EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6304 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Combined Study of Continuities and Boundedness in Neutrosophic Pseudo-Normed Linear Spaces Pandiselvi. M1, Jeyaraman. M1, Mohammad Akram2,∗ 1 PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivagangai, Affiliated to Alagappa University, Karaikudi, Tamil Nadu, India 2 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia Abstract. In this work, we investigate various notions of neutrosophic boundedness and continuity within the framework of neutrosophic pseudo normed linear spaces. Specifically, we introduce and analyze different types of neutrosophic continuity such as pointwise, uniform, and sequential neutrosophic continuity and examine their interrelationships. We also explore several forms of neutrosophic boundedness and establish connections between boundedness and continuity under neutrosophic settings. Illustrative examples are provided to demonstrate the applicability of the introduced concepts. 2020 Mathematics Subject Classifications: 03E72, 46A19, 03B52 Key Words and Phrases: Strongly neutrosophic continuity, Weakly neutrosophic continuity, Sequentially neutrosophic continuity, Uniformly neutrosophic bounded. 1. Introduction In 1992, Felbin [1] introduced the concept of fuzzy norms on linear spaces. Subse- quently, Xiao and Zhu [2] extended this framework by investigating the topological prop- erties of fuzzy normed linear spaces. Bag and Samanta proposed another form of fuzzy norm [3] and further advanced the theory by developing notions such as weak and strong fuzzy boundedness, weak and sequential fuzzy continuity, fuzzy continuity, and the fuzzy norm of linear operators with respect to an associated fuzzy norm [4]. The concept of fuzzy pseudo norms was introduced by S. Nadaban [5]. Dinda et al. [6] explored intu- itionistic fuzzy pseudo normed linear spaces and demonstrated that these possess a more general structure than intuitionistic fuzzy normed spaces. The foundation for intuition- istic fuzzy sets, as a generalization of fuzzy sets, was laid by Atanassov [7], while J. H. Park [8] introduced the notion of intuitionistic fuzzy metric spaces and examined several ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6304 Email addresses: mpandiselvi2612@gmail.com (Pandiselvi. M), jeya.math@gmail.com (Jeyaraman. M), akramkhan 20@rediffmail.com, akram@iu.edu.sa (M. Akram) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 2 of 15 of their fundamental properties. In 1998, Smarandache [9] proposed neutrosophic logic and neutrosophic sets. Building on this, Kirisci and Simsek [10] introduced the concept of neutrosophic metric spaces, which account for degrees of membership, non-membership, and indeterminacy (neutrality). In this context, Jeyaraman et al. [11] investigated mul- tivalued mappings in Hausdorff neutrosophic metric spaces and established several fixed point results. Their work emphasizes the structural richness of neutrosophic metric spaces and provides a foundation for various applications in fixed point theory. The present work focuses on the study of neutrosophic boundedness and various forms of neutrosophic continuity for linear operators within neutrosophic pseudo normed spaces—a generalization of neutrosophic normed spaces. Section 3 highlights the concept of neutrosophic continuities and the intra-relationships among their various types. Section 4 delves into different notions of neutrosophic boundedness. Finally, the interconnections between different forms of continuity and boundedness are discussed, following an initial investigation of intra-relations among the various types of neutrosophic boundedness. 2. Preliminaries This section recalls essential definitions and concepts. These preliminaries form the foundation for the results developed in subsequent sections Definition 1. [12] Let F be a linear space over a field R. A mapping ∥·∥ : F → R is named to be a pseudo norm on F if it holds the following assertions: 1. ∥ϖ∥ ≥ 0, for all ϖ ∈ F, 2. ∥ϖ∥ = 0 ⇔ ϖ = 0, 3. ∥kϖ∥ ≤ ∥ϖ∥, for all ϖ ∈ F, for all k ∈ K with |k| ≤ 1, 4. The strong triangle inequality ∥ϖ +w∥ ≤ ∥ϖ∥+ ∥w∥, for all ϖ,w ∈ F. Definition 2. Let F is a vector space over a field R, and η, ρ, ς are neutrosophic sets on F× R× R, if it meets the following conditions for every ϖ,w ∈ F and o, φ ∈ R (n1) 0 ≤ η(ϖ,φ) ≤ 1; 0 ≤ ρ(ϖ,φ) ≤ 1; 0 ≤ ς(ϖ,φ) ≤ 1; (n2) η(ϖ,φ) + ν(ϖ,φ) + ρ(ϖ,φ) ≤ 3; (n3) for all φ ∈ R with φ ≤ 0, η(ϖ,φ) = 0; (n4) for all φ ∈ R+, η(ϖ,φ) = 1 ⇔ ϖ = φ; (n5) for all φ ∈ R+, η(σϖ,φ) ≥ η(ϖ,φ) if |σ| ≤ 1 for all σ ∈ F; (n6) η(ϖ +w, o+ φ) ≥ min(η(ϖ, o), η(w, φ)) for all o, φ ∈ R+; (n7) lim φ→∞ η(ϖ,φ) = 1; (n8) if there exists 0 < δ < 1 such that η(ϖ,φ) > δ, ∀φ ∈ R+ then ϖ = 0; (n9) η(ϖ, ·) is left continuous on R, for all ϖ ∈ F; (n10) for all φ ∈ R with φ ≤ 0, ρ(ϖ,φ) = 1; (n11) for all φ ∈ R+, ρ(ϖ,φ) = 0 ⇔ ϖ = φ; (n12) for all φ ∈ R+, ρ(σϖ,φ) ≤ ρ(ϖ,φ) if |σ| ≤ 1 for all σ ∈ F; (n13) ρ(ϖ +w, o+ φ) ≤ max(ρ(ϖ, o), ρ(w, φ)) for all o, φ ∈ R+; (n14) lim φ→∞ ρ(ϖ,φ) = 0; Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 3 of 15 (n15) If there exists 0 < δ < 1 such that ρ(ϖ,φ) < δ, ∀φ ∈ R+ then ϖ = 0; (n16) ρ(ϖ, ·) is left continuous on R, for all ϖ ∈ F; (n17) for all φ ∈ R with φ ≤ 0, ς(ϖ,φ) = 1; (n18) for all φ ∈ R+, ς(ϖ,φ) = 0 ⇔ ϖ = 0; (n19) for all φ ∈ R+, ς(σϖ,φ) ≤ ς(ϖ,φ) if |σ| ≤ 1 for all σ ∈ F; (n20) ς(ϖ +w, o+ φ) ≤ max(ς(ϖ, o), ς(w,φ)) for all o, φ ∈ R+; (n21) lim φ→∞ ς(ϖ,φ) = 0; (n22) If there exists 0 < δ < 1 such that ς(ϖ,φ) < δ, ∀φ ∈ R+ then ϖ = 0; (n23) ς(ϖ, ·) is left continuous on R, for all ϖ ∈ F; Then the 4-tuple (F, η, ρ, ς) is named to be a Neutrosophic Pseudo Normed Linear Space [NPNLS]. Note 1. r ∗ s = r and r♢s = r,∀r ∈ [0, 1] is satisfied only when r ∗ s = max{r, s} and r♢s = max{r, s}. Definition 3. Let (F, η, ρ, ς) be NPNLS. A sequence {rn} converges to r ∈ F if and only if lim φ→∞ η(rn − r, φ) = 1, lim φ→∞ ρ(rn − r, φ) = 0 and lim φ→∞ ς(rn − r, φ) = 0. Theorem 2. Let (F, η, ρ, ς) be NPNLS. Then for any 0 < δ < 1 the functions ∥ϖ∥δ , ∥ϖ∥∗δ : F → [0,∞) defined as ∥ϖ∥δ = ∧{φ > 0 : η(ϖ,φ) ≥ δ} is ascending family of pseudo norm on F. ∥ϖ∥∗δ = ∧{φ > 0 : ρ(ϖ,φ) ≤ δ and ς(ϖ,φ) ≤ δ} is a descending family pseudo norm on F. Theorem 3. Let (F, η, ρ, ς) be NPNLS and let η ′ (ϖ,φ) = { ∨{0 < δ < 1 : ∥ϖ∥δ ≤ φ} ifφ > 0 0 ifφ ≤ 0 , ρ ′ (ϖ,φ) = { ∧{0 < δ < 1 : ∥ϖ∥∗δ ≤ φ} ifφ > 0 1 ifφ ≤ 0 and ς ′ (ϖ,φ) = { ∧{0 < δ < 1 : ∥ϖ∥∗δ ≤ φ} ifφ > 0 1 ifφ ≤ 0 then (1). (η ′ , ρ ′ , ς ′ ) is a neutrosophic pseudo norm on F. (2). η = η ′ , ρ = ρ ′ and ς = ς ′ , where ∥ϖ∥δ is an ascending family of pseudo norms and ∥ϖ∥∗δ is descending family of pseudo norms defined in Theorem (2). 3. Neutrosophic continuities of an operators on Neutrosophic Pseudo Normed Linear Spaces This section investigates various forms of continuity for operators on NPNLS, which are essential for understanding operator behavior in neutrosophic settings. Definition 4. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) be NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is named to be neutrosophic continuous at ϖ0 if for any given ϵ > 0 and 0 < δ < 1 there exist γ = γ(δ, ϵ) such that for all ϖ ∈ F, η1(ϖ −ϖ0, γ) > 1− α ⇒ η2(Υ(ϖ)−Υ(ϖ0), γ) > 1− δ Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 4 of 15 ρ1(ϖ −ϖ0, γ) < α ⇒ ρ2(Υ(ϖ)−Υ(ϖ0), γ) < δ and ς1(ϖ −ϖ0, γ) < α ⇒ ς2(Υ(ϖ)−Υ(ϖ0), γ) < δ. Definition 5. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) be NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is referred as sequentially neutrosophic con- tinuous at ϖ0 if for any sequence {ϖn}, ϖn ∈ F and φ > 0, lim φ→∞ η1(ϖn −ϖ0, φ) = 1 ⇒ lim φ→∞ η2(Υ(ϖn)−Υ(ϖ0), φ) = 1, lim φ→∞ ρ1(ϖn −ϖ0, φ) = 0 ⇒ lim φ→∞ ρ2(Υ(ϖn)−Υ(ϖ0), φ) = 0 and lim φ→∞ ς1(ϖn −ϖ0, φ) = 0 ⇒ lim φ→∞ ς2(Υ(ϖn)−Υ(ϖ0), φ) = 0. Theorem 4. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is sequentially neutro- sophic continuous at ϖ0 ∈ F then it is sequentially neutrosophic continuous on F, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Let {ϖn} be a sequence in F and ϖn → ϖ. Then for all φ > 0, lim φ→∞ η1(ϖn −ϖ,φ) = 1, lim φ→∞ ρ1(ϖn −ϖ,φ) = 0 and lim φ→∞ ς1(ϖn −ϖ,φ) = 0. Therefore, lim φ→∞ η1((ϖn −ϖ +ϖ0)−ϖ0, φ) = 1, lim φ→∞ ρ1((ϖn −ϖ +ϖ0)−ϖ0, φ) = 0 and lim φ→∞ ς1((ϖn −ϖ +ϖ0)−ϖ0, φ) = 0. Since Υ is sequentially neutrosophic continuous at ϖ0, for all φ > 0 we have lim φ→∞ η1(Υ(ϖn −ϖ +ϖ0)−Υ(ϖ0), φ) = 1, lim φ→∞ ρ1(Υ(ϖn −ϖ +ϖ0)−Υ(ϖ0), φ) = 0 and lim φ→∞ ς1(Υ(ϖn −ϖ +ϖ0)−Υ(ϖ0), φ) = 0 ⇒ lim φ→∞ η1(Υ(ϖn)−Υ(ϖ) + Υ(ϖ0)−Υ(ϖ0), φ) = 1, lim φ→∞ ρ1(Υ(ϖn)−Υ(ϖ) + Υ(ϖ0)−Υ(ϖ0), φ) = 0 and lim φ→∞ ς1(Υ(ϖn)−Υ(ϖ) + Υ(ϖ0)−Υ(ϖ0), φ) = 0, since Υ is linear. lim φ→∞ η1(Υ(ϖn)−Υ(ϖ), φ) = 1, lim φ→∞ ρ1(Υ(ϖn)−Υ(ϖ), φ) = 0 and lim φ→∞ ς1(Υ(ϖn)−Υ(ϖ), φ) = 0. Since ϖ ∈ F was chosen arbitrarily, it follows that Υ is sequentially neutrosophic contin- uous on F. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 5 of 15 Theorem 5. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is sequentially neutro- sophic continuous at ϖ0 ∈ F then it is sequentially neutrosophic continuous if and only if it is neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Suppose Υ be neutrosophic continuous at ϖ0 ∈ F, {ϖn} be a sequence in F and ϖn → ϖ0. Then for any given ϵ > 0, 0 < δ < 1 there exist γ = γ(δ, ϵ) and α = α(δ, ϵ) such that for all ϖ ∈ F, η1(ϖ −ϖ0, γ) > 1− α ⇒ η2(Υ(ϖ)−Υ(ϖ0), γ) > 1− δ ρ1(ϖ −ϖ0, γ) < α ⇒ ρ2(Υ(ϖ)−Υ(ϖ0), γ) < δ and ς1(ϖ −ϖ0, γ) < α ⇒ ς2(Υ(ϖ)−Υ(ϖ0), γ) < δ. Since {ϖn} converges to ϖ0 there exists n0 ∈ N such that for all n0 ≥ n, η1(ϖ −ϖ0, γ) > 1− α, ρ1(ϖ −ϖ0, γ) < α and ς1(ϖ −ϖ0, γ) < α and since Υ is neutrosophic continuous at ϖ0 ∈ F, we have η2(Υ(ϖ)−Υ(ϖ0), ϵ) > 1− δ, ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) < δ and ς2(Υ(ϖ)−Υ(ϖ0), ϵ) < δ. Hence, Υ(ϖn) → Υ(ϖ0). Conversely, suppose Υ be not neutrosophic continuous at ϖ0 ∈ F. Then there exist w ∈ F such that for any given ϵ > 0, 0 < δ < 1 there exist γ > 0 and α ∈ (0, 1), η1(w−ϖ0, γ) > 1− α ⇒ η2(Υ(w)−Υ(ϖ0), γ) > 1− δ, ρ1(w−ϖ0, γ) < α ⇒ ρ2(Υ(w)−Υ(ϖ0), γ) < δ and ς1(w−ϖ0, γ) < α ⇒ ς2(Υ(w)−Υ(ϖ0), γ) < δ. Hence for γ = α = 1 n+1 there exist wn for n = 1, 2, . . . , such that η1(wn −ϖ0, γ) = η1 ( wn −ϖ0, 1 n+ 1 ) > 1− 1 n+ 1 ⇒ η2(Υ(wn)−Υ(ϖ0), ϵ) ≤ 1− δ, ρ1(wn −ϖ0, γ) = ρ1 ( wn −ϖ0, 1 n+ 1 ) < 1 n+ 1 ⇒ ρ2(Υ(wn)−Υ(ϖ0), ϵ) ≥ δ and ς1(wn −ϖ0, γ) = ς1 ( wn −ϖ0, 1 n+ 1 ) < 1 n+ 1 ⇒ ς2(Υ(wn)−Υ(ϖ0), ϵ) ≥ δ. Therefore, lim n→∞ η1(wn −ϖ0, γ) = 1 ⇒ lim n→∞ η2(Υ(w)n −Υ(ϖ0), ϵ) ̸= 1, lim n→∞ ρ1(wn −ϖ0, γ) = 0 ⇒ lim n→∞ ρ2(Υ(w)n −Υ(ϖ0), ϵ) ̸= 0and lim n→∞ ς1(wn −ϖ0, γ) = 0 ⇒ lim n→∞ ς2(Υ(w)n −Υ(ϖ0), ϵ) ̸= 0. Hence Υ is not sequentially neutrosophic continuous at ϖ0. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 6 of 15 Definition 6. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) be NPNLS. We say that the mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutrosophic continuous at ϖ0 if, for each positive real number ϵ, one can find a δ ∈ (0, 1) such that a certain set of conditions is satisfied for all elements ϖ in F. η2(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ η1(ϖ −ϖ0, γ), ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ρ1(ϖ −ϖ0, γ) and ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ς1(ϖ −ϖ0, γ). Theorem 6. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutrosophic continuous at ϖ0 ∈ F then it is strongly neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Given that Υ possesses strong neutrosophic continuity at the point ϖ0, then corresponding to each ϵ > 0, there exists a positive number γ such that the condition η2 ( Υ(ϖ)−Υ(ϖ0), ϵ ) ≥ η1 ( ϖ −ϖ0, γ ) is satisfied for all ϖ ∈ F, ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ρ1(ϖ −ϖ0, γ) and ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ς1(ϖ −ϖ0, γ). Taking w ∈ F we have ϖ +ϖ0 −w ∈ F. Therefore replacing ϖ by ϖ +ϖ0 −w. We have, η2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≥ η1(ϖ +ϖ0 −w−ϖ0, γ), ρ2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≤ ρ2(ϖ +ϖ0 −w−ϖ0, γ) and ς2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≤ ς1(ϖ +ϖ0 −w−ϖ0, γ). Therefore, η2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0)), ϵ) ≥ η1(ϖ −w, γ), ρ2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0)), ϵ) ≤ ρ1(ϖ −w, γ) and ς2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0)), ϵ) ≤ ς1(ϖ −w, γ). Hence, η2(Υ(ϖ)−Υ(w), ϵ) ≥ η1(ϖ −w, γ), ρ2(Υ(ϖ)−Υ(w)), ϵ) ≤ ρ1(ϖ −w, γ) and ς2(Υ(ϖ)−Υ(w)), ϵ) ≤ ς1(ϖ −w, γ). Hence Υ is strongly neutrosophic continuous at w. Since w ∈ F is arbitrary, Υ is strongly neutrosophic continuous on F. Definition 7. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) be NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is referred as weakly neutrosophic continuous at ϖ0 if for any given ϵ > 0 there exists 0 < γ such that for all ϖ ∈ F, η1(ϖ −ϖ0, γ) ≥ δ ⇒ η2(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ δ, ρ1(ϖ −ϖ0, γ) ≤ δ ⇒ ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ and ς1(ϖ −ϖ0, γ) ≤ δ ⇒ ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 7 of 15 Theorem 7. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutrosophic continuous at ϖ0 ∈ F then it is weakly neutrosophic continuous on F, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Since Υ is weakly neutrosophic continuous at ϖ0, for given ϵ > 0 there exists 0 < δ and 0 < γ such that for all ϖ ∈ F, η1(ϖ −ϖ0, γ) ≥ δ ⇒ η2(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ δ, ρ1(ϖ −ϖ0, γ) ≤ δ ⇒ ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ and ς1(ϖ −ϖ0, γ) ≤ δ ⇒ ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ. Taking w ∈ F we have ϖ+ϖ0 −w ∈ F. Therefore replacing ϖ by ϖ+ϖ0 −w. We have, η1(ϖ −ϖ0, γ) ≥ δ ⇒ η2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≥ δ ⇒ η2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0), ϵ) ≥ δ ⇒ η2(Υ(ϖ)−Υ(w)−Υ(ϖ0), ϵ) ≥ δ, ρ1(ϖ −ϖ0, γ) ≥ δ ⇒ ρ2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≤ δ ⇒ ρ2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0), ϵ) ≤ δ ⇒ ρ2(Υ(ϖ)−Υ(w)−Υ(ϖ0), ϵ) ≤ δ and ς1(ϖ −ϖ0, γ) ≥ δ ⇒ ς2(Υ(ϖ +ϖ0 −w)−Υ(ϖ0), ϵ) ≤ δ ⇒ ς2(Υ(ϖ) + Υ(ϖ0)−Υ(w)−Υ(ϖ0), ϵ) ≤ δ ⇒ ς2(Υ(ϖ)−Υ(w)−Υ(ϖ0), ϵ) ≤ δ. Since w ∈ F is arbitrary, Υ is weakly neutrosophic continuous on F. Example 1. Let (F, ∥·∥) be a pseudo normed linear space and η, ρ, ς : F ×R → [0, 1] be defined by η(ϖ,φ) =  1 if φ > 0, ∥ϖ∥ < φ φ φ+∥ϖ∥ if φ > 0, ∥ϖ∥ ≥ φ 0 if φ ≤ 0 , ρ(ϖ,φ) =  0 if φ > 0, ∥ϖ∥ < φ ∥ϖ∥ φ+∥ϖ∥ if φ > 0, ∥ϖ∥ ≥ φ 1 if φ ≥ 0 and ς(ϖ,φ) =  0 if φ > 0, ∥ϖ∥ < φ ∥ϖ∥ φ if φ > 0, ∥ϖ∥ ≥ φ 1 if φ ≥ 0 then (F, η, ρ, ς) is a NPNLS. Let Υ : (F, η, ρ, ς) → (F, η, ρ, ς) be a linear operator defined by Υ(ϖ) = ϖ3 1+ϖ . Let ϖ0 ∈ F then for each ϖ ∈ F, ϵ > 0 there exists 0 < δ < 1 and 0 < γ, η(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ δ Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 8 of 15 ⇒ ϵ ϵ+ ∥Υ(ϖ)−Υ(ϖ0)∥ ≥ δ ⇒ ϵ ϵ+ ∥∥∥ ϖ3 1+ϖ − ϖ3 0 1+ϖ0 ∥∥∥ ≥ δ ϵ ∥(1 +ϖ)(1 +ϖ0)∥ ϵ ∥(1 +ϖ)(1 +ϖ0)∥+ ∥∥ϖ3 +ϖ0ϖ3 −ϖ3 0 −ϖξ30 ∥∥ ≥ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥ ∥1 +ϖ +ϖ0 +ϖξ0∥+ ∥∥(ϖ −ϖ0)(ϖ2 +ϖξ0 +ϖ2 0) +ϖξ0(ϖ +ϖ0)(ϖ −ϖ0) ∥∥ ≥ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥ ∥1 +ϖ +ϖ0 +ϖξ0∥+ ∥∥(ϖ −ϖ0)(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ≥ δ ϵ ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ ϵ ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ + ∥(ϖ −ϖ0)∥ ≥ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ ≥ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ + δ ∥(ϖ −ϖ0)∥ , ϵ ≥ δ.ϵ+ δ ∥(ϖ −ϖ0)∥ ∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ∥1 +ϖ +ϖ0 +ϖξ0∥ ≥ δ.ϵ+ δ ∥(ϖ −ϖ0)∥ , since ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ ∥1+ϖ+ϖ0+ϖξ0∥ ≥ 1. γ ≥ δ.γ + δ ∥(ϖ −ϖ0)∥, by taking ϵ = γ. ⇒ γ γ+∥(ϖ−ϖ0)∥ ≥ δ ⇒ η(ϖ −ϖ0, γ) ≥ δ. ρ(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ ∥Υ(ϖ)−Υ(ϖ0)∥ ϵ+ ∥Υ(ϖ)−Υ(ϖ0)∥ ≤ δ ⇒ ∥∥∥ ϖ3 1+ϖ − ϖ3 0 1+ϖ0 ∥∥∥ ϵ+ ∥∥∥ ϖ3 1+ϖ − ϖ3 0 1+ϖ0 ∥∥∥ ≤ δ ∥∥ϖ3 +ϖ0ϖ 3 −ϖ3 0 −ϖξ30 ∥∥ ϵ ∥(1 +ϖ)(1 +ϖ0)|+ ∥∥ϖ3 +ϖ0ϖ3 −ϖ3 0 −ϖξ30 ∥∥ ≤ δ∥∥(ϖ −ϖ0)(ϖ 2 +ϖξ0 +ϖ2 0) +ϖξ0(ϖ +ϖ0)(ϖ −ϖ0) ∥∥ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥+ ∥∥(ϖ −ϖ0)(ϖ2 +ϖξ0 +ϖ2 0) +ϖξ0(ϖ +ϖ0)(ϖ −ϖ0) ∥∥ ≤ δ∥∥(ϖ −ϖ0)(ϖ 2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ ∥1 +ϖ +ϖ0 +ϖξ0∥+ ∥∥(ϖ −ϖ0)(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ≤ δ ∥ϖ −ϖ0∥ ∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ∥1 +ϖ +ϖ0 +ϖξ0∥+ ∥ϖ −ϖ0∥ ∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ≤ δ ∥(ϖ −ϖ0)∥ ϵ ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ + ∥(ϖ −ϖ0)∥ ≤ δ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ + δ ∥(ϖ −ϖ0)∥ ≤ ∥(ϖ −ϖ0)∥ ∥(ϖ −ϖ0)∥ (1− δ) ≤ δϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ≤ δ.ϵ, Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 9 of 15 since ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ ≤ 1. ∥(ϖ −ϖ0)∥ − δ ∥(ϖ −ϖ0)∥ ≤ δγ, by taking ϵ = γ ⇒ ∥(ϖ −ϖ0)∥ ≤ δ(γ + ∥(ϖ −ϖ0)∥) ∥(ϖ−ϖ0)∥ γ+∥(ϖ−ϖ0)∥ ≤ δ ⇒ ρ(ϖ −ϖ0, γ) ≤ δ. ς(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ δ ⇒ ∥Υ(ϖ)−Υ(ϖ0)∥ ϵ ≤ δ ⇒ ∥∥∥ ϖ3 1+ϖ − ϖ3 0 1+ϖ0) ∥∥∥ ϵ ≤ δ∥∥ϖ3 +ϖ0ϖ 3 −ϖ3 0 −ϖξ30 ∥∥ ϵ ∥(1 +ϖ)(1 +ϖ0)∥ ≤ δ∥∥(ϖ −ϖ0)(ϖ 2 +ϖξ0 +ϖ2 0) +ϖξ0(ϖ +ϖ0)(ϖ −ϖ0) ∥∥ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥ ≤ δ∥∥(ϖ −ϖ0)(ϖ 2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥ ≤ δ ∥ϖ −ϖ0∥ ∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥ ≤ δ ∥(ϖ −ϖ0)∥ ϵ ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ ≤ δ δ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ + δ ∥(ϖ −ϖ0)∥ ≤ ∥(ϖ −ϖ0)∥ ∥(ϖ −ϖ0)∥ δ ≤ δϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ ≤ δ.ϵ, since ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)| ≤ 1. ⇒ δ ∥(ϖ −ϖ0)∥ ≤ δγ, by taking ϵ = γ ⇒ ∥(ϖ −ϖ0)∥ ≤ δγ ∥(ϖ−ϖ0)∥ γ ≤ δ ⇒ ς(ϖ −ϖ0, γ) ≤ δ. Thus for every η(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ η(ϖ −ϖ0, γ) ϵ ϵ+ ∥∥∥ ϖ3 1+ϖ − ϖ3 0 1+ϖ0 ∥∥∥ ≥ δ δ + ∥(ϖ −ϖ0)∥ ϵ ϵ+ ∥(ϖ3+ϖ3ϖ0−ϖ3 0−ϖξ30)∥ ∥(1+ϖ)(1+ϖ0)∥ ≥ γ γ + ∥(ϖ −ϖ0)∥ ϵ ∥(ϖ −ϖ0)∥ ∥1 +ϖ +ϖ0 +ϖξ0∥ ≥ γ ∥(ϖ −ϖ0)∥ ∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20) ∥∥ γ ≤ ϵ ∥1 +ϖ +ϖ0 +ϖξ0∥∥∥(ϖ2 +ϖξ0 +ϖ2ϖ0 +ϖξ20 ∥∥ . Now inf { ∥1+ϖ+ϖ0+ϖξ0∥ ∥(ϖ2+ϖξ0+ϖ2ϖ0+ϖξ20)∥ } = 0, for all ϖ ∈ F. Therefore, γ = 0, which is not possible. This shows that Υ is not strongly neutrosophic continuous. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 10 of 15 Theorem 8. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutro- sophic continuous then it is sequentially neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Let {ϖn} be a sequence in F and ϖn → ϖ0. lim n→∞ η1(ϖn − ϖ0, φ) = 1, lim n→∞ ρ1(ϖn − ϖ0, φ) = 0 and lim n→∞ ς1(ϖn − ϖ0, φ) = 0 for all φ > 0. Now since Υ is strongly neutrosophic continuous at ϖ0 ∈ F. Then for any given ϵ > 0, there exist γ = γ(ϵ) > 0 such that for all ϖ ∈ F, η2(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ η1(ϖ −ϖ0, γ), ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ρ1(ϖ −ϖ0, γ) and ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ ς1(ϖ −ϖ0, γ). lim n→∞ η2(Υ(ϖ)−Υ(ϖ0), ϵ) ≥ lim n→∞ η1(ϖn −ϖ0, φ) = 1 ⇒ lim n→∞ η2(Υ(ϖ)−Υ(ϖ0), ϵ) = 1 lim n→∞ ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ lim n→∞ ρ1(ϖn −ϖ0, φ) = 0 ⇒ lim n→∞ ρ2(Υ(ϖ)−Υ(ϖ0), ϵ) = 0 lim n→∞ ς2(Υ(ϖ)−Υ(ϖ0), ϵ) ≤ lim n→∞ ς1(ϖn −ϖ0, φ) = 0 ⇒ lim n→∞ ς2(Υ(ϖ)−Υ(ϖ0), ϵ) = 0. Since ϵ is arbitrary small positive number, Υ is sequentially neutrosophic continuous. Example 2. Consider the NPNLS (F, η, ρ, ς) as in Example (1) and the linear operator Υ is defined by Υ(ϖ) = ϖ3 1+ϖ . Let {ϖn} be a sequence in F and ϖn → ϖ0. lim n→∞ η1(ϖn −ϖ0, φ) = 1, lim n→∞ ρ1(ϖn −ϖ0, φ) = 0 and lim n→∞ ς1(ϖn −ϖ0, φ) = 0 ⇒ lim n→∞ φ φ+ ∥(ϖn −ϖ0)∥ = 1, lim n→∞ ∥(ϖn −ϖ0)∥ φ+ ∥(ϖn −ϖ0)∥ = 0 and lim n→∞ ∥(ϖn −ϖ0)∥ φ = 0. ⇒ lim n→∞ ∥(ϖn −ϖ0)∥ = 0 (3.1) η(Υ(ϖn)−Υ(ϖ0), φ) = φ φ+ ∥∥∥∥ ϖ3 n 1+ϖn − ϖ3 0 1+ϖ0 ∥∥∥∥ = φ φ+ ∥(ϖ3 n+ϖ3 nϖ0−ϖ3 0−ϖnϖ3 0)∥ ∥(1+ϖn)(1+ϖ0)∥ ⇒ φ∥(1+ϖn)(1+ϖ0)∥ φ∥(1+ϖn)(1+ϖ0)∥+∥(ϖn−ϖ0)∥∥(ϖ2 n+ϖnϖ0+ϖ2 nϖ0+ϖnϖ2 0)∥ = 1 as n → ∞ by Equation (3.1). And, ρ(Υ(ϖn)−Υ(ϖ0), φ) = ∥∥∥∥ ϖ3 n 1+ϖn − ϖ3 0 1+ϖ0 ∥∥∥∥ φ+ ∥∥∥∥ ϖ3 n 1+ϖn − ϖ3 0 1+ϖ0 ∥∥∥∥ = ∥(ϖ3 n+ϖ3 nϖ0−ϖ3 0−ϖnϖ3 0)∥ ∥(1+ϖn)(1+ϖ0)∥ φ+ ∥(ϖ3 n+ϖ3 nϖ0−ϖ3 0−ϖnϖ3 0)∥ ∥(1+ϖn)(1+ϖ0)∥ = ∥(ϖn−ϖ0)∥∥(ϖ2 n+ϖnϖ0+ϖ2 nϖ0+ϖnϖ2 0)∥ φ∥(1+ϖn)(1+ϖ0)∥+∥(ϖn−ϖ0)∥∥(ϖ2 n+ϖnϖ0+ϖ2 nϖ0+ϖnϖ2 0)∥ = 0 as n → ∞ by Equation (3.1). Also, ρ(Υ(ϖn)−Υ(ϖ0), φ) = ∥∥∥∥ ϖ3 n 1+ϖn − ϖ3 0 1+ϖ0 ∥∥∥∥ φ = ∥(ϖ3 n+ϖ3 nϖ0−ϖ3 0−ϖnϖ3 0)∥ ∥(1+ϖn)(1+ϖ0)∥ φ = ∥(ϖn−ϖ0)∥∥(ϖ2 n+ϖnϖ0+ϖ2 nϖ0+ϖnϖ2 0)∥ φ∥(1+ϖn)(1+ϖ0)∥ = 0 as n → ∞ by Equation (3.1). It follows that Υ exhibits sequential neutrosophic continuity at ϖ0 ∈ F, and thus this property extends over the entire space F. Nonetheless, Example (1) clearly illustrates that Υ does not satisfy the criteria for strong neutrosophic continuity. Corollary 1. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutro- sophic continuous then it is neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 11 of 15 Proof. The corollary is a direct consequence of Theorem (5) and Theorem (8). 4. Neutrosophic Boundedness of Operators on Neutrosophic Pseudo Normed Linear Space This section aims to generalize classical notions of boundedness for operators within neutrosophic pseudo normed linear spaces, offering new perspectives in the neutrosophic framework. Definition 8. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is said to be strongly neutrosophic bounded if for all ϖ ∈ F and φ ∈ R+, η2(Υ(ϖ), φ) ≥ η1(ϖ,φ), ρ2(Υ(ϖ), φ) ≤ ρ1(ϖ,φ) and ς2(Υ(ϖ), φ) ≤ ς1(ϖ,φ). Definition 9. Let (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is said to be weakly neutrosophic bounded if for any 0 < δ < 1, for all ϖ ∈ F and φ ∈ R+, η1(ϖ,φ) ≥ δ ⇒ η2(Υ(ϖ), φ) ≥ δ, ρ1(ϖ,φ) ≤ 1− δ ⇒ ρ2(Υ(ϖ), φ) ≤ 1− δ and ς1(ϖ,φ) ≤ 1− δ ⇒ ς2(Υ(ϖ), φ) ≤ 1− δ. Theorem 9. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutrosophic bounded then it is weakly neutrosophic bounded, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. The result can be readily derived from the definitions of strong and weak neutrosophic boundedness for linear operators. Definition 10. Let (F, η1, ρ1, ς1), (G, η2, ρ2, ς2) be NPNLS. A mapping Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is named to be uniformly neutrosophic bounded if there exists ϵ > 0, 0 < γ < 1 such that ∥Υϖ̃∥2δ ≤ ∥ϖ∥1δ, that ∥Υϖ̃∥2 ∗ δ ≤ ∥ϖ∥1 ∗ δ , where ∥·∥1δ and ∥·∥2δ are ascending family of pseudo norms and ∥·∥1 ∗ δ and ∥·∥2 ∗ δ are de- scending family of pseudo norm defined by ∥ϖ∥1δ = ∧{φ > 0 : η1(ϖ,φ) ≥ δ}, ∥Υ(ϖ)∥2δ = ∧{φ > 0 : η2(ϖ,φ) ≥ δ}, ∥ϖ∥1 ∗ δ = ∧{φ > 0 : ρ1(ϖ,φ) ≤ 1− δ and ς1(ϖ,φ) ≤ 1− δ}, ∥Υϖ̃∥2 ∗ δ = ∧{φ > 0 : ρ2(ϖ,φ) ≤ 1− δ and ς2(ϖ,φ) ≤ 1− δ}. Theorem 10. Consider the neutrosophic pseudo normed linear spaces (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2). A linear mapping Υ : F → G is strongly neutrosophic bounded if and only if it satisfies the condition of uniform neutrosophic boundedness relative to the corresponding δ-norms, where δ is any fixed real number in the interval (0, 1). Proof. Suppose Υ is strongly neutrosophic bounded. Then for all ϖ ∈ F and φ ∈ R+, ⇒ η2(Υ(ϖ), φ) ≥ η1(ϖ,φ), ρ2(Υ(ϖ), φ) ≤ ρ1(ϖ,φ) and ς2(Υ(ϖ), φ) ≤ ς1(ϖ,φ). (4.1) ∥ϖ∥1δ = ∧{o > 0 : η1(ϖ,φ) ≥ o}. Hene, there exist o0 > φ such that η1(ϖ, o0) ≥ o. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 12 of 15 There exist o0 > φ such that η2(Υ(ϖ), o0) ≥ o. (by Equation (4.1) ∥Υ(ϖ)∥2δ ≤ o0 < φ. Thus, ∥Υ(ϖ)∥2δ ≤ ∥ϖ∥1δ . Also, let ∥ϖ∥1 ∗ δ > φ ⇒ ∧{φ > 0 : ρ1(ϖ,φ) ≤ δ and ς1(ϖ,φ) ≤ δ} > φ Hence there exist o0 > φ such that ρ1(ϖ, o0) ≤ o and ς1(ϖ, o0) ≤ o There exist o0 > φ such that ρ2(Υ(ϖ), o0) ≤ o and ς2(Υ(ϖ), o0) ≤ o (by Equation (4.1) ∥Υ(ϖ)∥2δ ≥ o0 > φ. Thus, ∥Υ(ϖ)∥2 ∗ δ ≥ ∥ϖ∥1 ∗ δ . Hence Υ is uniformly neutrosophic bounded. Conversely, suppose Υ is uniformly neutro- sophic bounded with respect to to corresponding δ-norms. Then 0 < δ < 1, ∥Υ(ϖ)∥2δ ≤ ∥ϖ∥1δ , ∥Υ(ϖ)∥2 ∗ δ ≥ ∥ϖ∥1 ∗ δ (4.2) Let η1(ϖ,φ) > r ⇒ ∨ { 0 < δ < 1 : ∥ϖ∥1δ ≤ φ } > r. Hence there exists 0 < δ0 < 1 such that δ0 > r and ∥ϖ∥1δ0 ≤ φ There exists 0 < δ0 < 1 such that δ0 > r and ∥Υϖ̃∥2δ0 ≤ φ (by equation (4.2) η2(Υ(ϖ), φ) ≥ δ0 > r. Therefore, η2(Υ(ϖ), φ) ≥ η1(ϖ,φ). Let ρ1(ϖ,φ) < s ⇒ ∧{0 < δ < 1 : ∥ϖ∥1 ∗ δ ≥ φ} < s. Hence there exists 0 < δ0 < 1 such that δ0 < s and ∥ϖ∥1 ∗ δ0 ≤ φ Therefore there exists 0 < δ0 < 1 such that δ0 < s and ∥Υ(ϖ)∥2 ∗ δ ≤ φ (by equation (4.2) ρ2(Υ(ϖ), φ) ≤ δ0 < s. Therefore, ρ2(Υ(ϖ), φ) ≤ ρ1(ϖ,φ). Let ς1(ϖ,φ) < s ⇒ ∧{0 < δ < 1 : ∥ϖ∥1 ∗ δ ≥ φ} < s. Therefore there exists 0 < δ0 < 1 such that δ0 < s and ∥ϖ∥1 ∗ δ0 ≤ φ Hence there exists 0 < δ0 < 1 such that δ0 < s and ∥Υ(ϖ)∥2 ∗ δ ≤ φ (by equation (4.2)) ς2(Υ(ϖ), φ) ≤ δ0 < s. Therefore, ς2(Υ(ϖ), ϵ) ≤ ς1(ϖ,φ). Hence Υ is strongly neutrosophic bounded. Theorem 11. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutro- sophic bounded if and if it is strongly neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Suppose Υ is strongly neutrosophic bounded then for all ϖ ∈ F and ϵ ∈ R+, we have ⇒ η2(Υ(ϖ), ϵ) ≥ η1(ϖ, ϵ), ρ2(Υ(ϖ), ϵ) ≤ ρ1(ϖ, ϵ) and ς2(Υ(ϖ), ϵ) ≤ ς1(ϖ, ϵ). η2(Υ(ϖ − ϑ), ϵ) ≥ η1(ϖ − ϑ, ϵ), ρ2(Υ(ϖ − ϑ), ϵ) ≤ ρ1(ϖ − ϑ, ϵ) and ς2(Υ(ϖ − ϑ), ϵ) ≤ ς1(ϖ − ϑ, ϵ). η2(Υ(ϖ)−Υ(ϑ), ϵ) ≥ η1(ϖ − ϑ, γ), ρ2(Υ(ϖ)−Υ(ϑ), ϵ) ≤ ρ1(ϖ − ϑ, γ) and ς2(Υ(ϖ)−Υ(ϑ), ϵ) ≤ ς1(ϖ − ϑ, γ). Therefore Υ is strongly neutrosophic continuous at ϑ and hence by Theorem (6) Υ is strongly neutrosophic continuous on F. Conversely, suppose Υ is strongly neutrosophic continuous on F. Then Υ is strongly neutrosophic continuous at any point of F, say ϑ, for all ϖ ∈ F take ϵ = φ = δ, then η2(Υ(ϖ)−Υ(ϑ), φ) ≥ η1(ϖ − ϑ, φ), ρ2(Υ(ϖ)−Υ(ϑ), φ) ≤ ρ1(ϖ − ϑ, φ) and ς2(Υ(ϖ)−Υ(ϑ), φ) ≤ ς1(ϖ − ϑ, φ). Hence, η2(Υ(ϖ), φ) ≥ η1(ϖ,φ), ρ2(Υ(ϖ), φ) ≤ ρ1(ϖ,φ) and ς2(Υ(ϖ), φ) ≤ ς1(ϖ,φ). Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 13 of 15 If ϖ = ϑ, φ > 0 then η2(Υ(ϑ), φ) = η2(ϑ, φ) = 1 = η1(ϑ, φ), ρ2(Υ(ϑ), φ) = ρ2(ϑ, φ) = 0 = ρ1(ϑ, φ) and ς2(Υ(ϑ), φ) = ς2(ϑ, φ) = 0 = ς1(ϑ, φ). For any ϖ,φ ≤ 0,⇒ η2(Υ(ϖ), φ) = 0 = η1(ϖ,φ), ρ2(Υ(ϖ), φ) = 1 = ρ1(ϖ,φ) and ς2(Υ(ϖ), φ) = 0 = ς1(ϖ,φ). Hence for all ϖ ∈ F and φ ∈ R, Υ is strongly neutrosophic bounded. Corollary 2. If a linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neu- trosophic bounded then it is sequentially neutrosophic bounded, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. The corollary arises as a consequence of Theorem (8) and Theorem (11). Corollary 3. A linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is strongly neutro- sophic bounded then it is neutrosophic continuous, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Theorem 12. A linear operator Υ : (F, η1, ρ1, ς1) → (G, η2, ρ2, ς2) is weakly neutro- sophic continuous if and if it is weakly neutrosophic bounded, where (F, η1, ρ1, ς1) and (G, η2, ρ2, ς2) are NPNLS. Proof. Suppose Υ is weakly neutrosophic bounded. Then for any 0 < δ < 1, ϖ ∈ F and φ ∈ R+, η1(ϖ,φ) ≥ δ ⇒ η2(Υ(ϖ), φ) ≥ δ, ρ1(ϖ,φ) ≤ δ ⇒ ρ2(Υ(ϖ), φ) ≤ δ and ς1(ϖ,φ) ≤ δ ⇒ ς2(Υ(ϖ), φ) ≤ δ. η1(ϖ − ϑ, φ) ≥ δ ⇒ η2(Υ(ϖ − ϑ), φ) ≥ δ, ρ1(ϖ − ϑ, φ) ≤ δ ⇒ ρ2(Υ(ϖ − ϑ), φ) ≤ δ and ς1(ϖ − ϑ, ϵ) ≤ δ ⇒ ς2(Υ(ϖ − ϑ), ϵ) ≤ δ. η1(ϖ− ϑ, φ) ≥ δ ⇒ η2(Υ(ϖ)−Υ(ϑ), φ) ≥ δ, ρ1(ϖ− ϑ, φ) ≤ δ ⇒ ρ2(Υ(ϖ)−Υ(ϑ), φ) ≤ δ and ς1(ϖ − ϑ, ϵ) ≤ δ ⇒ ς2(Υ(ϖ)−Υ(ϑ), ϵ) ≤ δ, where ϵ = φ = δ. Therefore, Υ is weakly neutrosophic continuous at ϑ and hence by Theorem (7), Υ is weakly neutrosophic continuous. Conversely, suppose Υ is weakly neutrosophic continuous on F. Then Υ is weakly neutro- sophic continuous at any point of F, say ϑ, for all ϖ ∈ F take ϵ = φ = δ, then η1(ϖ − ϑ, φ) ≥ δ ⇒ η2(Υ(ϖ)−Υ(ϑ), φ) ≥ δ, ρ1(ϖ − ϑ, φ) ≤ δ ⇒ ρ2(Υ(ϖ)−Υ(ϑ), φ) ≤ δ ς1(ϖ − ϑ, φ) ≤ δ ⇒ ς2(Υ(ϖ)−Υ(ϑ), φ) ≤ δ. η1(ϖ,φ) ≥ δ ⇒ η2(Υ(ϖ), φ) ≥ δ, ρ1(ϖ,φ) ≤ δ ⇒ ρ2(Υ(ϖ), φ) ≤ δ and ς1(ϖ,φ) ≤ δ ⇒ ς2(Υ(ϖ), φ) ≤ δ. If ϖ = ϑ, φ > 0 then ⇒ η2(Υ(ϑ), φ) = 1 = η1(ϑ, φ), ρ2(Υ(ϑ), φ) = 0 = ρ1(ϑ, φ) and ς2(Υ(ϑ), φ) = 0 = ς1(ϑ, φ). Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 14 of 15 For any ϖ,φ ≤ 0,⇒ η2(Υ(ϖ), φ) = 0 = η1(ϖ,φ), ρ2(Υ(ϖ), φ) = 1 = ρ1(ϖ,φ) and ς2(Υ(ϖ), φ) = 0 = ς1(ϖ,φ). Hence for any 0 < δ < 1, for all ϖ ∈ F and φ ∈ R, Υ is weakly neutrosophic bounded. 5. Conclusions In this paper, we explored various forms of neutrosophic continuity and boundedness in NPNLS, establishing significant relationships between these properties. We introduced new characterizations and provided illustrative examples to validate the theoretical de- velopments. These contributions not only enhance the foundational understanding of operator behavior in NPNLS but also extend the existing framework of neutrosophic neu- trosophic normed linear spaces. Future work may involve studying these notions in more generalized settings, such as neutrosophic b-normed or intuitionistic fuzzy normed spaces. Additionally, exploring applications of these results in fields like decision theory, control systems, or differential equations could provide valuable real-world insights. Acknowledgements The authors are grateful to the Deanship of Graduate Studies and Scientific Research, Islamic University of Madinah, Saudi Arabia for supporting this research work. References [1] C. Felbin. Finite dimensional fuzzy normed linear space. Fuzzy Sets and Systems, 48:239–248, 1992. [2] J. Xiao and X. Zhu. On linearly topological structure and property of fuzzy normed linear space. Fuzzy Sets and Systems, 125:153–161, 2002. [3] T. Bag and S. K. Samanta. Finite dimensional fuzzy normed linear space. Journal of Fuzzy Mathematics, 11(3):687–705, 2003. [4] T. Bag and S. K. Samanta. Fuzzy bounded linear operators. Fuzzy Sets and Systems, 151:513–547, 2005. [5] S. Nadaban. Fuzzy pseudo-norms and fuzzy f-spaces. Fuzzy Sets and Systems, 282:99– 114, 2016. [6] B. Dinda and T. K. Samanta. Intuitionistic fuzzy continuity and uniform convergence. Journal of Open Problems in Computer Science and Mathematics, 3(1):8–26, 2010. [7] K. T. Atanassov. Intuitionistic Fuzzy Sets. Springer, 1999. [8] J. H. Park. Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 22(5):1039– 1046, 2004. [9] F. Smarandache. Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, Rehoboth, 1998. [10] M. Kirisci and N. Simsek. Neutrosophic metric spaces. Mathematical Sciences, 14:241–248, 2020. Pandiselvi. M, Jeyaraman. M andMohammad Akram / Eur. J. Pure Appl. Math, 18 (3) (2025), 6304 15 of 15 [11] M. Jeyaraman, Hassen Aydi, and M. De La Sen. New results for multivalued mappings in hausdorff neutrosophic metric spaces. Axioms, 11:724, 2022. [12] H. H. Schaefer and M. P. Wolff. Topological Vector Spaces. Springer, 1999.