Certain convergences for intuitionistic fuzzy sets

Abstract

In this paper, the characterization of Γ-convergence for the first countable topological spaces, characterization of convergence in supremum metric in general setting and some mutual relation between these convergences are discussed. The Γ-convergence is defined as the Kuratowaski-Painlevé convergence of the endographs of the intuitionistic fuzzy sets. The supremum metric is the supremum of Hausdroff distance among the η-cuts of the intuitionistic fuzzy sets. To study these convergences is an important part of the theoretical fundamentals for intuitionistic fuzzy set theory. Some results are given as an application to variational analysis.

Keywords

Intuitionistic fuzzy sets Pointwise convergence Hausdroff metric Supremum metric

1 Introduction

Atanassov [2] gave the notion of intuitionistic fuzzy set (IFS), which is an extension of Zadeh’s fuzzy set [26]. It enriches fuzzy set theory with a notion of indeterminacy expressing hesitation or abstention. IFS deals with more complicate and ambiguous real life problems, allowing more freedom. An intuitionistic fuzzy set (IFS) is known as function from some set X to $T$ , where $T = {(μ, ν) \in [0, 1]^{2} : μ + ν \leq 1}$ . For detail knowledge on IFS see the book by Atanassov [3]. Dubois et al. [11] pointed some terminological difficulties in fuzzy set theory and intuitionistic fuzzy set theory in their remarkable paper. IFS has received more attention and has been applied not only in theoretical fields (differential equation, variational analysis, fixed point theory, etc.) but also in practical fields (artificial intelligence, engineering, medical, similarity measure, decision making, etc.). Ban [5] gave the theory of intuitionistic fuzzy measure and its applications. Latter Atanassov [4] in his book gave the detailed survey on the theory and applications of intuitionistic fuzzy set theory and also described new directions and research problems.

To discuss different types of convergences for fuzzy sets is fundamental in fuzzy set theory [10 , 25]. For IFS in literature, there is notion of statistical convergence in the setting of norm spaces [1 , 24]. Intuitionistic fuzzy norms deal the situations where norms of the vectors cannot be found exactly. But there are some other important convergences like Γ-convergence, although it is a basic notion of variational analysis has been used in various other fields, in [15] long-term convergence of finite state mean-field games are explred with the use of Γ-convergence, in [9] for variable exponent Sobolev space some critical embeddings are proved by using Γ-convergence and in [22] large derivatives of Gibbs measures are studied by considering Γ-convergence of energy time functionals.

To model above discussed scenarios in intuitionistic fuzzy set theory In [7] different types of convergences like Γ-convergence, pointwise convergence and convergence in supremum metric are designed and relations among them are studied in detail. Latter in [8] a detailed discussion is done on the compatible topology of Γ-convergence. The main aim of this paper is to characterize Γ-convergence and convergence in supremum metric for practical use. This work provides the theoretical frame work for solving the problems of variational analysis like maximization and minimization with intuitionistic fuzzy set theory. For this purpose towards the end of this paper some results are given as applications to variational analysis.

This paper is organized as follows. In Section 2, some basic concepts and auxiliary results are given to understand our proposal. In Section 3, we discussed the characterization of Γ-convergence and its importance in variation analysis. Section 4 is devoted for the detailed study on convergence in supremum metric.

2 Basic Concepts

At first let us recall some of the definitions from [7]. The following order is used in $T$ : for α = (μ_α, ν_α) , $β = (μ_{β}, ν_{β}) \in T$ , $α \leq_{T} β$ if and only if μ_α ≤ μ_β and ν_α ≥ ν_β. Also, for $B \subseteq T$ $sup B = (sup_{η \in B} μ_{η}, inf_{η \in B} ν_{η})$ and $inf B = (inf_{η \in B} μ_{η}, sup_{η \in B} ν_{η})$ exist in $T$ , therefore $(T, \leq_{T})$ is a complete lattice with (0, 1) as the smallest and (1, 0) as the largest element. To explore different types of convergences of intuitionistic fuzzy sets two pseudo metrics are defined, as follows. For α, β in $T$ :

$d_{lower} (α, β) = max (μ_{α} - μ_{β}, ν_{β} - ν_{α}, 0);$ (2.1)

$d_{lower} (α, β) = max (μ_{β} - μ_{α}, ν_{α} - ν_{β}, 0);$ (2.2)

According to above defined pseudo metrics d_lower and d_upper lower and upper convergences are defined on $T$ , respectively.

Another way to explore $T$ is to study it by the topologies induced by order $\leq_{T}$ defined below.

$\begin{matrix} τ_{\leq_{T}} = {U \subseteq T : if α, β \in T, α \in U and \\ α \leq_{T} β then β \in U} . \end{matrix}$ (2.3)

$\begin{matrix} τ_{\geq_{T}} = {U \subseteq T : if α, β \in T, α \in U and \\ α \geq_{T} β then β \in U} . \end{matrix}$ (2.4)

The topologies $τ_{\leq_{T}}$ and $τ_{\geq_{T}}$ are called lower and upper topologies on $T$ , respectively. Fig. 1 presents the discerption of open sets in these topologies.

Fig.1

(a) Open set in τ_≤, (b) Open set in τ_≥.

Also, $T$ is explored by the standard Euclidean metric d_e of $ℝ^{2}$ . B_d (x, r) stands for an open ball with radius r in arbitrary metric space (X, d). For a subset B of X B_d (B, r) = ∪ _x∈BB_d (x, r) and for x ∈ X $N_{x}$ is defined to be the set of all neighborhoods of x. Hausdroff extended pseudo metric denoted H_d is the basis to define supremum metric as it is applied between the η-cuts of two IFSs, therefore its definition is given here. Take two subsets B and C in X then

$H_{d} (B, C) = max (e_{d} (B, C), e_{d} (C, B)),$ (2.5) where $e_{d} (B, C) = {\begin{matrix} sup_{a \in B} d (a, C) & if B≠ ∅ \\ 0, & if C=∅, \end{matrix}$ (2.6)

is the excess of B over C.

H_d (B, C) can be characterize as $H_{d} (B, C) = max {\begin{matrix} inf {ε > 0 : B \subseteq B_{d} (C, ε)}, \\ inf {ε > 0 : C \subseteq B_{d} (B, ε)} \end{matrix}}$ (2.7) where inf is +∞ if no such ε exists.

Let (X, τ) be a topological space, the set of all IFSs on X is denoted as $IF (X)$ . For g in $IF (X)$ its η-cut is known as $[g]^{η} : = {x \in X : η \leq_{T} g (x)}$ for $η \in T$ , its endograph or hypograph known as $endg : = {(x, η) \in X \times T : η \leq_{T} g (x)}$ and epigraph is known as $epig : = {(x, η) \in X \times T : g (x) \leq_{T} η}$ . We will work on endg in topological space $(X \times T, τ \times τ_{d_{e}})$ . Furthermore support of g is known as $Supp (g) = {x \in X : g (x) >_{T} (0, 1)}$ .

Take a net {B_ω} _ω∈Ω in X then

A subset of Ω is said to be residual if it contains all indices at or beyond some index λ.

A subset of Ω is said to be cofinal if it contains some indices at or beyond each index λ.

the lower limit of {B_ω} _ω∈Ω is the set $Li B_{ω} = {x \in X : U \cap B_{ω} \neq \emptyset residually, \forall U \in N_{x}};$

the upper limit of {B_ω} _ω∈Ω is the set $Ls B_{ω} = {x \in X : U \cap B_{ω} \neq \emptyset cofinally, \forall U \in N_{x}} .$

A net {B_ω} _ω∈Ω is said to be lower (resp. upper) Kuratowski-Painlevé convergent to B ⊆ X if B ⊆ LiB_ω (resp. LsB_ω ⊆ B). See [6, page 2, 145] B for details. Kuratowski-Painlevé convergence is very important set convergence that is used to define Γ-convergence.

Definition 2.1. Take g in $IF (X)$ . Then:

lower limit of g is defined as $\underline{g} (x) = \underset{y \to x}{lim inf} g (y) = sup_{U \in N_{x}} inf_{y \in U} g (y);$

upper limit of g is defined as $\bar{g} (x) = \underset{y \to x}{lim sup} g (y) = inf_{U \in N_{x}} sup_{y \in U} g (y) .$

The functions $\bar{g}$ and $\underline{g}$ are important functions in discussing the continuity of g.

Definition 2.2. Let $g \in IF (X)$ . Then at x in X:

g is said to be lower semicontinuous if $g (x) \leq_{T} \underline{g} (x);$

g is said to be upper semicontinuous if $\bar{g} (x) \leq_{T} g (x);$

g is said to be continuous if and only if $g (x) = \bar{g} (x) = \underline{g} (x)$ .

After defining continuity different types of convergences for a net {g_ω} _ω∈Ω in $IF (X)$ are discussed.

Definition 2.3. Let {g_ω} _ω∈Ω be a net in $IF (X)$ . Then for g in $IF (X)$ .

at x in X {g_ω} _ω∈Ω is said to be Lower pointwise convergent to g if $\underset{ω \in Ω}{lim inf} g_{ω} (x) \geq_{T} g (x);$

at x in X {g_ω} _ω∈Ω is said to be Upper pointwise convergent to g if $\underset{ω \in Ω}{lim sup} g_{ω} (x) \leq_{T} g (x);$

at x in X {g_ω} _ω∈Ω is said to be Pointwise convergent to g if $lim_{ω \in Ω} g_{ω} = \underset{ω \in Ω}{lim inf} g_{ω} (x) = \underset{ω \in Ω}{lim sup} g_{ω} (x) = g (x) .$

{g_ω} _ω∈Ω is said to be lower pointwise, upper pointwise and pointwise convergent to g, if the above hold for all x in X, respectively.

Definition 2.4. Let {g_ω} _ω∈Ω be a net in $IF (X)$ . Then:

if for all ε > 0 there exists ω_ε in Ω such that for all x ∈ X and for all ω ≥ ω_ε $d_{lower} (g (x), g_{ω} (x)) < ε,$ then {g_ω} _ω∈Ω is said to be lower uniformly convergent to g in $IF (X)$ ;

if for all ε > 0 there exists ω_ε in Ω such that for all x ∈ X and for all ω ≥ ω_ε $d_{upper} (g (x), g_{ω} (x)) < ε,$ then {g_ω} _ω∈Ω is said to be upper uniformly convergent to g in $IF (X)$ ;

if {g_ω} _ω∈Ω is both upper and lower uniformly convergent to g then {g_ω} _ω∈Ω is said to be uniformly convergent to g in $IF (X)$ .

Definition 2.5. Let {g_ω} _ω∈Ω be a net in $IF (X)$ . Then for g in $IF (X)$ :

if there exists ω₀ in Ω such that for all x ∈ X and for all ω ≥ ω₀ $g (x) \leq_{T} g_{ω} (x),$ then {g_ω} _ω∈Ω is said to be convergent to g $τ_{\leq_{T}}$ uniformly;

if there exists ω₀ in Ω such that for all x ∈ X and for all ω ≥ ω₀ $g (x) \geq_{T} g_{ω} (x),$ then {g_ω} _ω∈Ω is said to be convergent to g $τ_{\geq_{T}}$ uniformly.

3 Characterization of Γ-convergence

Originally Γ-convergence is the notion of variational analysis, but it has been widely used in other field as well like in studying the embedding of Sobolev spaces [9], in game theory [15]. In [7] Γ-convergence is designed for intuitionistic fuzzy frame work so that if the similar scenarios are modeled with IFSs then we have necessary tools for obtaining solutions of such problems.

Definition 3.1. Let {g_ω} _ω∈Ω be a net in $IF (X)$ . Then for g in $IF (X)$

{g_ω} _ω∈Ω is said to be Γ^--convergent to g if the corresponding net of endographs {endg_ω} _ω∈Ω is lower Kuratowski-Painlevé convergent to endg in topological space $(X \times T, τ \times τ_{d_{e}})$ i.e. endg ⊆ Liendg_ω;

{g_ω} _ω∈Ω is said to be Γ⁺-convergent to g if the corresponding net of endographs {endg_ω} _ω∈Ω is upper Kuratowski-Painlevé convergent to endg in topological space $(X \times T, τ \times τ_{d_{e}})$ i.e. Lsendg_ω ⊆ endg;

{g_ω} _ω∈Ω is said to be Γ-convergent to g if the corresponding net of endographs {g_ω} _ω∈Ω is both Γ^- and Γ⁺-convergent to g i.e. Liendg_ω = Lsendg_ω = endg.

The counter part epi convergence is defined in a similar manner if in above definition endographs are replaced by epigraphs.

Natural question arises whether or not Γ-convergence is topological. In [8] the compatible topology of Γ-convergence is studied in detail for a locally compact Hausdroff topological space X, Γ = Γ⁺ ∧ Γ^- and problem lies with Γ⁺-convergence as it is not topological, whereas Γ^- is always topological any easy to manipulate topologically, in fact the topology of Γ^--convergence is Fell topology.

An other aspect is when to use Γ-convergence and its counter part epi-convergence. Basically it is dependent upon the nature of the problem for problems related to maximization Γ-convergence and for minimization epi-convergence is usually used.

To practically determine the Γ-convergence two functions were defined in [7] given below:

$(Li g_{ω}) (x) = inf_{U \in N_{x}} \underset{ω \in Ω}{lim inf} M_{g_{ω}} (U)$ 3.1 and $(Ls g_{ω}) (x) = inf_{U \in N_{x}} \underset{ω \in Ω}{lim sup} M_{g_{ω}} (U),$ 3.2 where $M_{g_{ω}} (U) = sup_{y \in U} g_{ω} (y) .$

The above functions can be simplified in a metric space (X, d) as follows by replacing arbitrary neighbourhoods by open balls. $(Li g_{ω}) (x) = inf_{ε > 0} \underset{ω \in Ω}{lim inf} M_{g_{ω}} (B_{d} (x, ε)) .$ 3.3 $(Ls g_{ω}) (x) = inf_{ε > 0} \underset{ω \in Ω}{lim sup} M_{g_{ω}} (B_{d} (x, ε)) .$ 3.4

Lemma 3.2. [7, Lemma 4.4] BT Let {g_ω} _ω∈Ω be a net in $IF (X)$ , then $Liend g_{ω} = endLi g_{ω} and Lsend g_{ω} = endLs g_{ω} .$

By virtue of the above lemma Γ-convergence is completely determined as follows:

{g_ω} _ω∈Ω is Γ^--convergent to g if and only if $g (x) \leq_{T} (Li g_{ω}) (x)$ ;

{g_ω} _ω∈Ω is Γ⁺-convergent to g if and only if $g (x) \geq_{T} (Ls g_{ω}) (x)$ ;

{g_ω} _ω∈Ω is Γ-convergent to g if and only if g (x) = (Lig_ω) (x) = Lsg_ω.

We have some nice characterization of Γ^--convergence and Γ⁺-convergence when (X, τ) is a first countable space. First we give some useful lemmas.

Lemma 3.3. (Cf. [6, Lemma 5.2.7]B) Consider (X, τ) a first countable topological space and ${g_{n}}_{n \in ℕ}$ in $IF (X)$ . Then (x, η) ∈ Liendg_n if and only if there is some $N \in ℕ$ such that for n ≥ N we have (x_n, η_n) ∈ endg_n and {(x_n, η_n)} converges to (x, η).

Lemma 3.4. (Cf. [6, Lemma 5.2.8]B) Consider (X, τ) a first countable topological space and ${g_{n}}_{n \in ℕ}$ in $IF (X)$ . Then (x, η) ∈ Lsendg_n if and only if we have positive integers n₁ < n₂ < n₃ < . . . such that we have (x_{n
_z}, η_{n
_z}) ∈ endg_{n
_z} and {(x_{n
_z}, η_{n
_z})} converges to (x, η).

Theorem 3.1. Let ${g_{n}}_{n \in ℕ}$ be a sequence in $IF (X)$ where (X, τ) is a first countable topological space. Then for g in $IF (X)$ :

for any x ∈ X there is some sequence ${x_{n}}_{n \in ℕ}$ convergent x such that $g (x) \leq_{T} \underset{n \to \infty}{lim inf} g_{n} (x_{n})$ if and only if ${g_{n}}_{n \in ℕ}$ is Γ^--convergent to g;

for any x ∈ X, for all sequences ${x_{n}}_{n \in ℕ}$ convergent x we have $\underset{n \to \infty}{lim sup} g_{n} (x_{n}) \leq_{T} g (x)$ if and only if ${g_{n}}_{n \in ℕ}$ is Γ⁺-convergent to g.

Proof. (1): Take ${g_{n}}_{n \in ℕ}$ with residually endg ⊆ Liendg_n. For any x ∈ X, choose η = g (x), we have (x, η) ∈ Liendg_n. Now by Lemma 3 there is some sequence ${(x_{n}, η_{n})}_{n \in ℕ}$ convergent to (x, η). Also (x_n, η_n) ∈ endg_n. Hence $\underset{n \to \infty}{lim inf} g (x_{n}) \geq_{T} \underset{n \to \infty}{lim inf} η_{n} = g (x)$ .

Conversely, we assumed the existence of convergent sequence ${x_{n}}_{n \in ℕ}$ for any x ∈ X such that $g (x) \leq_{T} \underset{n \to \infty}{lim inf} g_{n} (x_{n})$ . Take (x, η) ∈ endg then there is some $\overset{´}{n} \in ℕ$ such that for all $n \geq \overset{´}{n}$ we have $η \leq_{T} g (x) \leq_{T} g_{n} (x_{n})$ . It gives us (x_n, η) ∈ endg_n and from the fact that x_n → x, we concluded (x, η) ∈ Liendg_n.

(2): Take ${g_{n}}_{n \in ℕ}$ with cofinally endg_n ⊆ endg. Let ${x_{n}}_{n \in ℕ}$ be a sequence convergent to x in X and $g (x) <_{T} \underset{n \to \infty}{lim sup} g_{n} (x_{n})$ , this means we take η such that $g (x) <_{T} η <_{T} \underset{n \to \infty}{lim sup} g_{n} (x_{n})$ . Which means (x, η) ∈ Lsendg_n, but on the other hand Lsendg_n ⊆ endg, so $η \leq_{T} g (x)$ . That leads to contradiction for our assumption $g (x) <_{T} \underset{n \to \infty}{lim sup} g_{n} (x_{n})$ . Lastly Lemma 3 ensures the existence of at least one sequence ${x_{n}}_{n \in ℕ}$ in X that is convergent to x in X with $\underset{n \to \infty}{lim sup} g_{n} (x_{n}) \leq_{T} g (x)$ .

Conversely, let for all x ∈ X, for all sequences ${x_{n}}_{n \in ℕ}$ convergent x we have $\underset{n \to \infty}{lim sup} g_{n} (x_{n}) \leq_{T} g (x)$ . Take (x, η) ∈ Lsendg_n and Consider the constant sequence {x}. Since, $\underset{n \to \infty}{lim sup} g_{n} (x) \leq_{T} g (x)$ , that implies for all $n \in ℕ$ there is some $\overset{´}{n} \geq n$ such that $g_{\overset{´}{n}} (x) \leq_{T} g (x)$ i.e. $η \leq_{T} g (x)$ .

■

In variational analysis for a net {g_ω} _ω∈Ω of functions, it is important that what happened to maximizers of these functions. For example, in some sense {g_ω} _ω∈Ω converges to a function g then what about the net of {M_{g
_ω} (X)} does it also converges to M_g (X)? On the other hand let say $x_{ω} \in \underset{X}{Argmax} g_{ω} = {x \in X : g_{ω} (x) = M_{g_{ω}} (X)}$ for all ω ∈ Ω and x_ω → x, does $x \in \underset{X}{Argmax} g$ ? These and their dual questions for minimizers are discussed in [17, 23], but for lower semicontinuous functions. We try to answer these questions in the context of intuitionistic fuzzy sets.

Proposition 3.6. (Cf. [23, Proposition 7.29(b)]RW) Let {g_ω} _ω∈Ω be a net in $IF (X)$ . Then for all $U \in N_{x},$ $M_{g} (U) \leq_{T} \underset{ω \in Ω}{lim inf} M_{g_{ω}} (U)$ for all x ∈ X if and only if {g_ω} _ω∈Ω is Γ^--convergent to g.

Proof. It is straightforward from the Equation 3.1 and Lemma 3. ■

Proposition 3.7. (Cf. [17, Theorem 8.6.6]L) Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a first countable topological space (X, τ) that is Γ-convergent to some g. is a first countable topological space and a sequence ${g_{n}}_{n \in ℕ}$ in $IF (X)$ Γ-convergent to g. For a sequence ${x_{n}}_{n \in ℕ}$ in X convergent to x, If $x_{n} \in \underset{X}{Argmax} g_{n}$ for all $n \in ℕ$ then $x \in \underset{X}{Argmax} g$ .

Proof. By Theorem 3 we have $\underset{n \to \infty}{lim sup}$ $g_{n} (x_{n}) \leq_{T} g (x)$ . As $x_{n} \in \underset{X}{Argmax} g$ that means $\underset{n \to \infty}{lim sup} M_{g_{n}} (X) \leq_{T} g (x)$ , but also Proposition 3 gives us that $M_{g} (X) \leq_{T} \underset{n \to \infty}{lim inf} M_{g_{n}} (X)$ . Hence $M_{g} (X) \leq_{T} g (x)$ . ■

These results showed some advantages to use Γ-convergence, it maintains many stability properties. Whereas pointwise convergence unable to do so. The relation between pointwise convergence and Γ-convergence is discussed in [7, Theorem 4.5]BT.

4 Characterization of convergence in supremum metric

Hausdroff metric H_d between two subsets of a metric space is widely used for different purposes. Based on H_d the supremum metric D_∞ is defined by taking Hausdroff distance between η-cuts of two IFS, thus enlarging the scope of solving more wide range of problems with IFS theory.

The aim of this section is to characterize the convergence in supremum metric to get a way of establishing convergence practically. Let (X, d) be a metric space and take two IFSs g, h in $IF (X)$ . $\begin{matrix} D_{\infty} (g, h) = sup_{η \in T} H_{d} ([g]^{η}, [h]^{η}) \\ D_{\infty} (g, h) = sup_{η \in T} max {\begin{matrix} sup_{x \in [g]^{η}} d (x, [h]^{η}), \\ sup_{x \in [h]^{η}} d (x, [g]^{η}) \end{matrix}} . \end{matrix}$ (4.1)

To ease our work on D_∞ is split into parts lower quasi-metric $D_{\infty}^{-}$ and upper quasi-metric $D_{\infty}^{+}$ known as

$D_{\infty}^{-} (g, h) = sup_{η \in T} e_{d} ([g]^{η}, [h]^{η}) = sup_{η \in T} [sup_{x \in [g]^{η}} d (x, [h]^{η})] .$ (4.2)

$D_{\infty}^{+} (g, h) = sup_{η \in T} e_{d} ([h]^{η}, [g]^{η}) = sup_{η \in T} [sup_{x \in [h]^{η}} d (x, [g]^{η})] .$ (4.3)

The above metrics are not symmetric for this reason they are called quasi-metrics. Now $D_{\infty} = D_{\infty}^{-} \lor D_{\infty}^{+}$ , therefore a net in convergent in D_∞ if and only if it is convergent in both $D_{\infty}^{-}$ and $D_{\infty}^{+}$ . Further, convergence in $D_{\infty}^{-}$ and $D_{\infty}^{+}$ can be characterized by Equation 2.7 as follows.:

for ε > 0 there exists ω_ε ∈ Ω such that [g] ^η ⊆ B_d ([g_ω] ^η, ε) for all ω ≥ ω_ε and for all $η \in T$ if and only if {g_ω} _ω∈Ω is $D_{\infty}^{-}$ -convergent to $g \in IF (X)$ ;

for ε > 0 there exists ω_ε ∈ Ω such that [g_ω] ^η ⊆ B_d ([g] ^η, ε) for all ω ≥ ω_ε and for all $η \in T$ if and only if {g_ω} _ω∈Ω is $D_{\infty}^{+}$ -convergent to $g \in IF (X)$ .

A net {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{-}$ when $g (x) \leq_{T} inf_{ω \in Ω} g_{ω} (x)$ and it is convergent to g in $D_{\infty}^{+}$ when $g (x) \geq_{T} sup_{ω \in Ω} g_{ω} (x)$ , therefore convergence in supremum metric is some how related to convergence in $τ_{\leq_{T}}$ that will be discussed in coming subsections. Another way to express $D_{\infty}^{-}$ and $D_{\infty}^{+}$ by using the characterization of $e_{d} (B, C) = sup_{x \in X} [d (C, x) - d (B, x)]$ given in [6, Lemma 1.5.1]B, as follows:

$D_{\infty}^{-} (g, h) = sup_{η \in T} sup_{x \in X} [d ([h]^{η}, x) - d ([g]^{η}, x)];$ (4.4)

$D_{\infty}^{+} (g, h) = sup_{η \in T} sup_{x \in X} [d ([g]^{η}, x) - d ([h]^{η}, x)] .$ (4.5)

The above discussion leads to following result.

Theorem 4.1. Let {g_ω} _ω∈Ω be net in $IF (X)$ for a metric space (X, d). Then for all ε > 0 there is some ω_ε ∈ Ω that for all ω ≥ ω_ε for all x ∈ X and for all $η \in T$ we have $| d ([g]^{η}, x) - d ([g_{ω}]^{η}, x) | < ε,$ if and only if {g_ω} _ω∈Ω is convergent to g in D_∞.

4.1 Characterization of Convergence in

D_{\infty}^{-}

Our discussion on D_∞-convergence is divided in two parts $D_{\infty}^{-}$ -convergence and $D_{\infty}^{+}$ -convergence. A set theoretic approach is taken to characterize $D_{\infty}^{-}$ . Consider a nonempty set X and its power set $P (X)$ . Let {B_ω} _ω∈Ω be a net in $P (X)$ we say $B \in P (X)$ is lower limit if

$B \subseteq \underset{ω \in Ω}{lim inf} B_{ω} : = ⋃_{ω_{0} \in Ω} (⋂_{ω \geq ω_{0}} B_{ω}) .$ (4.6)

Definition 4.2. Consider a nonempty set X and a net of set functions {q_ω} _ω∈Ω $q_{ω} : X \to P (T)$ . For a set function $q : X \to P (T)$ if there is some ω₀ ∈ Ω such that for all x ∈ X $q (x) \subseteq ⋂_{ω \geq ω_{0}} q_{ω} (x),$ then {q_ω} _ω∈Ω is defined to be lower uniformly convergent to set function q.

Before moving on to the main theorem, we like to discuss the closure in $τ_{\leq_{T}}$ . Note that for A ⊆ X, $η \in {\bar{g (A)}}^{τ_{\leq_{T}}}$ if and only if there is some y ∈ A such that $η \leq_{T} g (y) .$ The Figure 2 showed the closure in $τ_{\leq_{T}}$ for two important cases.

Fig.2

Closure in $τ_{\leq_{T}}$

Theorem 4.3. Let {g_ω} _ω∈Ω be a net in $IF (X)$ with (X, d) a metric space. For ε > 0 consider $g_{ω}^{ε} : X \to P [(T, τ_{\leq_{T}})]$ defined as $g_{ω}^{ε} (x) : = {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq_{T}}}$ . Then for all ε > 0 net of set functions ${g_{ω}^{ε}}_{ω \in Ω}$ is lower uniformly convergent to g if and only if the net {g_ω} _ω∈Ω is $D_{\infty}^{-}$ -convergent to $g \in IF (X)$ .

Proof. Let {g_ω} _ω∈Ω be a net $D_{\infty}^{-}$ -convergent to g. For ε > 0 there is some ω_ε ∈ Ω such that for all ω ≥ ω_ε and for all $η \in T$ [g] ^η ⊆ B_d ([g_ω] ^η, ε) hold. In particular for η = g (x) we have y_ω ∈ B_d (x, ε) with $g (x) \leq_{T} g_{ω} (y_{ω})$ for all ω ≥ ω_ε. Therefore ${g (x)} \subseteq ⋂_{ω \geq ω_{ε}} {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq}}$ .

Conversely, assumed that ${g_{ω}^{ε}}_{ω \in Ω}$ is lower uniformly convergent to g. Let x ∈ [g] ^η for $η \in T$ . By hypothesis for ε > 0, there exist ω_ε ∈ Ω for which ${g (x)} \subseteq ⋂_{ω \geq ω_{ε}} {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq}}$ . Therefore, we have y_ω ∈ B_d (x, ε) such that $η \leq_{T} g (x) \leq_{T} g_{ω} (y_{ω})$ . Hence [g] ^η ⊆ B_d ([g_ω] ^η, ε). ■

Corollary 4.4. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Then for all x in X $g (x) \in ⋂_{ε > 0} \underset{ω \in Ω}{lim inf} {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq}},$ if {g_ω} _ω∈Ω is $D_{\infty}^{-}$ -convergent to $g \in IF (X)$ .

In view of the above the following holds. $\begin{matrix} sup ⋂_{ε > 0} \underset{ω \in Ω}{lim inf} {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq_{T}}} \\ = inf_{ε > 0} \underset{ω \in Ω}{lim inf} M_{g_{ω}} (B_{d} (x, ε)) . \end{matrix}$

From this simplification and by the Definition of (Lig_ω) 3.3 it is evident that convergence in $D_{\infty}^{-}$ implies Γ^--convergence, that has already been proved in [7]. After characterizing convergence in $D_{\infty}^{-}$ we discuss about the net {M_{g
_ω} (B_d (· , ε))} _ω∈Ω, indeed convergence of {g_ω} _ω∈Ω in $D_{\infty}^{-}$ implies $τ_{\leq_{T}}$ -convergence of {M_{g
_ω} (B_d (· , ε))} _ω∈Ω, converse is not true in general. But with extra condition of upper semicontinuity on IFSs following result prove the equivalence.

Theorem 4.5. Let {g_ω} _ω∈Ω be a net of compactly supported upper semicontinuous IFSs in $IF (X)$ for a metric space (X, d). Then {M_{g
_ω} (B_d (· , ε))} _ω∈Ω is convergent to g $τ_{\leq_{T}}$ uniformly for all ε > 0 if and only if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{-}$ .

Proof. We already proved the necessity in [7, Theorem 5.1]BT, here we only need to prove the sufficiency.

Suppose {M_{g
_ω} (B_d (· , ε))} _ω∈Ω is convergent to g $τ_{\leq_{T}}$ uniformly i.e. for ε > 0 we can find ω_ε ∈ Ω in a way for all ω ≥ ω_ε we have $g (x) \leq_{T} M_{g_{ω}} (B_{d} (x, ε / 2))$ . If $g (x) <_{T}$ M_{g
_ω} (B_d (x, ε/2)) for some ω ≥ ω_ε then there is some y_ω ∈ B_d (x, ε) for which $g (x) <_{T} g_{ω} (y_{ω})$ and that implies $g (x) \in {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq}} .$ Now for some ω ≥ ω_ε, if g (x) = M_{g
_ω} (B_d (x, ε/2)) with $g (x) >_{T} (0, 1)$ (the case g (x) = (0, 1) is trivial). There exist a sequence ${b_{n}}_{n \in ℕ}$ in B_d (x, ε/2) with $lim_{n \to \infty} g_{ω} (b_{n}) = g (x)$ with $g_{ω} (b_{n}) >_{T} (0, 1)$ for all $n \in ℕ$ . Since g_ω has compact support, without loss of generality we can assume that ${b_{n}}_{n \in ℕ}$ is convergent to b in $\bar{B_{d} (x, ε / 2)}$ . But by the upper semicontinuity of g_ω we have $\underset{n \to \infty}{lim sup} g_{ω} (b_{n}) = g (x) \leq_{T} g_{ω} (b)$ . Therefore $g (x) \in {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq_{T}}}$ . Hence by Theorem 4.1 {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{-}$ . ■

Corollary 4.6. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Then for any ε > 0 we can find ω_ε ∈ Ω in a way that for all x ∈ X and for all ω ≥ ω_ε $g (x) \leq_{T} M_{g_{ω}} (B_{d} (x, ε))$ if and only if {g_ω} _ω∈Ω in convergent to g in D^-. Furthermore, if for some ω ≥ ω_ε, $g (x) = M_{g_{ω}} (B_{d} (x, ε)) then \underset{B_{d} (x, ε)}{Argmax} g_{ω} \neq \emptyset .$

For a sequence of intuitionistic fuzzy sets we have the following characterization for convergence in $D_{\infty}^{-}$ for first countable spaces.

Theorem 4.7. Let ${g_{n}}_{n \in ℕ}$ be a sequence in $IF (X)$ for a metric space (X, d). Then there exits a sequence ${h_{n}}_{n \in ℕ}$ where each h_n : X → X that converges uniformly to id_X and ${g_{n} \circ h_{n}}_{n \in ℕ}$ is convergent to g $τ_{\leq_{T}}$ uniformly if and only if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{-}$ .

Proof. Assume that ${g_{n}}_{n \in ℕ}$ is convergent to g in $D_{\infty}^{-}$ . For each $z \in ℕ$ there exists $n_{z} \in ℕ$ for which for all x in X [g] ^g(x) ⊆ B_d ([g_n] ^g(x), 1/z) for all n ≥ n_z. we can assume ${n_{z}}_{z \in ℕ}$ to be strictly increasing without loss. As x ∈ [g] ^g(x), so for all n ≥ n_z we can choose x_n,z ∈ B_d (x, 1/z) such that $g (x) \leq_{T} g_{n} (x_{n, z})$ . For any $n \in ℕ$ we define $h_{n} (x) : = {\begin{matrix} x_{0}, & when n< n1; \\ x_{n, z}, & when nz ≤ n < nz+1, \end{matrix}$ where x₀ is some fixed element of X. First of all it is evident that the above sequence is convergent to id_X $τ_{\leq_{T}}$ uniformly, as d (x, h_n (x)) <1/z for any given z, for all n ≥ n_z and for all x ∈ X. Secondly, $g (x) \leq_{T} g_{n} (h_{n} (x))$ for all n ≥ n_z and for all x ∈ X.

Conversely, we assumed the existence of a sequence of self maps. This means, for ε > 0, there is some n_ε such that for all n ≥ n_ε and for all x ∈ X, we have d (x, h_n (x)) < ε and $g (x) \leq_{T} g_{n} (h_{n} (x))$ . Hence if x ∈ [g] ^η then x ∈ B_d ([g_n] ^η, ε) for all n ≥ n_ε. Also n_ε does not depend on η, so we have the desired result. ■

The following proposition will give the relation between $D_{\infty}^{-}$ -convergence and maximizers as an application to variational analysis.

Proposition 4.8. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Then $M_{g} (X) \leq_{T} M_{g_{ω}} (X)$ residually if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{-}$ .

Proof. It follows from the fact that $D_{\infty}^{-}$ implies Γ^--convergence and Proposition 3.

4.2 Characterization of convergence in

D_{\infty}^{+}

A similar set theoretical approach is used to characterize $D_{\infty}^{+}$ -convergence as we did for $D_{\infty}^{-}$ -convergence. But first we want to discuss the connection between convergence in $D_{\infty}^{+}$ and Γ⁺-convergence. In general there is no relationship between these convergences. However we are able to prove with extra condition of uniformly upper semicontinuity on limit function.

Theorem 4.9. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Consider a net {g_ω} _ω∈Ω in $IF (X)$ and g is uniformly upper semicontinuous IFS. Then:

if {g_ω} _ω∈Ω is convergent to g $τ_{\geq_{T}}$ uniformly then {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ ;

if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ then {g_ω} _ω∈Ω is upper uniformly convergent to g;

if {g_ω} _ω∈Ω is upper uniformly convergent to g then {g_ω} _ω∈Ω is Γ⁺-convergent to g.

Proof. (1): We suppose that {g_ω} _ω∈Ω is convergent to g $τ_{\geq_{T}}$ uniformly. This means we can find some $\overset{´}{ω}$ in Ω such that for all x in X and for all $ω \geq \overset{´}{ω}$ we have $g_{ω} (x) \leq_{T} g (x)$ . Then it is clear that for all η in $T$ and for all $ω \geq \overset{´}{ω}, ε > 0$ [g_ω] ^η ⊆ B_d ([g] ^η, ε).

(2): Assume that {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ . By the upper semicontinuity of g for ε > 0 there exist a δ_ε such that $d_{upper} (g (x), g (y)) < ε for all x \in X for all y \in B_{d} (x, δ_{ε}) .$

Hence, for all x in X $d_{upper} (g (x), M_{g} (B_{d} (x, δ_{ε}))) < ε .$

From [7, Theorem 5.9(2)]BT, $D_{\infty}^{+}$ -convergence implies that for all ε > 0 {g_ω} _ω∈Ω is convergent to {M_g (B_d (· , ε))} $τ_{\geq_{T}}$ uniformly. Therefore for this δ_ε we have some ω_{δ
_ε} ∈ Ω in a way that for all ω ≥ ω_{δ
_ε} $g_{ω} (x) \leq_{T} M_{g} (B_{d} (x, δ_{ε})) for all x \in X .$

Therefore $d_{upper} (g (x), g_{ω} (x)) < ε,$ for all x in X and for all ω ≥ ω_{δ
_ε} .

(3): Let {g_ω} _ω∈Ω be a net convergent to g upper uniformly. For ε₀ > 0 there is some ω_{ε
₀} ∈ Ω in a way for all x in X and for all ω ≥ ω_{ε
₀} $d_{upper} (g (x), g_{ω} (x)) < ε_{0} .$ Which means for any ε > 0, we have $d_{upper} (M_{g} (B_{d} (x, ε)), M_{g_{ω}} (B_{d} (x, ε))) < ε_{0},$ therefore by taking ε₀ → 0 we can deduce that $\underset{ω \in Ω}{lim sup} M_{g_{ω}} (B_{d} (x, ε)) \leq_{T} M_{g} (B_{d} (x, ε)) .$ Hence, $Ls g_{ω} (x) = inf_{ε > 0} \underset{ω \in Ω}{lim sup} M_{g_{ω}} (B_{d} (x, ε))$ $\leq_{T} inf_{ε > 0} M_{g} (B_{d} (x, ε))$ $= \underset{y \to x}{lim sup} g (y) \leq_{T} g (x) .$

Corollary 4.10. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d) and g be a uniformly upper semicontinuous IFS. Then {g_ω} _ω∈Ω is Γ⁺ convergent to g if $IF (X)$ is convergent to g in $D_{\infty}^{+}$ .

The reverse implications of Theorem 4.2 are do not hold for arbitrary nets. The examples given below reveal this fact.

Example 4.11 (Converse of (1) is false) Let ${g_{n}}_{n \in ℕ}$ be a sequence with each g_n is upper semicontinuous IFS on [0, 1] defined as $g_{n} (x) = {\begin{matrix} (x + \frac{1}{5 n}, \frac{9}{10} - x - \frac{1}{5 n}) & if 12n≤ x ≤ 910- 14n; \\ (0, 1), & otherwise, \end{matrix}$ for all $n \in ℕ$ . See the behaviour of ${g_{n}}_{n \in ℕ}$ in Fig. 3.

Fig.3

(a) n = 2, (b) n = 10.

Let

$g (x) = {\begin{matrix} (x, \frac{9}{10} - x), & if 0≤ x≤ 9/10; \\ (0, 1), & otherwise . \end{matrix}$ be the intuitionistic fuzzy set on [0, 1]. Given ε > 0, take $n_{ε} \in ℕ$ with 1/5n_ε < ε. Let $η \in T - {(0, 1)}$ , n ≥ n_ε and x ∈ [g_n] ^η therefore $η \leq_{T} g_{n} (x) = (x + \frac{1}{5 n}, \frac{9}{10} - x - \frac{1}{5 n}) = g (x + \frac{1}{5 n}) .$ Since $x + \frac{1}{5 n} \in B_{d_{e}} (x, ε)$ then x ∈ B_{d
_e} ([g] ^η, ε). Consequently ${g_{n}}_{n \in ℕ}$ is convergent to g in $D_{\infty}^{+}$ . On the other hand ${g_{n}}_{n \in ℕ}$ is unable to converge to g $τ_{\geq_{T}}$ uniformly, since $g (x) <_{T} g_{n} (x)$ for all $x \in [\frac{1}{2}, \frac{13}{20}]$ .

Example 4.12 (Converse of (2) is false) Define a sequence of IFSs on $ℝ$ as $g_{n} (x) = (\frac{1}{3} + \frac{2}{n + 2}, \frac{17}{30} - \frac{2}{n + 2})$ . Let $g (x) = (\frac{1}{3}, \frac{17}{30})$ for all $x \in ℝ$ . It is easy to see that $d_{upper} (g (x), g_{n} (x)) < \frac{2}{n + 2}$ and hence ${g_{n}}_{n \in ℕ}$ is convergent to g upper uniformly. Whereas, for a given $n \in ℕ$ and ε > 0 there exists a η in a way that $(\frac{1}{3}, \frac{17}{30}) <_{T} η <_{T} (\frac{1}{3} + \frac{2}{n + 2}, \frac{17}{30} - \frac{2}{n + 2})$ so $[g_{n}]^{η} = ℝ$ but B_{d
_e} ([g] ^η, ε) =∅. Hence ${g_{n}}_{n \in ℕ}$ is unable to converge g in $D_{\infty}^{+}$ .

Example 4.13 (Converse of (3) is false) For each $n \in ℕ$ , define g_n: $[0, 1] \to T$ given by $g_{n} (x) = {\begin{matrix} (\frac{1}{3}, \frac{1}{2}) & if x ≥ 1n; \\ (1, 0), & otherwise . \end{matrix}$

Let $g (x) = (\frac{1}{3}, \frac{1}{2})$ for all x ∈ [0, 1].

Since $inf_{ε > 0} \underset{n \in ℕ}{lim sup} M_{g_{n}} (B_{d} (x, ε)) = (\frac{1}{3}, \frac{1}{2})$ for all x ∈ [0, 1], hence ${g_{n}}_{n \in ℕ}$ is convergent to g $τ_{\geq_{T}}$ uniformly. Nevertheless, given 0 < ε < 2/3 and $n \in ℕ$ then d_upper (g (1/(n + 1)) , g_n (1/(n + 1))) =2/3 > ε, therefore ${g_{n}}_{n \in ℕ}$ is unable to converge g upper uniformly.

Now, we come to the main goal of this subsection. Let {B_ω} _ω∈Ω be a net in $P (X)$ . Then a subset $B \in P (X)$ is said to be upper limit of {B_ω} _ω∈Ω if

$\underset{ω \in Ω}{lim sup} B_{ω} : = ⋂_{ω_{0} \in Ω} (⋃_{ω \geq ω_{0}} B_{ω}) \subseteq B .$

Definition 4.14. Consider a nonempty set X and a net of set functions {q_ω} _ω∈Ω $q_{ω} : X \to P (T)$ . For a set function $q : X \to P (T)$ if there is some ω₀ ∈ Ω such that for all x ∈ X ⋃_ω≥ω₀q_ω (x) ⊆ q (x) ,

then {q_ω} _ω∈Ω is defined to be upper uniformly convergent to set function q.

The net {g_ω} _ω∈Ω can also be consider as net of set functions as {{g_ω (x)}} _ω∈Ω.

Theorem 4.15. Let {g_ω} _ω∈Ω be a net in $IF (X)$ with (X, d) a metric space. For ε > 0 consider $g_{ω}^{ε} : X \to P [(T, τ_{\leq_{T}})]$ defined as $g_{ω}^{ε} (x) : = {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq_{T}}}$ . Then for all ε > 0 net {g_ω} _ω∈Ω treated as set functions is upper uniformly convergent to g^ε if and only if the net {g_ω} _ω∈Ω is $D_{\infty}^{+}$ -convergent to $g \in IF (X)$ .

Proof. Let {g_ω} _ω∈Ω be a net convergent to g in D⁺. For ε > 0 there is some ω_ε ∈ Ω such that for all ω ≥ ω_ε and for all η in $T$ we have [g_ω] ^η ⊆ B_d ([g] ^η, ε) for all $η \in T$ . Take η = g_ω (x), [g_ω] ^g_ω(x) ⊆ B_d ([g] ^g_ω(x), ε) implies g_ω (x) ∈ g^ε (x). Therefore

⋃_{ω≥ω_ε} {g_ω (x)} ⊆ g^ε (x) forallx ∈ X .

Conversely, let {g_ω} _ω∈Ω be a net of set functions that is upper uniformly convergent to g^ε for all ε > 0. For ε > 0 there exists ω_ε ∈ Ω in a way that for all ω ≥ ω_ε and for all x in X g_ω (x) ∈ g^ε (x). For any x ∈ [g_ω] ^η, we know that g_ω (x) ∈ g^ε (x), therefore we have y_ω ∈ B_d (x, ε) such that $g_{ω} (x) \leq_{T} g (y_{ω})$ . Hence for all $η \in T$ and for all ω ≥ ω_ε [g_ω] ^η ⊆ B_d ([g] ^η, ε). ■

Corollary 4.16. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Then for g in $IF (X)$

$\underset{ω \in Ω}{lim sup} end g_{ω} \subseteq end \bar{g},$

if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$

Proof. Assume that {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ . Then {g_ω} _ω∈Ω is convergent to g^ε upper uniformly by Theorem 4.2. It implies that for any ε > 0, we have ω_ε ∈ Ω in a way that $g_{ω} (x) \leq_{T} M_{g} (B_{d} (x, ε))$ for all ω ≥ ω_ε and for all x ∈ X. Therefore, ⋃_{ω≥ω_ε}endg_ω ⊆ endM_g (B_d (x, ε)). Hence,

$\underset{ω \in Ω}{lim sup} end g_{ω} \subseteq ⋂_{ε > 0} ⋃_{ω \geq ω_{ε}} end g_{ω}$

$\subseteq ⋂_{ε > 0} end M_{g} (B_{d} (\cdot, ε)) = end \bar{g} .$ ■

Theorem 4.17. Let {g_ω} _ω∈Ω be a net of compactly supported upper semicontinuous IFSs in $IF (X)$ for a metric space (X, d). Then for a upper semi continuous IFS g, {g_ω} _ω∈Ω is convergent to M_g (B_d (· , ε)) $τ_{\geq_{T}}$ uniformly for all ε > 0 if and only if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ .

Proof. By virtue of [7, Theorem 5.9]BT we just need to prove sufficiency. We assumed that {g_ω} _ω∈Ω is convergent to M_g (B_d (· , ε)) $τ_{\geq_{T}}$ uniformly. For any ε > 0 we have some ω_ε ∈ Ω in a way that for all ω ≥ ω_ε and for all x ∈ X $g_{ω} (x) \leq_{T} M_{g} (B_{d} (x, ε / 2))$ . If for some ω ≥ ω_ε we have $g_{ω} (x) <_{T} M_{g} (B_{d} (x, ε / 2))$ then it is clear that we can find y_ω ∈ B_d (x, ε/2) in a way that $g_{ω} (x) \leq_{T} g (y_{ω})$ . Therefore g_ω (x) ∈ g^ε (x) .

On the other hand for some ω ≥ ω_ε if we assumed that $(0, 1) <_{T} g_{ω} (x) = M_{g} (B_{d} (x, ε))$ (as for g_ω (x) = (0, 1) proof is trivial), there exit a sequence ${b_{n}}_{n \in ℕ}$ in B_d (x, ε/2) in a way that $lim_{n \to \infty} g (b_{n}) = g_{ω} (x)$ and $g (b_{n}) >_{T} (0, 1)$ for all $n \in ℕ .$ Without loss of generality, we can assume that ${b_{n}}_{n \in ℕ}$ is convergent to $b \in \bar{B_{d} (x, ε / 2)}$ , because g has compact support. But also g is upper semicontinuous, which implies that $\underset{n \to \infty}{lim sup} g (b_{n}) = g_{ω} (x) \leq_{T} g (b)$ . Hence g_ω (x) ∈ g^ε (x). ■

Corollary 4.18. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Then for any ε > 0 we can find ω_ε ∈ Ω in a way that for all x ∈ X and for all ω ≥ ω_ε

$g_{ω} (x) \leq_{T} M_{g} (B_{d} (x, ε))$

if and only if {g_ω} _ω∈Ω in convergent to g in D⁺. Furthermore, if for some ω ≥ ω_ε,

$g_{ω} (x) = M_{g} (B_{d} (x, ε)) then \underset{B_{d} (x, ε)}{Argmax} g \neq \emptyset .$

Now we discuss the relationship between $d_{\infty}^{+}$ -convergence and maximizers.

Proposition 4.19. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d) and g in $IF (X)$ . Then $M_{g_{ω}} (X) \leq_{T} M_{g} (X)$ residually if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ .

Proof. It follows from [7, Theorem 5.9(2)]BT. ■

4.3 Convergence in D_∞

Finally the convergence in D_∞ is characterized in the form of following theorems.

Theorem 4.20. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d) and g in $IF (X)$ . For ε > 0 consider $g_{ω}^{ε} : X \to P [(T, τ_{\leq_{T}})]$ defined as $g_{ω}^{ε} (x) : = {\bar{g_{ω} (B_{d} (x, ε))}}^{τ_{\leq_{T}}}$ . Then {g_ω} _ω∈Ω is convergent to g in D_∞ if and only if:

for all ε > 0 net of set functions ${g_{ω}^{ε}}_{ω \in Ω}$ is convergent to g lower uniformly;

for all ε > 0 net {g_ω} _ω∈Ω treated as set functions is convergent to g^ε upper uniformly.

Theorem 4.21. Let {g_ω} _ω∈Ω be a net of compactly supported upper semicontinuous IFSs in $IF (X)$ for a metric space (X, d). Then for a upper semicontinuous IFS g, the net {g_ω} _ω∈Ω is convergent to g in D_∞ if and only if:

for all ε > 0 {M_{g
_ω} (B_d (· , ε))} _ω∈Ω is convergent to g $τ_{\leq_{T}}$ uniformly;

for all ε > 0 {g_ω} _ω∈Ω is convergent to M_g (B_d (· , ε)) $τ_{\geq_{T}}$ uniformly.

Last but not the least, we discussed some explicit results as an application to variational analysis in the context of convergence in D_∞.

Proposition 4.22. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d) and g in $IF (X)$ . Then M_{g
_ω} (X) = M_g (X) residually if {g_ω} _ω∈Ω is convergent to g in D_∞.

Proof. It follows from Propositions 4.1 and 4.2. ■

Proposition 4.23. Let {g_ω} _ω∈Ω be a net in $IF (X)$ for a metric space (X, d). Let {g_ω} _ω∈Ω is convergent to a upper semi continuous IFS g in D_∞. If the net {x_ω} _ω∈Ω is convergent to $\overset{´}{x}$ and $x_{ω} \in \underset{X}{Argmax} g_{ω}$ for all ω ∈ Ω then $\overset{´}{x} \in \underset{X}{Argmax} g$ .

Proof. By [7, Theorem 5.9(2)]BT we know that, if {g_ω} _ω∈Ω is convergent to g in $D_{\infty}^{+}$ then {g_ω} _ω∈Ω is convergent to M_g (B_d (· , ε)) $τ_{\geq_{T}}$ uniformly for all ε > 0. Therefore for given ε > 0 we can find ω_ε ∈ Ω in a way that for all x in X and for all ω ≥ ω_ε

$g_{ω} (x) \leq_{T} M_{g} (B_{d} (x, ε / 2)) .$

By hypothesis {x_ω} _ω∈Ω is convergent to $\overset{´}{x}$ , so without loss of generality assume that

$d (\overset{´}{x}, x_{ω}) < ε / 2 for all ω \geq ω_{ε} .$

Therefore, $B_{d} (x_{ω}, ε / 2) \subseteq B_{d} (\overset{´}{x}, ε)$ for all ω ≥ ω_ε. Since $x_{ω} \in \underset{X}{Argmax} g_{ω}$ , we have

$g_{ω} (x_{ω}) = M_{g_{ω}} (X) \leq_{T} M_{g} (B_{d} (x_{ω}, ε / 2))$

$\leq_{T} M_{g} (B_{d} (\overset{´}{x}, ε)) \leq_{T} M_{g} (X),$

for all ω ≥ ω_ε. By Proposition 4.3 M_{g
_ω} (X) = M_g (X) residually, therefore

$M_{g} (B_{d} (\overset{´}{x}, ε)) = M_{g} (X) residually .$

By the upper semicontinuity of g $g (\overset{´}{x}) = \underset{x \to \overset{´}{x}}{lim sup} g (x) = inf_{ε > 0} M_{g} (B_{d} (\overset{´}{x}, ε)) = M_{g} (X) .$ ■

Now we show that the above result result is not true in general, consider the constant {g_ω} _ω∈Ω on the interval [0, 1] with g_ω (x) = (1, 0) and the IFS $g (x) = {\begin{matrix} (1, 0), & when 0< x≤ 1; \\ (1 / 2, 1 / 4), & when x=0 . \end{matrix}$ With the above definitions it is straight forward to see that {g_ω} _ω∈Ω is convergent to g in D_∞. But on the other hand for all ω ∈ Ω $0 \in \underset{X}{Argmax} g_{ω}$ whereas $0 \notin \underset{X}{Argmax} g$ .

Footnotes

Acknowledgment

The work was supported by the National Science Centre, Decision No. 2016/23/N/HS4/019. The authors would like to thank the editors and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper.

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