n}t^{n}H^{n}(x),(x\in K)} is continuous and sublinear for every t ≥ 0 {t\geq 0} and F t ∘ F s ( x ) ⊆ ∑ n = 0 ∞ c n H n ( x ) {F^{t}\circ F^{s}(x)\subseteq\sum_{n=0}^{\infty}c_{n}H^{n}(x)} , x ∈ K {x\in K} , where c n = ∑ k = 0 n a k a n - k t k s n - k {c_{n}=\sum_{k=0}^{n}a_{k}a_{n-k}t^{k}s^{n-k}} , t , s ≥ 0 {t,s\geq 0} .
A strong open mapping theorem for surjections from cones onto Banach spaces
Advances in Mathematics, 2014
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.
Right inverses of surjections from cones onto Banach spaces
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; an improved version of the usual Open Mapping Theorem is then a special case. As another consequence, a stronger version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.
Cornell University - arXiv, 2010
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A NOTE ON VARIOUS TYPES OF CONES AND FIXED POINT RESULTS IN CONE METRIC SPACES
Various types of cones in topological vector spaces are discussed. In particular, the usage of (non)-solid and (non)-normal cones in fixed point results is presented. A recent result about normable cones is shown to be wrong. Finally, a Geraghty-type fixed point result in spaces with cones which are either solid or normal is obtained.