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Journals / Sinop Üniversitesi Fen Bilimleri Dergisi / 2026 / Cilt: 11 Sayı: 1

Bridging Quasilinear and Bilinear Operators with Soft Operator Theory via Soft Biquasilinear Mappings

Esnek Biquasilineer Dönüşümler Aracılığıyla Esnek Operatör Teorisinde Quasilineer ve Bilineer Operatörler Arasında Köprü Kurma

Berivan Demirci(BATMAN ÜNİVERSİTESİ)0009-0003-3384-5704
Hacer Bozkurt(BATMAN ÜNİVERSİTESİ)0000-0002-2216-2516
Pages
458–482
Publication Date
2026-06-26
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

These results demonstrate the advantages and applicability of the soft framework, thereby laying the foundation for further theoretical developments and practical applications.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, the relationship between soft bilinear operators and classical bilinear operators is illustrated using an example.
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Key Findings & Data:We denote the set of all soft biquasilinear functions by Λ(Q ̃^2,(Ω_C (R)) ̃), and we show that when equipped with a suitably defined norm, this set forms a soft normed quasilinear space and, moreover, a Banach quasilinear space.
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Conclusion & Impact:These results demonstrate the advantages and applicability of the soft framework, thereby laying the foundation for further theoretical developments and practical applications.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Esnek küme · esnek quasilineer uzay · esnek normlu quasilineer uzay

Abstract

In this paper, the relationship between soft bilinear operators and classical bilinear operators is illustrated using an example. We then introduce soft biquasilinear operators, which extend the concept of classical biquasilinear operators to the soft setting. Several soft biquasilinear operators are examined, leading to important findings. In addition, as in classical functional analysis, we observe that every soft quasilinear inner product gives rise to a soft biquasilinear operator. Moreover, the relationships among these operators are summarized in a small table. We denote the set of all soft biquasilinear functions by Λ(Q ̃^2,(Ω_C (R)) ̃), and we show that when equipped with a suitably defined norm, this set forms a soft normed quasilinear space and, moreover, a Banach quasilinear space. Additionally, the symmetry and positivity of soft biquasilinear operators are analyzed, and several related theorems are established, thereby providing a solid foundation for further developments in soft operator theory and soft functional analysis. These results demonstrate the advantages and applicability of the soft framework, thereby laying the foundation for further theoretical developments and practical applications.

Özet

Bu çalışmada, esnek bilineer operatörler ile klasik bilineer operatörler arasındaki ilişki bir örnek kul-lanılarak gösterilmektedir. Daha sonra, klasik biquasilineer operatörler kavramını esnek ortama genişleten esnek biquasilineer operatörleri tanıtıyoruz. Birkaç esnek biquasilineer operatör incelenerek önemli bulgulara ulaşılmaktadır. Ek olarak, klasik fonksiyonel analizde olduğu gibi, her esnek quasilineer iç çarpımın bir esnek biquasilineer operatör oluşturduğu gösterilmektedir. Bunun yanında, söz konusu operatörler arasındaki ilişkiler küçük bir tablo hâlinde özetlenmiştir. Tüm esnek biquasilineer fonksiyonların kümesini Λ(Q ̃^2,(Ω_C (R)) ̃) ile gösteriyoruz ve uygun şekilde tanımlanmış bir norm ile donatıldığında bu kümenin bir esnek normlu quasilineer uzay ve dahası bir Banach quasilineer uzay oluşturduğunu gösteriyoruz. Ek olarak, esnek biquasilineer operatörlerin simetrisi ve pozitifliği analiz edilmekte ve ilgili birkaç teorem oluşturulmaktadır; böylece, esnek operatör teorisi ve esnek fonksiyonel analizinde daha ileri gelişmeler için sağlam bir temel sağlanmaktadır. Bu sonuçlar, esnek yaklaşımın avantajlarını ve uygulanabilirliğini ortaya koymakta; böylece daha ileri teorik gelişmeler ve pratik uygulamalar için temel oluşturmaktadır.

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Cite This Article

Demirci, B., Bozkurt, H. (2026). Bridging Quasilinear and Bilinear Operators with Soft Operator Theory via Soft Biquasilinear Mappings. Sinop Üniversitesi Fen Bilimleri Dergisi, Cilt: 11 Sayı: 1, 458–482. https://doi.org/10.33484/sinopfbd.1847643

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