Example 2.3 Let $\mathcal {I}$ be an ideal on $\mathbf {N}$ such that $2\mathbf {N}\subseteq \mathcal {I}$ . Note that such ideals exist, e.g., the family of subsets of $\mathbf {N}$ containing finitely many odd integers, or the maximal ideals extending $2\mathbf {N}$ . Now, let $X=\mathbf {R}$ , and define and $u(x)=x^3$ , for each $x \in \mathbf {R}$ . Moreover, set $\mathscr {C}:=\{\boldsymbol {y} \in \mathscr {F}_{\mathrm {K}}: y_0=1\}$ . Then, the continuous correspondence $\Phi $ has a unique fixed point, i.e., $\mathrm {Fix}(\Phi )=\{0\}$ . It is easily seen that the system $\langle X, \Phi , u, \mathcal {I}, \mathscr {C}\rangle $ satisfies conditions (A1)–(A4). Moreover, also condition (A5) holds: indeed, notice that $F=\{x \in \mathbf {R}: u(x) \ge u(0)\}=[0,\infty )$ . Then, setting $T(r)=r$ , for all $r \in \mathbf {R}$ , we obtain that $Ty<Tx$ for all $(x,y) \neq (0,0)$ with $x \in F$ and $y \in \Phi (x)$ , i.e., for all $x>0$ and .

At this point, let $\boldsymbol {x}=(x_0,x_1,\ldots )\in \mathscr {C}$ be the sequence defined by $x_n=(-1)^{n}$ , for all $n \in \mathbf {N}$ . Then, $\boldsymbol {x} \in \mathscr {C}$ and $u(\boldsymbol {x})=\boldsymbol {x}$ , so that $\mathcal {I}\text {-}\lim u(\boldsymbol {x})=1$ . Because $|u(y_n)|\le 1$ , for all $\boldsymbol {y} \in \mathscr {C}$ and $n \in \mathbf {N}$ , it follows that $\boldsymbol {x}$ is $\mathcal {I}$ -optimal. In addition, the sequence $\boldsymbol {y}$ defined by $y_n=2^{-n}$ , for all $n \in \mathbf {N}$ , belongs to $\mathscr {C}$ , and it is convergent (in the classical sense) to $0$ . Hence, all hypotheses of Corollary 2.2 hold. However, the sequence $\boldsymbol {x}$ is clearly not $\mathcal {I}$ -convergent to $0$ .

Example 2.5 Set $X=\mathbf {R}$ , $\mathcal {I}=\mathcal {Z}$ , $\mathscr {C}=\mathscr {F}_{\mathrm {K}}$ , $\Phi (x):=[-2x,-\frac {x}{2}]$ , and $u(x)=x$ , for all $x \in \mathbf {R}$ . It is not difficult to see that conditions (A1)–(A6) hold, and $\mathrm {Fix}(\Phi )=\{0\}$ (cf. Example 2.3).

At this point, let $\boldsymbol {x}=(x_0,x_1,\ldots )$ be the sequence for which $\text{for all }n x_n=(-1)^nz_n$ , where $\boldsymbol {z}=(z_0,z_1,\ldots )$ is defined as it follows:

Here, for each $k\ge 1$ , the block $B_k$ has $2k-1+k!$ terms and its middle part is made by $k!$ consecutive terms equal to

.

Let us show that $\boldsymbol {x}$ is $\mathcal {Z}$ -optimal: first of all, using the fact that

$$ \begin{align*}\textstyle \sum_{k\le n-1}|B_k| \le \sum_{k\le n-1}3\cdot (k-1)!=o(n!) \text{ as }n\to \infty, \end{align*} $$

thus $\mathcal {Z}\text {-}\lim \boldsymbol {x}=0$ (cf. also [Reference Leonetti and Tringali17, Lemma 1]). In particular, $\mathcal {Z}\text {-}\liminf \boldsymbol {x}=\mathcal {Z}\text {-}\liminf u(\boldsymbol {x})=0$ . Let us suppose, for the sake of contradiction, that there exists a sequence $\boldsymbol {y}\in \mathscr {F}_{\mathrm {K}}$ such that $\kappa :=\mathcal {I}\text {-}\liminf \boldsymbol {y}>0$ . Hence, $\kappa $ is a statistical cluster point of $\boldsymbol {y}$ (see Corollary 3.3 below). It follows that

However, by construction, we have that $y_ny_{n+1}<0$ whenever $y_n\neq 0$ . Therefore,

, for all $n \in A$ . Considering that $\mathcal {Z}$ is a translation invariant ideal, it follows that $\boldsymbol {y}$ is a bounded sequence such that

. We conclude by Lemma 3.1 (ii) below that the sequence $\boldsymbol {y}$ has a negative statistical cluster point, contradicting the standing hypothesis that $\mathcal {I}\text {-}\liminf \boldsymbol {y}>0$ .

Hence, $\boldsymbol {x}$ is $\mathcal {Z}$ -optimal. However, because the length of block $B_k$ is odd for each $k\ge 2$ , it follows that $\liminf \boldsymbol {x}=-1$ and $\limsup \boldsymbol {x}=1$ .

Example 2.6 Set $X=\mathbf {R}$ , let $\mathcal {I}$ be a translation invariant ideal on $\mathbf {N}$ , and let $u: \mathbf {R}\to \mathbf {R}$ be a strictly increasing continuous function. In addition, let $\{\phi _1,\ldots ,\phi _k\}$ be an iterated function system on $\mathbf {R}$ , that is, a finite number of contractions on $\mathbf {R}$ , and define the correspondence $\Phi : \mathbf {R}\rightrightarrows \mathbf {R}$ by

$$ \begin{align*}\forall x \in \mathbf{R}, \quad \Phi(x):=\{\phi_1(x),\ldots,\phi_k(x)\}. \end{align*} $$

Accordingly, let $\mathscr {C}$ be an arbitrary subset of bounded feasible sequences such that

(2.4) $$ \begin{align} \forall i=1,\ldots,k, \exists x \in \mathbf{R}, \quad (x, \phi_i(x), \phi_i^2(x),\ldots) \in \mathscr{C}. \end{align} $$

(It is remarkable that there exists a unique nonempty compact set $S\subseteq \mathbf {R}$ , called attractor, such that $\lim _n H^n(\{x_0\})=S$ , for all $x_0 \in \mathbf {R}$ , in the Hausdorff metric, where H stands for Hutchinson operator defined by $H(A):= \bigcup _{x \in A}\Phi (x)$ , for all $A\subseteq \mathbf {R}$ ; see [Reference Hutchinson13].)

For each $i=1,\ldots ,k$ , let $\eta _i$ be the fixed point of $\phi _i$ , hence the restriction of u on $\mathrm {Fix}(\Phi )=\{\eta _1,\ldots ,\eta _k\}$ is maximized at the unique point $\eta ^\star =\max \{\eta _1,\ldots ,\eta _k\}$ . In particular, conditions (A1)–(A4) hold. At this point, note that $F= [\eta ^\star , \infty )$ and

$$ \begin{align*} \begin{aligned} \forall i=1,\ldots,k, \forall \eta>\eta_i, \quad \phi_i(\eta)-\eta_i \le |\phi_i(\eta)-\phi_i(\eta_i)|< \eta-\eta_i, \end{aligned} \end{align*} $$

hence $\phi _i(\eta )<\eta $ whenever $\eta>\eta _i$ . In particular, $\max \Phi (\eta ^\star )=\eta ^\star $ and $\max \Phi (\eta )<\eta $ , for all $\eta>\eta ^\star $ . Therefore, condition (A5) holds letting T be the identity map. Finally, let j be an index such that $\eta _{j}=\eta ^\star $ . Then, it follows by (2.4) and the Banach contraction theorem that there exists $x \in \mathbf {R}$ such that $(x, \phi _j(x), \phi _j^2(x), \ldots ) \in \mathscr {C}$ and $\lim _n \phi ^n_j(x)=\eta ^\star $ .

We conclude by Corollary 2.2 that, if a sequence $\boldsymbol {x}$ in the constraint set $\mathscr {C}$ is $\mathcal {I}$ -optimal, then $\mathcal {I}\text {-}\lim \boldsymbol {x}=\eta ^\star $ . In particular, in the special case $\mathscr {C}=\mathscr {F}_{\mathrm {K}}$ , $u(x)=x$ , and $\mathcal {I}=\mathrm {Fin}$ , we obtain that: if a bounded feasible sequence $\boldsymbol {x}$ maximizes its smallest accumulation point, then it is convergent to the maximal fixed point $\eta ^\star $ (this could be obtained, of course, also by a direct method.)

Example 5.1 Let X be the Hilbert space $\ell _2$ of square summable real sequences, i.e., sequences $\boldsymbol {x}=(x_0,x_1,\ldots )$ such that

$$ \begin{align*}\textstyle \|\boldsymbol{x}\|:=\sqrt{\sum_{i\ge 0}x_i^2}<\infty. \end{align*} $$

Fix a sequence $\boldsymbol {x}^\star \in \ell _2$ , and define $\mathscr {C}=\{(\boldsymbol {x}^{(n)}) \in \mathscr {F}_{\mathrm {K}}: \boldsymbol {x}^{(0)}=\boldsymbol {x}^\star \}$ . Let also $\mathcal {I}$ be an arbitrary translation invariant ideal, and, for each $\boldsymbol {x} \in \ell _2$ , set $u(\boldsymbol {x})=x_0$ and

(5.1)

where . First of all, let us show that $\Phi $ is well defined. To do this, fix $\boldsymbol {x} \in \ell _2$ and let us prove that $\Phi (\boldsymbol {x})\subseteq \ell _2$ . If $\boldsymbol {y} \in \Phi (\boldsymbol {x})$ is equal to $\frac {1}{2}\boldsymbol {x}$ , then it clearly belongs to $\ell _2$ . Otherwise, because $\ell _2$ is a vector space, it is sufficient to show that $\boldsymbol {z}=\boldsymbol {y}-\boldsymbol {x}=(-x_0-\sum _{i\ge 1}x_i^2,z_1,z_2,\ldots ) \in \ell _2$ , where , for all $i\ge 1$ . Therefore,

$$ \begin{align*} \begin{aligned} \textstyle \sum_{i\ge 0}z_i^2\le z_0^2+\sum_{i\ge 1}\left(|x_i|+\frac{1}{i}\right)^2 &\textstyle \le z_0^2+\sum_{i\ge 0}x_i^2+\sum_{i\ge 1}\frac{1}{i^2}+2\sum_{i\ge 1}\frac{|x_i|}{i} \\ &\textstyle \le z_0^2+\sum_{i\ge 0}x_i^2+\sum_{i\ge 1}\frac{1}{i^2}+2\sqrt{\sum_{i\ge 1}x_i^2 \cdot \sum_{i\ge 1}\frac{1}{i^2}}<\infty, \end{aligned} \end{align*} $$

where the last $\le $ follows by the Cauchy–Schwarz inequality. In addition, $\Phi (\boldsymbol {x})$ is compact. To this aim, because $\{\frac {1}{2}\boldsymbol {x}\}$ is compact, it is sufficient to show that a translation of the first set in the definition (5.1) of $\Phi (\boldsymbol {x})$ is compact. Let us define

(5.2)

Let $\boldsymbol {a}$ be the sequence defined by ${a}_0:=-x_0-\sum _{i\ge 1}x_i^2$ and ${a}_i:=|x_i|+\frac {1}{i}$ . Note that $\boldsymbol {a} \in \ell _2$ and that $\varphi (\boldsymbol {x})$ is a closed subset of $\{\boldsymbol {z} \in \ell _2: |z_i| \le {a}_i \text { for all }i\ge 0\}$ . However, the latter set is compact thanks to [Reference Dunford and Schwartz7, p. 453], hence $\varphi (\boldsymbol {x})$ is compact too. To sum up, $\Phi (\boldsymbol {x})$ is compact subset of $\ell _2$ which is nonempty (because it contains $\frac {1}{2}\boldsymbol {x}$ ).

At this point, let us show that $\Phi $ is continuous. Reasoning as above, it is sufficient to show that the correspondence $\varphi $ defined in (5.2) is continuous at $\boldsymbol {x}$ . Assume that $\varphi (\boldsymbol {x})\neq \emptyset $ , i.e., , for all $i\ge 1$ , otherwise the claim is trivial.

First, let us show that $\varphi $ is upper hemicontinuous. Fix $\varepsilon>0$ and define the open set $U_\varepsilon :=\{\boldsymbol {z} \in \ell _2: \exists \boldsymbol {y} \in \varphi (\boldsymbol {x}), \|\boldsymbol {z}-\boldsymbol {y}\|<\varepsilon \}$ . We need to find a constant $\delta>0$ such that, for each $\boldsymbol {x}^\prime \in \ell _2$ , if $\|\boldsymbol {x}-\boldsymbol {x}^\prime \|<\delta $ , then $\varphi (\boldsymbol {x}^\prime ) \subseteq U_\varepsilon $ . Hence, fix also $\boldsymbol {x}^\prime \in \ell _2$ such that $\|\boldsymbol {x}-\boldsymbol {x}^\prime \|<\delta $ for a suitable $\delta>0$ that will be chosen later and pick $\boldsymbol {y}^\prime \in \varphi (\boldsymbol {x}^\prime )$ . In particular, $|x_i-x_i^\prime |<\delta $ , for all $i \ge 0$ . Similarly, we can assume without loss of generality that , for all $i\ge 1$ , otherwise $\varphi (\boldsymbol {x}^\prime )=\emptyset $ . Then,

(5.3) $$ \begin{align} \begin{aligned} \textstyle \forall \boldsymbol{y} \in \varphi(\boldsymbol{x}), \quad |y_0-y_0^\prime| &\textstyle \le |x_0-x_0^\prime|+|\sum_{i\ge 1}x_i^2-\sum_{i\ge 1}(x_i^\prime)^2\,|\\ &\textstyle \le |x_0-x_0^\prime|+|x_0^2-(x_0^\prime)^2|+|\|\boldsymbol{x}\|^2-\|\boldsymbol{x}^\prime\|^2\,| \\ &\textstyle \le \delta+\delta |x_0+x_0^\prime|+\delta(\|\boldsymbol{x}\|+\|\boldsymbol{x}^\prime\|) \\ &\textstyle \le \delta\left(1+2 (\|\boldsymbol{x}\|+\|\boldsymbol{x}^\prime\|)\right) \\ &\textstyle \le \delta\left(1+4\|\boldsymbol{x}\|+2\delta\right). \end{aligned} \end{align} $$

Now, recall that, for each integer $i\ge 1$ , we have and . In particular, there exists $\boldsymbol {y} \in \varphi (\boldsymbol {x})$ such that $|y_i-y_i^\prime |\le |x_i-x_i^\prime |$ , for all $i\ge 1$ . It follows that

(5.4) $$ \begin{align} \textstyle \sum_{i\ge 1}(y_i-y_i^\prime)^2\le \sum_{i\ge 1}(x_i-x_i^\prime)^2 \le \|\boldsymbol{x}-\boldsymbol{x}^\prime\|^2 \le \delta^2. \end{align} $$

Putting together the above estimates, we obtain that, for each given $\boldsymbol {y}^\prime \in \varphi (\boldsymbol {x}^\prime )$ , there exists $\boldsymbol {y} \in \varphi (\boldsymbol {x})$ such that

(5.5) $$ \begin{align} \textstyle \|\boldsymbol{y}-\boldsymbol{y}^\prime\|=\sqrt{|y_0-y_0^\prime|^2+\sum_{i\ge 1}(y_i-y_i^\prime)^2}\le \delta \sqrt{\left(1+4\|\boldsymbol{x}\|+2\delta\right)^2+1}<\varepsilon, \end{align} $$

where the last inequality holds if $\delta $ is sufficiently small.

Second, let us show that $\varphi $ is lower hemicontinuous. To this aim, fix an arbitrary $\boldsymbol {y} \in \varphi (\boldsymbol {x})$ and some $\varepsilon>0$ . We claim that there exists $\delta>0$ such that if $\|\boldsymbol {x}-\boldsymbol {x}^\prime \|<\delta $ , then $\|\boldsymbol {y}-\boldsymbol {y}^\prime \|<\varepsilon $ , for some $\boldsymbol {y}^\prime \in \varphi (\boldsymbol {x}^\prime )$ . Note that estimates (5.3) and (5.4) hold simmetrically also in this case, with the conclusion that we have exactly the same upper bound computed in (5.5) for $\|\boldsymbol {y}-\boldsymbol {y}^\prime \|$ .

Therefore, $\Phi $ is a nonempty compact-valued continuous correspondence on $\ell _2$ , $\mathcal {I}$ is translation invariant, and u is clearly continuous. Hence, conditions (A1)–(A3) hold.

The set of fixed points of $\Phi $ is neither convex nor compact, and it is equal to

$$ \begin{align*}\textstyle \mathrm{Fix}(\Phi)=\{\boldsymbol{x} \in \ell_2: x_0=-\sum_{i\ge 1}x_i^2 \text{ and }x_j\le 0 \text{ for all }j\ge 1\}. \end{align*} $$

It follows that the restriction of u on $\mathrm {Fix}(\Phi )$ has a unique maximizer, which is the zero sequence $\boldsymbol {0}$ of $\ell _2$ , hence condition (A4) holds. Moreover, we have that

$$ \begin{align*}F=\{\boldsymbol{x} \in \mathscr{C}: u(\boldsymbol{x}) \ge u(\boldsymbol{0})\}=\{\boldsymbol{x} \in \mathscr{C}: x_0 \ge 0\}. \end{align*} $$

Fix sequences $\boldsymbol {x} \in F$ and $\boldsymbol {y} \in \Phi (\boldsymbol {x})$ such that $(\boldsymbol {x}, \boldsymbol {y}) \neq (\boldsymbol {0}, \boldsymbol {0})$ . Note that $\Phi (\boldsymbol {0})=\{\boldsymbol {0}\}$ , hence $\boldsymbol {x}\neq \boldsymbol {0}$ (in particular, if $x_0=0$ , then $x_i\neq 0$ , for some $i\ge 1$ ). Thus, setting $T=u$ (which is a continuous linear functional on $\ell _2$ ), we obtain that $T\boldsymbol {y}=-\sum _{i\ge 1}x_i^2<x_0=T\boldsymbol {x}$ , i.e., condition (A5) holds. Finally, note that the sequence $(\boldsymbol {x}^\star , \frac {1}{2}\boldsymbol {x}^\star , \frac {1}{4}\boldsymbol {x}^\star , \ldots )$ is convergent to $\boldsymbol {0}$ and belongs to $\mathscr {C}$ (indeed, it is starts at $\boldsymbol {x}^\star $ , it is feasible, and its image is contained in the compact set $\{k\boldsymbol {x}^\star : k \in [0,1]\}$ ).

We conclude by Corollary 2.2 that each $\mathcal {I}$ -optimal sequence $(\boldsymbol {x}^{(n)}) \in \mathscr {C}$ in the system $\langle \, \ell _2, \Phi , u, \mathcal {I}, \mathscr {C}\rangle $ is necessarily $\mathcal {I}$ -convergent to $\boldsymbol {0}$ .