defined over Sobolev spaces is connected to the convexity of the potential w. In the scalar case, that is for functions u with domain or range in R, the functional I is weakly W 1,p lower semi-continuous (weakly * W ) if and only if w is convex, provided it is continuous and satisfies some growth conditions. The notion which replaces convexity in the vector case is quasi-convexity (introduced b...

In this article, we establish sufficient conditions for the solution sets of parametric generalized quasi-variational relation problems with the stability properties such as the upper semicontinuity, lower semi-continuity, the Hausdorff lower semicontinuity, continuity, Hausdorff continuity, and closedness. Our results improve recent existing ones in the literature. Mathematics Subject Classifi...

Sufficient conditions ensuring weak lower semi-continuity of the optimal value function are presented. To this end, refined inner semi-continuity properties of set-valued maps are introduced which meet the needs of the weak topology in Banach spaces. The results are applied to prove the existence of solutions in various worst-case robust optimal control problems governed by semilinear elliptic ...

where x stands for the three coordinate functions, ^(w , u)} ff (#\ u), x{u, u), and X stands for their three Jacobians. Various theorems of the following general type have been proved by McShane, Caccioppoli, Cimmino, and Radó: If (i) the triples of functions xn{u , u), i = 1, 2; n = 0, 1, 2, • • • , satisfy certain conditions; (ii) the surfaces defined by x = xn(u , u), i = l,2;n = l,2, • • •...

Blake A. Allison⇤, Adib Bagh†, and Jason J. Lepore‡. October 5, 2016 Abstract. We provide sufficient conditions for a game with discontinuous payoffs to be weakly reciprocally upper semi-continuous in mixed strategies. These conditions are imposed on the individual payoffs and not on their sum, and they can be readily verified in a large class of games even when the sum of payoffs in such games...

This note provides conditions under which Zadeh extension of a map between L -topological spaces preserves semicontinuity and continuity of functions with values in the L-unit interval. Invariance of complete L-regularity under continuous open–closed surjections is established among others. © 2012 Elsevier B.V. All rights reserved.

In this paper, the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector equilibrium problem involving set-valued mappings are established by using a new proof method which is different from the ones used in the literature.