Let $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$ . In addition, assume that the restriction of u on the fixed points of $\Phi $ has a unique maximizer $\eta ^\star $ . Then, we consider feasible paths $(x_0,x_1,\ldots )$ with values in X such that $x_{n+1} \in \Phi (x_n)$ , for all $n\ge 0$ . Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots )$ which maximizes the smallest $\mathcal {I}$ -cluster point of the sequence $(u(x_0),u(x_1),\ldots )$ is necessarily $\mathcal {I}$ -convergent to $\eta ^\star $ .

We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.

令 $\Phi $ 是从一个范数向量空间X到它自身的对应，令 $u: X\to \mathbf {R}$ 是一个函数，并且令 $\mathcal {I}$ 是 $\mathbf上的一个理想{N}$ 。此外，假设u对 $\Phi $ 不动点的限制具有唯一的最大化器 $\eta ^\star $ 。然后，我们考虑可行路径 $(x_0,x_1,\ldots )$ ，其值在X中使得 $x_{n+1} \in \Phi (x_n)$ ，对于所有 $n\ge 0$ 。在某些附加条件下，我们证明了以下收费结果：每条可行路径 $(x_0,x_1,\ldots )$ 最大化序列 $(u(x_0),u(x_1),\ldots )$ 的最小 $\mathcal {I}$ -cluster point必然是 $\mathcal {I }$ -收敛到 $\eta ^\star $ 。

我们提供的例子一方面证明了我们结果的假设是正确的，另一方面证明了我们正在包括以前没有在相关文献中考虑的新案例。