¹ PG and Research Department of Mathematics, Muthurangam Government Arts College (Autonomous), Vellore, India
² Department of Mathematics, Wallaga University, Nekemte, Ethiopia
³ Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman 19328, Jordan

Krishnamoorthy Kavitha
Roles: Conceptualization, Writing – Original Draft Preparation

Prakasam Muralikrishna
Roles: Conceptualization, Writing – Review & Editing

Gamachu Adugna Ganati
Roles: Writing – Review & Editing

Aseel Smerat
Roles: Writing – Review & Editing

∗:[0,1]×[0,1]→[0,1] in binary exists t− norm if it accomplishes the listed below factors:

∗ can be stated that in lieu of continuous t− norm if it can be considered continuous.

⋄:[0,1]×[0,1]→[0,1] is a binary operation constitutes a t− co-norm if the aforementioned accomplishes the ones mentioned underneath factors

Examine the set that is not empty V along regarding the operations in algebra +,⋅ fulfill, (V,+) is a group, as well as with respect to scalar multiplication,

The triplet (V,+,⋅) constitutes a vector space.

Over a vector space X , the fuzzy set N∗ in X×[0,∞) comprises a fuzzy norm with regard to X While it accomplishes these specifications,

(X,N∗,∗) is therefore referred to represents fuzzy normed linear space (FNLS).

Hesitant Fuzzy Normed Linear Space (HFNLS): Considering a vector space V∗ across the aforementioned field F,∗ is composed of t-norm, and H:V∗×[0,∞)→P∗[0,1] appears with regard to hesitant fuzzy set containing the following properties ∀x11,y11∈V∗ and t11,t2>0 .

As an a sequence {vn} along with HFNLS (V,H), referred to as converges regarding v1∈V , we could find N incorporating H(vn−v1,t11∗)⊃U∗∖S∗ ∀n11≥N (or) limn→∞H(vn−v1,t11∗)=U∗ .

A cauchy sequence is defined as a sequence {vn} in a HFNLS (V,H,∗) , if for every ∅∗⊂S∗⊂U∗,t∗>0 there exists a numerical value N together with

H(vm1−vn1,t11∗)⊃U∗∖S∗forallm1,n1≥N.(or)limn1→∞H(vn−v,t11∗)=U∗.

H′(P⋆,Q⋆) represents a vector space.

Observe that as a R1⋆,R2⋆∈H′(P⋆,Q⋆) and p∈P⋆,

(R1⋆+R2⋆)(p)=R1⋆(p)+R2⋆(p)(λ1R1⋆)(p)=λ1R1⋆(p)

Since R1⋆ and R2⋆ are WHFB, ∀S1∗∈P(0,1) , there exists MS1∗1,MS1∗2>0, so that ∀p∈P⋆,∀t∈ℝ We own,

H1(p,tMS1∗1)⊇S1∗⟹H2(R1⋆(p),t)⊇S1∗

and

H1(p,tMS1∗2)⊇S1∗⟹H2(R2⋆(p),t)⊇S1∗

Now, for any pair of scalars k1 and k2 ,that are not zero we have,

H1(p,t2|k1|MS1∗1)=H1(k1p,t2MS1∗1)⊇S1∗⟹H2(R1⋆(k1p),t2)⊇S1∗

and

H1(p,t2|k2|MS1∗2)=H1(k2p,t2MS1∗2)⊇S1∗⟹H2(R2⋆(k2p),t2)⊇S1∗

Choose MS!∗=2|k1|MS1∗1+2|k2|MS1∗2. Now,

H1(p,tMS1∗)=H1(p,t2|k1|MS1∗1+2|k2|MS1∗2)⊇S1∗⟹H1(p,t2|k1|MS1∗1)⊇S1∗andH1(p,t2|k2|MS1∗2)⊇S1∗⟹H2(R1⋆(k1p),t2)⊇S1∗andH2(R2⋆(k2p),t2)⊇S1∗⟹H2(k1R1⋆(p)+k2R2⋆(p),t)⊇S1∗⟹H2((k1R1⋆+k2R2⋆)(p),t)⊇S1∗⟹k1R1⋆+k2R2⋆∈H′(P⋆,Q⋆)

When k1 and k2 equal zero, clearly,

⟹k1R1⋆+k2R2⋆∈H′(P⋆,Q⋆)

As a result, H′(P⋆,Q⋆) is a linear space.

Consider a linear space Q⋆ such that Q⋆=ℝorℂ. Create a corresponding function H2:Q⋆×ℝ→[0,1] via H2(p11,t11)={∅∗,ift11≤|p11|U∗,ift11>|p11|

Afterwards, it is simple to confirm that H2 is a hesitant fuzzy norm on Q⋆ . Consequently, (Q⋆,H2) is a HFNLS.

(Q⋆,H2) fulfills this requirement (i.e;∀t11>0,H(p11,t11)≠∅∗impliesp11=0). For t11>0,H2(p11,t11)≠∅∗implies|p11|<t11,∀t11>0implies|p11|=0⟹p11=0.

Consequently, ∀t11>0,H2(p11,t11)≠∅∗⟹p11=0. Therefore, if ∥∥S1∗2 be the S1∗− norms of H2, Afterwards, We own ,

∥p11∥S1∗2=⋀{t11>0:H2(p11,t11)⊇S1∗},S1∗∈P(0,1)=⋀{t11>0:t11>|p11|}=|p11|

A SHFB linear operator described as (P⋆,H1) and (Q⋆,H2) where (Q⋆,H2) is by remark( 1 ) is referred to as SHFB linear functional over (P⋆,H1). We indicate by P⋆▫ the collection of every SHFB linear functionals over (P⋆,H1).

A WHFB linear operator described as (P⋆,H1) and (Q⋆,H2) where (Q⋆,H2) is by remark( 1 ) is referred to as WHFB linear functional over (P⋆,H1). We indicate by P⋆◊ the collection of every WHFB linear functionals over (P⋆,H1).

Assume that the HFNLS (P⋆,H1) satisfies

and

Suppose for the p11≠0,H(p11,•) constitutes a continuous function of ℝ along with strictly increasing upon the subset

Moreover, P⋆▫ be considered the vector space of all SHFB linear functionals over P⋆.

Allow R⋆∈P⋆▫ along with {∥∥S1∗1:S1∗∈P(0,1)} be the family of S1∗− norms of H1 ,We describe

∥R⋆∥S1∗∗=⋁p11∈P⋆,p11≠0|R⋆(p11)|∥p11∥′(U∗/S1∗),∀S1∗∈P(0,1)

Then {∥∥S1∗∗:S1∗∈P(0,1)} is an ascending family of norms on P⋆▫. Because R⋆ is strongly hesitant fuzzy bounded, there exists M>0 so that,

|R⋆(p11)|≤M∥p11∥′(U∗/S1∗),∀p11 along with thus ∥R⋆∥S1∗∗≤M,∀S1∗∈P(0,1), again we define,

H∗(R⋆,s11)={⋁{S1∗∈P(0,1):∥R⋆∥S1∗∗≤s11}for(R⋆,s11)≠(0,0)0for(R⋆,s11)=(0,0)

Next, by therorem 2.2 [2], Consequently H∗ constitutes a hesitant fuzzy norm on P⋆▫ also therefore (P⋆▫,H∗) is a HFNLS. We summon P⋆▫, the SHFDS of P⋆.

Presume according to the HFNLS (P⋆,H1) satisfies

and

Presume that for p11≠0,H(p11,•) is a continuous function of ℝ along with strictly increasing over the subset

In addition to P⋆◊ be consider the vector space of all WHFB linear functionals over P⋆. Allow R⋆∈P⋆◊ along with {∥∥S1∗1:S1∗∈P(0,1)} be the family of S1∗− norms of H1 , we describe

∥R⋆∥′S1∗=⋁p11∈P⋆,p11≠0|R⋆(p11)|∥p11∥′(U∗/S1∗),∀S1∗∈P(0,1)

Then {∥∥S1∗∗:S1∗∈P(0,1)} is an ascending family of norms over P⋆◊. Because R⋆ is WHFB, there exists MS1∗>0 so that,

|R⋆(p11)|≤MS1∗∥p11∥′(U∗/S1∗),∀p11 and thus ∥R⋆∥′S1∗≤MS1∗,∀S1∗∈P(0,1), again we define,

H◊(R⋆,s11)={⋁{S1∗∈P(0,1):∥R⋆∥′S1∗≤s11}for(R⋆,s11)≠(0,0)0for(R⋆,s11)=(0,0)

Afterwards by therorem 2.2, ² consequently H◊ constitutes a hesitant fuzzy norm on P⋆◊ alongside therefore (P⋆◊,H◊) is a HFNLS. We summon P⋆◊, the WHFDS of P⋆.

Assume that the hesitant fuzzy normed linear spaces (U,H1) fulfill (1) and (2) and {∥∥S1∗:S1∗∈P(0,1)} represent the family that corresponds S1∗− norms regarding the N on P⋆ described by

∥x11∥S1∗=⋀{t11:H1(x11,t11)⊇S1∗,S1∗∈P(0,1)}

Next, with regard to any increasing (or decreasing) sequence {S1n∗} underneath

P(0,1),S1n∗→S1∗∈P(0,1)⟹∥x11∥S1n∗→∥x11∥S1∗,∀x11∈P⋆

Suppose in which the HFNLS (P⋆,H1) satisfies (1) and (2) .Moreover, (Q⋆,H2) is the HFNLS described by remark ( 1 ). Allow H′ be the hesitant fuzzy norm on P⋆◊ described b

where {∥∥′S1∗:S1∗∈P(0,1)} refers to the ascending family of norms. If moreover,

∥∥′S1∗:P⋆◊→ℝ+ become a function described by

Afterwards ∥R⋆∥′S1∗=∥R⋆∥″S1∗.

According to theorem (2), for each increasing (or) decreasing sequence {S1n∗} inside P(0,1) , we have like a sequence.

If R⋆=0 afterwards ∥R⋆∥″S1∗=0=∥R⋆∥′S1∗,∀p∈P(0,1).

Consider R⋆≠0. Let S0∗∈P(0,1) and placing ∥R⋆∥′S0∗=t0 afterwards t0>0 arriving lemma (1) we obtain H◊(R⋆,t0)⊇S0∗. Now from (6) we own, ∥R⋆∥″S0∗≤t0=∥R⋆∥′S0∗. Thus

Following that, let r>∥R⋆∥″S0∗ which implies there exists t1<r so that H◊(R⋆,t11)⊇S0∗ ⟹ there exists t11<r so that

⋃{S1∗∈P(0,1):∥R⋆∥′S1∗≤t111}⊇S0∗.

If

⋃{S1∗∈P(0,1):∥R⋆∥′S1∗≤t11}=S0∗

there exists a sequence {S1n∗} in P(0,1) so that S1n∗→S0∗ and ∥R⋆∥′S1n∗≤t11,∀n.

If ⋃{S1∗∈P(0,1):∥R⋆∥′S1∗≤t11}⊇S0∗, the implication is that,

Now from (8) and (9) we own ∥R⋆(p11)∥′S0∗=∥R⋆(p11)∥″S0∗.

Since S0∗∈P(0,1) is arbitrary, consequently ∥R⋆(p11)∥′S1∗=∥R⋆(p11)∥″S1∗,∀R⋆∈U◊,∀S1∗∈P(0,1). ◻

Assume that the HFNLS (P⋆,H1) satisfies (1) and (2). Then the WHDS (P⋆◊,H◊) is a complete HFNLS.

Examine a cauchy sequence {Rg⋆} in (P⋆◊,H◊). So,

limg→∞H◊(Rg⋆−Rg+i⋆,t11)=U∗,∀t11>0,i=1,2,…

Next, according to property 1.1² and by applying the lemma mentioned above, we have

Now,

|Rg⋆(p11)−Rg+i⋆(p11)|=|(Rg⋆−Rg+i⋆)(p11)|≤∥Rg⋆−Rg+i⋆∥′S1∗∥p11∥′U∗∖S1∗,∀S1∗∈P(0,1),∀i∈P⋆.⟹|Rg⋆(p11)−Rg+i⋆(p11)|→0asg→∞,∀p11∈P⋆by(11)

Consequently {Rg⋆(p11)} is a cauchy sequence inside Q⋆. Because (Q⋆,∥∥) is complete, consequently, limn→∞Rg⋆(p11)=y is present in (Q⋆,∥∥). Because this holds true for all p in (P⋆,H1) and allow limn→∞Rg⋆(p11)=R⋆(p11). Then, it is simple to confirm that R⋆ is linear. Furthermore,

∥Rg⋆∥′S1∗−∥Rg+i⋆∥′S1∗≤∥Rg⋆−Rg+i⋆∥′S1∗→0,asg→∞,∀S1∗∈P(0,1)

via (11). Consequently the sequence {∥Rg⋆∥′S1∗} of real numbers that are positive is cauchy, for all S1∗∈P(0,1) accordingly, It corresponds to convergent. Consequently, the sequence {∥Rg⋆∥′S1∗} is bounded or every S1∗∈P(0,1) as a result there exists kS1∗>0 for every S1∗∈P(0,1) so that,

∥Rg⋆∥′S1∗≤kS1∗forn=1,2,…

Now,

|R⋆(p11)|=limg→∞|Rg⋆(p11)|≤kS1∗∥p11∥′U∗∖S1∗,∀S1∗∈P(0,1),∀p11∈P⋆.

It follows that R⋆ is bounded for each S1∗∈P(0,1) and hence R⋆ is WHFB. Thus R⋆∈P⋆◊. We now obtain from (11)

∥Rg⋆−Rg+p11⋆∥′S1∗→0,asg→∞,∀S1∗∈P(0,1)

Consequently, for any ϵ1>0, the existence of HS1∗(ϵ1) so that ∥Rg⋆−Rg+i⋆∥′S1∗<ϵ1,∀g≥HS1∗(ϵ1). After that,

|Rg⋆(p11)−Rg+i⋆(p11)|≤∥Rg⋆−Rg+i⋆∥′S1∗∥p11∥′U∗∖S1∗<ϵ1∥p11∥′U∗∖S1∗,∀g≥HS1∗(ϵ1),∀p11∈P⋆,i=1,2…

Letting i→∞ we obtain

|Rg⋆(p11)−R⋆(p11)≤ϵ1∥p∥′U∗∖S1∗,∀g≥HS1∗(ϵ1),∀p11∈P⋆,⟹⋁p11∈P⋆,p11≠0|(Rg⋆−R⋆)(p11)|∥p11∥′U∗∖S1n∗≤ϵ1,∀g≥HS1∗(ϵ1)⟹∥Rg⋆−R⋆∥′S1∗≤ϵ1,∀g≥HS1∗(ϵ1)∥Rg⋆−R⋆∥′S1∗→0asg→∞,∀S1∗∈P(0,1)⟹limg→∞H◊(Rg⋆−R⋆)=U∗,∀t>0.

⟹(P⋆◊,H◊) is a complete HFNLS.

Assume that the HFNLS (P⋆,H1) satisfies (1) and (Q⋆,H2) be the HFNLS defined by theorem (1). We describe P⋆S1∗▫ be considered the collection of all linear functionals described as (P⋆,H1) to (Q⋆,H2) which are bounded in relation to ∥∥S1∗1 along with ∥∥S1∗2 here ∥∥S1∗1 and ∥∥S1∗2 indicate the S1∗− norms of H1 and H2 correspondingly for S1∗∈P(0,1).

Let (Q⋆,H2) be the HFNLS described by theorem (1) and (P⋆,H1) be a HFNLS fulfilling (1)

Furthermore, R⋆∈P⋆S1∗▫⟹|R⋆(p11)|≤MS1∗∥p∥S1∗1 regarding a specific S1∗∈P(0,1) .

R⋆∈P⋆◊⟹|R⋆(p11)|≤MS1∗∥p11∥S1∗1,∀S1∗∈P(0,1).

Thus, it may be inferred from above that

Now,

R⋆∈∩P⋆S1∗▫⟹R⋆∈P⋆S1∗▫,∀S1∗∈P(0,1).

⟹ for each S1∗∈P(0,1), there exists NS1∗ so that

From (12) and (13) we have ∩S1∗P⋆▫=U◊

We denote by ∥fS1∗∥S1∗ the norm of fS1∗ with respect to the norm ∥∥S1∗ and ∥fS1∗∥U∗∖S1∗ denotes the norm of fS1∗ with respect to the norm ∥∥U∗∖S1∗ where ∥∥S1∗ is the S1∗− norm of H1 and fS1∗∈P⋆S1∗▫.

(Hahn - Banach) Let (P⋆,H1) be a HFNLS fulfills (1) and (2) in addition to W possess a subspace of P⋆. Let f∗ possess a SHFB linear functional defined over (W,H1). Next, over every S1∗∈P(0,1), there exists fS1∗∗∈P⋆U∗∖S1∗▫ It constitutes a expansion of f∗ (i.e; fS1∗∗(x)=f∗(x),∀x∈W ) along with if f∗≠0 subsequently H(f∗,∥fS1∗∗∥U∗∖S1∗)⊇S1∗ where H constitutes the hesitant fuzzy norm on W▫.

Considering that f∗:(W,H1)→(Q⋆,H2) is a SHFB linear functionals thus

∥f∗∥S1∗▫=⋁p∈W,x≠0|f∗(p11)|∥p11∥U∗∖S1∗1

is finite along with the function H determined by

is a hesitant fuzzy norm on W▫.

Let f:(W,∥∥U∗∖S1∗1)→(Q⋆,∥∥S1∗2) be uniformly bounded for all S1∗∈P(0,1) where ∥∥U∗∖S1∗1 denote the U∗∖S1∗− norms of H1 along with ∥∥S1∗2 is provided via

∥x∥S1∗2=|x|,∀x∈Q⋆(Q⋆=ℝorℂ).

According to the HBT upon normed linear space, each S1∗∈P(0,1) there is a linear functional that is bounded, say fS1∗∈P⋆U∗∖S1∗▫ which extends f in such a way that

Now, H(f,∥fS1∗∥U∗∖S1∗) =⋃{S2∗∈P(0,1):∥f∥S1∗▫≤∥fS1∗∥U∗∖S1∗}forf≠0.

⟹H(f,∥fS1∗∥U∗∖S1∗)⊇S1∗ by using (15)

Suppose f represents a linear functional of WHFB, afterwards the Hanh-Banach theorem’s results are comparable. Because f is WHFB indicates which f is bounded with respect to the S1∗− norms according to the H1 along with H1 for every S1∗∈P(0,1).

Let (P⋆,H1) be a HFNLS satisfying (1) and (2) and x0≠0∈P⋆. Then for each S1∗∈P(0,1), the existence of fS1∗∈P⋆U∗∖S1∗▫ so that

∥fS1∗∥U∗∖S1∗=U∗andfS1∗(p0)=∥p0∥U∗∖S1∗1.

Since (P⋆,H1) is a HFNLS satisfying (1), then (P⋆,∥∥U∗∖S1∗1) is a normed linear space for every S1∗∈P(0,1) by HBT upon the normed linear space, (P⋆,∥∥U∗∖S1∗1) there exists fS1∗∈P⋆U∗∖S1∗▫ such that ∥fS1∗∥U∗∖S1∗=U∗ and fS1∗(p0)=∥p0∥U∗∖S1∗1. ◻

If the hesitant fuzzy normed linear space (P⋆,H1) meets (1) and (2), then

x0≠0∈P⋆,

H1(x,⋁f∈P⋆U∗∖S1∗▫,f≠0|f(x)|∥f∥U∗∖S1∗)⊇U∗∖S1∗,∀S1∗∈P(0,1)

during which P⋆U∗∖S1∗▫ as stated in definition (12)

Let ∥∥U∗∖S1∗ be the corresponding U∗∖S1∗− norm of H1. Thus (P⋆,∥∥U∗∖S1∗) for each is a normed linear space S1∗∈P(0,1). Using the HBT upon (P⋆,∥∥U∗∖S1∗) We own,

Hence, for x≠0 we have

H1(x,tα)=⋃{S2∗∈P(0,1):∥x∥S2∗1≤tS1∗}⟹H1(x,tS1∗)⊇U∗∖S1∗

by (16)

i.e;H1(x,⋁f∈P⋆U∗∖S1∗▫,f≠0|f(x)|∥f∥U∗∖S1∗)⊇U∗∖S1∗,∀S1∗∈P(0,1).

Let us consider a HFNLS (P⋆,H) . We establish a set B(p,S1∗,t∗) as

B(p,S1∗,t∗)={y1:H(p−y1,t∗)⊃U∗∖S1∗}.

Let us consider a HFNLS (P⋆,H) . If we describe τ1={K⊂P⋆:p∈Kiff∃t>0and∅∗⊂S1∗⊂U∗sothatB(p,S1∗,t)⊂P⋆}, after which τ1 is a topology on (P⋆,H).

Let t0=min{t1,t2,…,tn} and S0∗=∩{S1∗,S2∗,…,Sn∗} then

B(p,S0∗,t0)⊂⋂nl=1Kl=K.ThusK∈τ1.

Given that x is any point in K , it follows that K∈τ1. As a consequence, τ1 is a topology on P⋆.

Members of τ1 are referred to as open sets in (P⋆,H).

Let the hesitant fuzzy normed linear space (P⋆,H) fulfill (1). An interior point with regard to some ∥∥S1∗ is defined as a point p∈O⊂P⋆ , if for a given ϵ>0, the existence of a subset L of O so that, y1∈L⟹∥p−y1∥S1∗<ϵ. ( ∥∥S1∗ denotes the S1∗− norms of H,∅∗⊂S1∗⊂U∗)

We describe

LS1∗(x,t∗)={y1∈P⋆:∥x−y1∥S1∗<t∗},t>0

as well as is referred to as an open ball in (P⋆,∥∥S1∗) where ∥∥S1∗ denotes the S1∗− norms of H and we indicate

LS1∗pl(p,mpl)={y1∈Ol:∥p−y1∥S1∗pl<mpl}sinceOl⊂P⋆

where the HFNLS (P⋆,N) satisfies (1).

Assume that the HFNLS (P⋆,H) satisfies (1). We specify

τ2={O⊂P⋆:if and only ifallpointsxofOareinterior points concerningafew∥∥S1∗,S1∗reliesonx,S1∗∈P(0,1)along with∥∥S1∗indicates theS1∗−norms ofH}

then τ2 is a topology on (P⋆,H).

Assume that the HFNLS (P⋆,H) satisfies (1) and (2). Consequently, τ1 and τ2 are equivalent.

Let O∈τ2. Every point p in O is therefore an interior point with regard to a ∥∥S1∗, here S1∗ depends on p. Let p0∈O. Consequently, p0 is an interior point concerning a few ∥∥S1p0∗. So there exists tp0>0, so that LS1p0∗(x0,tx0)⊂O.

i.e.,{y:∥p0−y∥S1p0∗<tp0}⊂O⟹{y:∥p0−y∥U∗∖S2p0∗<tp0}⊂O

here S1p0∗=U∗∖S2p0∗.

Now for y∈P⋆ with

H(p0−y,tp02)⊃U∗∖S2p0∗⟹∥p0−y∥U∗∖S2p0∗≤tp02≤tp0⟹B(p0,S2p0∗,tp02)⊂{y:∥p0−y∥U∗∖S2p0∗<tp0}⊂O.

Therefore, O can be written as a union of members of τ1 and hence O∈τ1. Next, let O∈τ1, Consequently, O can be written as a union of sets of the type B(p,S1∗,t(p)).

Examine y∈B(p,S10∗,t0),S10∗∈P(0,1),t0>0. Then H(p0−y,t0)⊃U∗∖S10∗. Substantially the continuity of H(p0−y,•). (Because of H fulfills ([2])), there exists t′<t0 so that, H(p0−y,t′(p0−y))⊃U∗∖S10∗.

Allow y0∈B(p0,S10∗,t0), consequently H(p0−y0,t0)⊃U∗∖S10∗.

Choose S20∗ so that, H(p0−y0,t0)⊇S20∗⊃U∗∖S10∗.

⟹∥p0−y0∥S20∗(p0,y0)<t0

by (17). Let y∈S20∗(y0,t00) where t00=t0−∥p0−y0∥S20∗2⊃∅∗.

Now

∥p0−y∥S20∗≤∥p0−y0∥S20∗+∥y0−y∥S20∗≤∥p0−y0∥S20∗+t0−∥p0−y0∥S20∗2=t0+∥p0−y0∥S20∗2≤t0

As a result, H(p0−y,t0)⊇S20∗⊃U∗∖S10∗,∀y∈S20∗(y0,t00) by (2).

⟹y∈B(p0,S10∗,t0). Thus y∈S20∗(y0,t00)⊂B(p0,S10∗,t0).

Consequently, y0 is an interior point with regard to ∥∥S20∗.

Thus, each element p∈B(p0,S20∗,t0) is an interior point along with so B(p0,S20∗,t0)∈τ2. Accordingly, O∈τ2.

Open Mapping Theorem on Hesitant Fuzzy Norm

Consider two S1∗ (∅∗⊂S1∗⊂U∗) complete hesitant fuzzy normed linear spaces (P⋆,H1) , (Q⋆,H2) that satisfy (1) and (2). Let R⋆:(P⋆,H1)→(Q⋆,H2) be a SHFB linear operator. If R⋆ is onto then R⋆ is an open mapping.

Considering that (P⋆,H1) and (Q⋆,H2) are S1∗− complete hesitant fuzzy normed linear spaces, for every S1∗∈P(0,1), as a result P⋆ and Q⋆ are complete normed linear spaces in relation to ∥∥S1∗1 and ∥∥S1∗2∀S1∗∈P(0,1), where ∥∥S1∗1 and ∥∥S1∗2 are S1∗− norms of H1 and H1 correspondingly. Again, because R⋆:(P⋆,H1)→(P⋆,H2) is a SHFB linear operator then, R⋆:(P⋆,∥∥S1∗1)→(Q⋆,∥∥S1∗2) constitutes a uniformly bounded linear operator (BLO) for all S1∗∈P(0,1), and so R⋆:(P⋆,∥∥S1∗1)→(Q⋆,∥∥S1∗2) is a BLO for every S1∗∈P(0,1). Thus R⋆:(P⋆,∥∥S1∗1)→(Q⋆,∥∥S1∗2) is a BLO from a banach space (P⋆,∥∥S1∗1) onto a banach space (Q⋆,∥∥S1∗2) for all S1∗∈P(0,1). Thus, using the OMT upon Banach spaces, it can be concluded that R⋆ is an open mapping from (P⋆,∥∥S1∗1) to (Q⋆,∥∥S1∗2) for all S1∗∈P(0,1).

Let O⊂P⋆ be open concerning (P⋆,τ1). Given that H1 fulfills (1) and (2), by remark (7) O is open in relation to (P⋆,τ2). Let x∈O, next, there is S1∗∈P(0,1) so that x constitutes an interior point of O in (P⋆,∥∥S1∗1). Because R⋆ is open from (P⋆,∥∥S1∗1) to (Q⋆,∥∥S1∗2). R⋆(x) is an interior point of R⋆(O) in (Q⋆,∥∥S1∗2). All points in R⋆(O) is therefore an interior point with regard to some ∥∥S1∗2. As a result, R⋆(O) is open in (Q⋆,τ2) and hence in (Q⋆,τ1) also. (Given that H2 satisfies (1) and (2). Therefore R⋆ is an open mapping. ◻

Given that R⋆ is WHFB, it is bounded with regard to ∥∥S1∗1 and ∥∥S1∗2 for every S1∗∈P(0,1). Therefore, if R⋆ is WHFB, then the previous theorem also holds. If two hesitant fuzzy normed linear spaces that satisfy (1) are (P⋆,H1) and (Q⋆,H2) . then (P⋆,∥∥S1∗1) and (Q⋆,∥∥S1∗2) are normed linear spaces for all S1∗∈P(0,1) where ∥∥S1∗1 and ∥∥S1∗2 denote the S1∗− norms of H1 and H2 correspondingly. If we specify

∥(p,y)∥S1∗=∥p∥S1∗1+∥y∥S1∗2,∀S1∗∈P(0,1)and∀(p,y)∈P⋆×Q⋆,

then {∥∥S1∗1:S1∗∈P(0,1)} is an ascending family of norms on P⋆×Q⋆.

Closed Graph Theorem on Hesitant Fuzzy Norm

Consider two S1∗− complete (∅∗⊂S1∗⊂U∗) hesitant fuzzy normed linear spaces (P⋆,H1) and (Q⋆,H2) fulfills (1) and (2) and R⋆:P⋆→Q⋆ have been a linear operator. If the graph of R⋆ be closed with respect to ∥∥S1∗=∥∥S1∗1+∥∥S1∗2,∀S1∗∈P(0,1), then R⋆ is WHFB.

Given that (P⋆,H1) and (Q⋆,H2) have been S1∗− complete fuzzy normed linear spaces for every S1∗∈P(0,1), subsequently P⋆ and Q⋆ are complete normed linear spaces in relation to ∥∥S1∗1 and ∥∥S1∗2 for each S1∗∈P(0,1) where ∥∥S1∗1 and ∥∥S1∗2 indicate the S1∗− norms of H1 and H2 appropriately. By utilizing S10∗∈P(0,1). Thus (P⋆,∥∥S10∗1) as well (Q⋆,∥∥S10∗2) are complete normed linear spaces. As is well known, (P⋆×Q⋆,∥∥0) is a normed linear space with respect to linear operations. ∥∥0 is given by ∥(x,y)∥S10∗=∥x∥S10∗1+∥y∥S10∗2. Now (P⋆,∥∥S10∗1) and (Q⋆,∥∥S10∗2) are complete normed linear spaces, As a result, (P⋆×Q⋆,∥∥S10∗) is a complete normed linear space.

Since the graph of R⋆ that is K(R⋆) is closed with respect to ∥∥S1∗ for all S1∗∈P(0,1), thus K(R⋆) is closed wiht respect to ∥∥S10∗. So K(R⋆) is a closed subset of P⋆×Q⋆ with respect to ∥∥S10∗. Thus (P⋆,∥∥S10∗1) and (Q⋆,∥∥S10∗2) are two Banach spaces and R⋆:(P⋆,∥∥S10∗1)→(Q⋆,∥∥S10∗2) is a closed linear mapping. Consequently, the closed graph theorem over normed linear space implies that, R⋆ is bounded with respect to ∥∥S10∗. Since S10∗∈P(0,1) is arbitrary, thus R⋆ is bounded with respect to ∥∥S1∗,S1∗∈P(0,1). Thus, R⋆ is weakly hesitant fuzzy bounded. ◻

Uniform Boundedness Principle Theorem on Hesitant Fuzzy Norm

Let (P⋆,H1) be an S1∗− complete (∅∗⊂S1∗⊂U∗) HFNLS satisfying (1) and (2) and (Q⋆,H2) be another S1∗− complete (∅∗⊂S1∗⊂U∗) HFNLS satisfying (1) and (2). Let Rn⋆:(P⋆,H1)→(Q⋆,H2) be a sequence of SHFB linear operators. If every one of x∈P⋆ and for each S1∗∈P(0,1), the existence of a constant CS1∗,x>0 (depending on x and S1∗ both) so that H2(Rn⋆(x),CS1∗,x)⊇S1∗,∀n=1,2,…,S1∗∈P(0,1), the existence of MS1∗>0 so that H(Rn⋆,MS1∗)⊇S1∗,∀n=1,2,…, here H is the hesitant fuzzy norm on F(P⋆,Q⋆).

Given that (P⋆,H1) and (Q⋆,H2) are HFNLS fulfillis (1), then (P⋆,∥∥S1∗1) and (Q⋆,∥∥S1∗2) are normed linear spaces for all S1∗∈P(0,1) where ∥∥S1∗1 and ∥∥S1∗2 denote the S1∗− norms of H1 and H2 correspondingly. Again, because (P⋆,H1) is S1∗− complete for all S1∗∈P(0,1) then by remark (8) consequently, (P⋆,∥∥S1∗1) is complete for each S1∗∈P(0,1). Since, Rn⋆:(P⋆,H1)→(Q⋆,H2) is SHFB for all n=1,2,… , thus Rn⋆:(P⋆,∥∥S1∗1))→(Q⋆,∥∥S1∗2) is bounded (Uniformly over S1∗∈P(0,1). )for all n=1,2,… by theorem (9).

Again H2(Rn⋆(x),CS1∗,x)⊇S1∗,∀S1∗∈P(0,1). Now from the definition we have

∥Rn⋆(x)∥S1∗2=⋀{t>0:H2(Rn⋆(x),t)⊇S1∗}⟹∥Rn⋆(x)∥S1∗2≤CS1∗,x,∀S1∗∈P(0,1)

As a result, we have

Thus, according to the Uniform Boundedness Principle Theorem upon normed linear spaces, each S1∗∈P(0,1), there exists a constant MS1∗>0 so that

∥Rn⋆∥S1∗≤MS1∗forn=1,2,…i.e;‖Rn⋆MS1∗‖S1∗≤1

Now from proposition (4.1 in chapter 3), we have

H(Rn⋆MS1∗,1)=⋁{S2∗∈P(0,1):‖Rn⋆MS1∗‖S1∗≤1}⊇S1∗

Therefore, H(Rn⋆,MS1∗)⊇S1∗,S1∗∈P(0,1)and∀n=1,2,…,

This study is a mathematical theory in nature, no experimental or observational datasets were generated or analyzed during the study.

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