We show that the solution set of a strictly convex combination of equilibrium problems is not necessarily contained in the corresponding intersection of solution sets of equilibrium problems even if the bifunctions defining the equilibrium problems are continuous and monotone. As a consequence, we show that some results given in some recent papers are not always true. Therefore different numerical methods for computing common solutions of families of equilibrium problems proposed in the literature may not converge under the monotonicity assumption. Finally, we prove that if the bifunctions are also parapseudomonotone, then the solution set of any strictly convex combination of a family of equilibrium problems is equivalent to the solution set of the intersection of the same family of equilibrium problems.

中文翻译：

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关于平衡问题组合的注释

我们表明，即使定义平衡问题的双功能是连续和单调的，平衡问题的严格凸组合的解集也不一定包含在平衡问题的解集的对应交集中。结果，我们证明了一些最新论文给出的结果并不总是正确的。因此，在单调性假设下，文献中提出的用于计算平衡问题族的公共解的不同数值方法可能不会收敛。最后，我们证明，如果双功能也是对伪单调，那么一个平衡问题系列的任何严格凸组合的解集都等于同一平衡问题系列的交集的解集。