By E. R. de Arantes e Oliveira (auth.)
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Ical point of a given functional on a given subset if the proper local minimizer of a certain functional of family ~ on a certain 3. The Convergence Theory for Variational Methods 64 isoconstrained subset. An analogous system of variational theorems can be establish.. ed f or space X I • This means that the approximation of a given point ~ of X which is a critical point for a given functional on a given subset 5 of X can be determined by determining the critical point of the same functional on the T -image of 5 in Xl.
Ed f or space X I • This means that the approximation of a given point ~ of X which is a critical point for a given functional on a given subset 5 of X can be determined by determining the critical point of the same functional on the T -image of 5 in Xl. The convergence of sequences of such approximations can be examined only if such critical points are also minimizers of a certain functional on isoconstrained subsets, because then all the theory can be applied. No conclusions can be drawn however if no minimum principles are known.
2 A second operator, ed, with domain X and range X I , A , can thus be considerwhich makes the intersec- tion of each isoconstrained and each isominimizing subset of X correspond to the intersection of the corresponding iso- constrained and isominimizing subsets of (0) As the zero element belongs both to x, . X and XI , T (0) = 0 . The isoconstrained subset of Xl which contains the zero element must thus be contained by the corresponding isoconstrained subset of X , i f T is conforming. Approximation Operator A 43 A is assumed also bounded and continuous, and also such that A(e) - e This means that on each element of ed and an isominiinizing subset of X X I cross an isoconstrainand the corresponding isoconstrained and isominimizing subsets of X~ (fig.