Orientor field equations in Banach spaces

• Published in: Journal of optimization theory and applications 1976-05, Vol.19 (1), p.141-164
• Main Authors: Kaiser, P. J., Suryanarayana, M. B.
• Format: Article
• Language: English
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/BF00934058

The paper presents lower closure theorems (with closure theorems as special cases) in Banach spaces, with reduced requirements on convexity and property (Q), the upper semicontinuity property for set-valued functions proposed by Cesari, of the sets in the orientor fields. More precisely, the range space B is considered as a product space B = B1 x B2, as it is assumed that only the sections of the relevant subsets Q(t,y) of B parallel to B1 are convex, and these sets satisfy property (Q) with respect to (y,b2), where b2 is a fixed point in B2. This property is denoted as property (Q-asterisk), and reduces to the intermediate property (Q-rho) for Euclidean spaces. Lower closure theorems are also given for Euclidean spaces with the intermediate property (Q-asterisk) required, not on the original sets of the orientor field, but on their intersections with spheres centered at the origin. It is then proved that the property (Q-asterisk) implies that the complementary sections automatically satisfy Kuratowski's upper semicontinuity property with respect to (y,b1), where b1 is a fixed point in B1.