1. Introduction
After L.A. Zadeh introduced in his famous paper [
1] the notion of fuzzy set, many researchers wanted to extend the classical mathematical knowledge in this new fuzzy context. The approach is both an internal, meaning developing mathematics from within, as well as an external one, namely responding to some needs and challenges coming from other domains, starting with engineering and even reaching economic and social domains. Therefore, in the paper [
2], starting from the classic notions of pseudo-norm, F-norm and F-space, we introduced the notions of fuzzy pseudo-norm, fuzzy F-norm and fuzzy F-space. An important result is that a fuzzy F-normed linear space is a metrizable topological linear space.
The goals of this research were established in 2016, in the paper [
2]: the study of fuzzy continuous mappings on fuzzy F-spaces and obtaining, in this more general setting, a fuzzy version for principles of functional analysis. The aims of this study are: to introduce different types of fuzzy continuity for mappings between fuzzy F-normed linear spaces and to investigate the relations between them. In this context, an important goal is to compare the newly introduced concepts with the classic continuity from F-normed linear space. We expect to prove that fuzzy continuity and topological continuity are equivalent. Other studied concepts such as strong fuzzy continuity and weak fuzzy continuity will be useful in future papers where we will study the boundedness of linear operators. Last but not least, an important motivation for this approach is the researchers’ interest in these spaces and their generalizations.
Thus, in 2019, Dinda, Ghosh and Samanta introduce the concept of intuitionistic fuzzy pseudo-normed linear space (see [
3]), and then they study, within this context, different types of intuitionistic fuzzy continuities and intuitionistic fuzzy boundedness as well as the relationships between them (see [
4]). In their paper [
5], they study the spectrum and the spectral properties of bounded linear operators in intuitionistic fuzzy pseudo-normed linear spaces.
In 2022, following the notion of fuzzy pseudo-normed linear spaces defined by Nădăban, Wu [
6] introduces the notion of fuzzy pseudo-semi-normed linear space, according to general t-norm.
The structure of the paper is as follows: after the preliminary section, in
Section 3, we introduce different types of continuity: fuzzy continuity, sequentially fuzzy continuity, and strong fuzzy continuity. We also establish relations between them. Such methods have been carried out by Bag and Samanta [
7], Nădăban [
8], and Sadeqi and Solaty Kia [
9] in the case of fuzzy normed linear spaces and by Dinda, Ghosh and Samanta [
4] in the case of intuitionistic fuzzy pseudo-normed linear spaces. As is known, in the functional analysis closed graph theorem, the open mapping theorem and the uniform boundedness principle are of great importance. We will present these theorems in
Section 4, in the context of fuzzy F-spaces. These principles were obtained in the frame of fuzzy normed linear space by Bag snd Samanta [
7] and Sadeqi and Solaty Kia [
9]. We will use a result obtained by Zabreiko [
10], which proved useful even in the classic case in order to prove many important theorems in functional analysis. We can use Zabreiko’s theorem because in [
2] we showed that fuzzy F-spaces are metrizable topological vector spaces.
3. Fuzzy Continuous Mappings
Let and be two FFNL-spaces with corresponding families of F-norms , respectively .
Definition 6. Let be two FFNL-spaces. A mapping is called fuzzy continuous (FC) with respect to and at if Definition 7. Let be two FFNL-spaces. A mapping is said to be uniformly fuzzy continuous (uniformly FC) with respect to and on a subset A of if Example 1. Let E be a vector space and p be a F-norm on E. Then,is a fuzzy F-norm on E (see [2]). Similarly,is a fuzzy F-norm on E. Let , . We will prove that f is uniformly FC on E with respect to and .
Let . We take . Let . Thus, . Hence, . As , we obtain that , namely . Thus, .
Remark 2. If f is uniformly FC on a subset A of , then f is FC on A.
Theorem 4 (Uniform continuity theorem). Let be two FFNL-spaces. Let A be a compact subset of . If is a FC mapping with respect to and on A, then f is uniformly FC with respect to and on A.
Proof. Let
and
. As
is FC on
A, for all
, there exist
,
such that
We can take .
Since A is compact and is an open covering of A, there exist in A such that . Let and , for .
Let
be arbitrary, such that
. As
, there exists
such that
, namely
. Hence,
Thus, .
Therefore, f is uniformly FC on A. □
Proposition 1. Let be two FFNL-spaces. Let be a linear operator. Then, T is FC with respect to and on if and only if T is FC with respect to and at a point .
Proof. “” It is clear.
“
” Let
be arbitrary. We will prove that
T is FC at
z. Let
. As
T is FC at
, there exist
such that
Replacing with , we have: Thus, T is FC at . □
Corollary 1. Let be two FFNL-spaces. Let be a linear operator. Then, T is FC with respect to and on if and only if Remark 3. We note that a mapping is said to be continuous at , if an open neighborhood of we have that is an open neighborhood of in . If f is continuous at each point of , then f is called continuous on .
Theorem 5. Let be two FFNL-spaces. Let . Then f is FC with respect to and at if and only if f is continuous at .
Proof. “⇒” Let
be an open neighborhood of
. It results that there exists
. As
f is FC at
we obtain that there exist
such that
Thus, . Hence , namely is an open neighborhood of in . Therefore, f is continuous in .
“⇐” Let
and
be an open neighborhood of
in
. As
f is continuous at
it result that
is an open neighborhood of
in
. Thus,
such that
, i.e.,
. Thus, for
, we have that
, i.e.,
Hence, f is FC at . □
Proposition 2. Let be two FFNL-spaces. Let be a linear operator and . Then T is continuous at if and only if are continuous at .
Proof. “⇒” Let
. We prove that
is continuous at
. Let
. As
T is continuous at
it result that there exist
such that
Thus, for all we have that , namely . Hence is continuous at .
“⇐”. Let . As is continuous at , we obtain that there exist an open neighborhood of such that we have that , namely . Thus we have that . Therefore T is continuous at . □
Definition 8. Let be two FFNL-spaces. A mapping is said to be sequentially fuzzy continuous (sequentially FC) with respect to and at if Theorem 6. Let be two FFNL-spaces. Let . Then, f is FC with respect to and at if and only if f is sequentially FC with respect to and at .
Proof. “⇒” Let , i.e., .
Let
. As
f is FC at
, we find that there exist
such that
As
, we obtain that, for
,
. Thus
. Therefore,
.
“⇐” We suppose that
f is not FC at
. Thus,
In particular, for
Let
. We take
. Thus, for
, we have:
Thus, . As , it results that , which is in contradiction to our assumption. □
Definition 9. Let be two FFNL-spaces. A mapping is said to be strongly fuzzy continuous (strongly FC) with respect to and at if Theorem 7. Let be two FFNL-spaces. Let . If f is strongly FC with respect to and at a point , then f is FC with respect to and at .
Proof. Let
. As
f is strongly FC at
, we find that there exists
such that
. We take
. Thus,
Hence, f is FC at . □
Proposition 3. Let be two FFNL-spaces. Let be a linear operator. Then, T is strongly FC with respect to and on if and only if T is strongly FC with respect to and at a point .
Proof. “⇒” It is clear.
“⇐” Let
be arbitrary. As
T is strongly FC at
, for
:
Replacing with , we have: Thus, T is strongly FC at . □
Corollary 2. Let be two FFNL-spaces. Let be a linear operator. Then, T is strongly FC with respect to and on if and only if Example 2. Clearly the mapping , by Example 1 is a linear operator. We will prove that T is strongly FC on E. Let . We take . Therefore, As , we obtain Hence, T is strongly FC on E.
Definition 10. Let be two FFNL-spaces. A mapping is said to be weakly fuzzy continuous (weakly FC) with respect to and at if Proposition 4. Let be two FFNL-spaces. Let be a linear operator. Then, T is weakly FC with respect to and on if and only if T is weakly FC with respect to and at a point .
Proof. “⇒” It is clear.
“⇐” Let
be arbitrary. We will prove that
T is weakly FC at
z. Let
. As
T is weakly FC at
, there exists
such that
Replacing with , we have: Thus, T is weakly FC at . □
Corollary 3. Let be two FFNL-spaces. Let be a linear operator. Then, T is weakly FC with respect to and on if and only if Theorem 8. Let be two FFNL-spaces. Let . If f is weakly FC with respect to and at a point , then f is FC with respect to and at .
Proof. Let
. As
f is weakly FC at
, it results that there exists
such that
. We take
. Thus,
Hence, f is FC at . □
Theorem 9. Let be two FFNL-spaces. Let . If f is strongly FC with respect to and at a point , then f is weakly FC with respect to and at .
Proof. Let . As f is strongly FC at , it results that there exists such that . If , then . Hence, f is weakly FC at . □
4. Principles of Fuzzy Functional Analysis
Based on the fact that fuzzy F-spaces are metrizable topological linear spaces and since fuzzy continuity and topological continuity are equivalent, we obtain the closed graph theorem and the open mapping theorem in this context. However, for the beauty of the proof, I made the proof for the next theorem.
Theorem 10 (Closed graph theorem). Let be two fuzzy F-spaces and be a linear operator. Then, T is FC on if and only if is closed in .
Proof. “⇒” Let . Let be a convergent sequence to . Thus, and . As T is FC at , we have that T is sequentially FC at . Hence, . Thus, . Therefore, and is closed.
“⇐”. Let . We define by . We prove that satisfies the assumptions from Zabreiko’s Lemma:
(1)
is subadditive. Indeed,
(3) Let
be a convergent series in
. If
, then
. We assume that
. Hence,
. Thus,
is a Cauchy sequence in
. As
is complete, we have that
is a convergent sequence in
. Let
. As
, we obtain that
. Thus,
. Hence,
Thus,
As satisfies the hypotheses of Zabreiko’s lemma, we find that is continuous. As are continuous at , using Proposition 2, it results that T is continuous at . Applying Proposition 1, we have that T is continuous on . □
Theorem 11 (Bounded inverse theorem). Let be two fuzzy F-spaces. If is a bijective FC linear operator, then is FC.
Theorem 12 (Open mapping theorem). Let be two fuzzy F-spaces. If is a surjective FC linear operator, then T is an open mapping.
In the case of the following two theorems, a certain type of uniform boundedness is involved, characteristic of fuzzy F-normed linear spaces and for this reason we will prove them. We will use a result obtained by Zabreiko [
10], which, even in the classic case, proved to be useful in order to demonstrate many important theorems in functional analysis.
Theorem 13 (Uniform bounded principle). Let be a fuzzy F-spaces, be a fuzzy F-normed linear spaces and be a family of FC linear operators from to . If , we have that , then and the convergence is uniform for .
Proof. Let and . We will prove that satisfies the hypotheses of Zabreiko’s lemma:
It is obvious that is subadditive;
Let and . Then . Applying (PN5), we obtain . Hence, .
Let
be an convergent series in
. Let
. As
is linear and an FC operator, we have that
. If
, the desired inequality takes place. If
, then
. Hence,
By Zabreiko’s lemma, we obtain that is continuous in . Hence, , uniform for . Thus, , uniform for , . Therefore, , uniform for . □
Theorem 14 (Banach–Steinhaus). Let be a fuzzy F-space, be a fuzzy F-normed linear space and be FC linear operators. If
- 1.
, we have ;
- 2.
a dense subset of such that is a Cauchy sequence, for all .
Then:
- 1.
is a Cauchy sequence, for all ;
- 2.
If is a fuzzy F-space and T is defined by , then T is a FC linear operator.
Proof. (1) Let
. As
M is dense in
, there exist
. As
, by previous theorem we have that
, uniform for
. Hence,
As
is a Cauchy sequence, we have that
Hence, is a Cauchy sequence, for all .
(2) As is complete, we have that is convergent. Let T defined by . It is obvious that T is linear. If , then , uniform by . Hence . Therefore T is continuous in . Thus T is continuous on . □
5. Conclusions and Further Works
In this paper, first, we introduced different types of continuity in fuzzy F-normed linear spaces and we established the relationships between them. Then, we obtained the principles of fuzzy functional analysis in the context of fuzzy F-spaces.
The present study will be followed by a detailed analysis of different types of boundedness in fuzzy F-normed spaces. Such concepts have been introduced by several authors in fuzzy normed linear spaces (see [
13]). Further on, motivated by the present papers in the context of probabilistic normed spaces (see [
14]), of those in the context of fuzzy normed linear space (see [
7,
9,
15,
16]) we will study various types of boundedness for linear operator between fuzzy F-spaces.
As is only natural, the first fixed point theorems in fuzzy context were established on fuzzy metric spaces. Interesting results were also obtained in the context of fuzzy normed linear spaces (see [
17]). In the framework of extended rectangular fuzzy b-metric space, in the paper [
18], the authors introduced the concept of Ćirić-type fuzzy contraction and obtained fixed point theorems for these fuzzy contractions. In the paper [
19], the generalization properties of contractive conditions of Ćirić-type in fuzzy metric space were investigated. In the paper [
20], some new fixed point results for nonlinear fuzzy set-valued
-contractions in the context of metric-like spaces are introduced. Our aim is to obtain in future papers fixed point theorems in fuzzy F-normed linear spaces.