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Convexity of values of vector integrals, theorems on measurable choice and variational problems

V. I. Arkin, V. L. Levin


Abstract: We give an account of applications of measurable many-valued mappings and theorems on convexity of finite-dimensional vector integrals to several variational problems. Theorems on convexity are carried over to vector integrals with values in function spaces, and with the help of these we obtain a aximum principle as a ecessary and sufficient extremum condition and an existence theorem for a on-linear variational problem with operator constraints of integral equality type, similar to Monge's problem on mass displacement.

UDC: 517.4+519.3+519.9

MSC: 28B05, 46G10, 26B25, 30C80, 47J20

Received: 17.01.1972

 English version: Russian Mathematical Surveys, 1972, 27:3, 21–85

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