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General mixed vector F-implicit complementarity problems in Banach spaces

Wu K.-Q., Huang N.-J.

Journal of Applied Analysis

2008

Vol. 14, nr 1

73-88

In this paper, we introduce a new class of general mixed vector F -implicit complementarity problems and general mixed vector F -implicit variational inequality problems, and study the equivalence between of them under certain assumptions in Banach spaces. We alsoderive some new existence theorems of solutions for the general lllixed vector F -implicit complementarity problems and the general mixed vector F -implicit variational inequality problems by using the FKKM theorem under some suitable assumptions without monotonicity. Moreover , we establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution mapping of the general mixed vector F -implicit variational inequality problems.

On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems

Pickenhain S., Lykina V., Wagner M.

Control and Cybernetics

2008

Vol. 37, no 2

451-468

This paper deals with infinite horizon optimal control problems, which are formulated in weighted Sobolev spaces ... [wzór] and weighted Lp-spaces ... [wzór]. We ask for the consequences of the interpretation of the integral within the objective as a Lebesgue or an improper Riemann integral. In order to justify the use of both types of integrals, various applications of infinite horizon problems are presented. We provide examples showing that lower semicontinuity may fail for objectives involving Lebesgue as well as improper Riemann integrals. Further we prove a lower semicontinuity theorem for an objective with Lebesgue integral under more restrictive growth conditions on the integrand.

On a lower semicontinuity of a certain multifunction

Puchała P.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2003

Vol. 2, nr 1

153--158

Let T be a metric space, X - n-dimensional Euclidean space and P : T → 2x - continuous multifunction with compact convex values. We will show that multifunction T ϶ t ϵ ext P(t) ϵ 2x is lower semicontinuous.