This work demonstrates necessary efficiency conditions in variational models, suggesting broader applications of new functionals.
In this paper, necessary and sufficient efficiency conditions in new multi-cost variational models are formulated and proved. To this end, we introduce a new notion of (ϑ0,ϑ1)−(σ0,σ1)−type−I functionals determined by multiple integrals. To better emphasize the significance of the suggested (ϑ0,ϑ1)−(σ0,σ1)−type−I functionals and how they add to previous studies, we mention that the (ϑ0,ϑ1)−(σ0,σ1)−type−I and generalized (ϑ0,ϑ1)−(σ0,σ1)−type−I−typeI assumptions associated with the involved multiple integral functionals cover broader and more general classes of problems, where the convexity of the functionals is not fulfilled or the functionals considered are not of simple integral type. In addition, innovative proofs are provided for the main results.
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Treanţă et al. (2025) studied this question.
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