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

Öz

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.

Anahtar Kelimeler

• Esnek küme
• esnek quasilineer uzay
• esnek normlu quasilineer uzay
• esnek quasilineer operatör
• esnek biquasilineer operatör

Destekleyen Kurum

Yazarlar, bu çalışmanın araştırma, yazarlık veya yayınlanması için herhangi bir mali destek almadıklarını beyan ederler.

Etik Beyan

Yazarlar bu çalışmanın herhangi bir etik kurul onayına veya özel izne ihtiyaç duymadığını beyan ederler.

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

Öz

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.

Anahtar Kelimeler

• Soft set
• soft quasilinear space
• soft normed quasilinear space
• soft quasilinear operator
• soft biquasilinear operator

Destekleyen Kurum

The authors declare that no financial support was received for the research, authorship, or publication of this study.

Etik Beyan

The authors declare that this study does not require any ethics committee approval or special permission.

Kaynakça

1. Molodtsov, D. (1999). Soft set theory first results. Computers & Mathematics with Applications, 37, 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
2. Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers and Mathematics with Applications, 45, 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
3. Das, S., & Samanta, S. K. (2012). Soft real sets, soft real numbers and their properties. Journal of Fuzzy Mathematics, 20(3), 551–576.
4. Das, S., Majumdar, P., & Samanta, S. K. (2015). On soft linear spaces and soft normed linear spaces. Annals of Fuzzy Mathematics and Informatics, 9, 91–109.
5. Das, S., & Samanta, S. K. (2013). On soft metric spaces. The Journal of Fuzzy Mathematics, 21, 207–213.
6. Das, S., & Samanta, S. K. (2013). On soft inner product spaces. Annals of Fuzzy Mathematics and Informatics, 6, 151–170.
7. Das, S., & Samanta, S. K. (2013). Soft linear operators in soft normed linear spaces. Annals of Fuzzy Mathematics and Informatics, 6(2), 295–314.
8. Das, S., & Samanta, S. K. (2014). Soft linear functionals in soft normed linear spaces. Annals of Fuzzy Mathematics and Informatics, 7(4), 629–651.

9. Bayramov, S., & Gunduz, C. (2013). Soft locally compact spaces and soft paracompact spaces. Journal of Mathematics and System Science, 3, 122–130.
10. Yazar, M. I., Bilgin, T., Bayramov, S., & Gunduz Aras, C. (2014). A new view on soft normed spaces. International Mathematical Forum, 9, 1149–1159. https://doi.org/10.12988/imf.2014.4470
11. Yazar, M. I., Aras, C. G., & Bayramov, S. (2019). Results on soft Hilbert spaces. TWMS Journal of Applied and Engineering Mathematics, 9(1), 159–164.
12. Bulak, F. O., & Bozkurt, H. (2013). Soft quasilinear operators in soft normed quasilinear spaces. Journal of Intelligent & Fuzzy Systems, 45, 4847–4856. https://doi.org/10.3233/JIFS-23003529
13. Gonci, M. S., & Bozkurt, H. (2024). Soft quasilinear inner product spaces. TWMS Journal of Applied and Engineering Mathematics 14(1), 122–133.
14. Bozkurt, H., & Yılmaz, Y. (2020). Biquasilinear functionals on quasilinear spaces and some related results. Erzincan University Journal of Science and Technology, 13(1), 95–104. https://doi.org/10.18185/erzifbed.664675
15. Yılmaz, Y. (2026). On quasilinear algebra of linear interval equations and interval Cramer’s rule. Mathematics, 14(6), 1018. https://doi.org/10.3390/math14061018
16. Bozkurt, H. (2020). Soft quasilinear spaces and soft normed quasilinear spaces. Adıyaman University Journal of Science, 10(2), 506–523. https://doi.org/ 10.37094/adyujsci.728934
17. Bayramov, S., Gunduz, C., & Coskun, A. E. (2025). 2-Normed structures on soft vector spaces. Filomat, 39(24), 8385–8394. https://doi.org/10.2298/FIL2524385B
18. Yolcu, A., Ozturk, T. Y., & Bayramov, S. (2024). A new approach to soft relations and soft functions. Computational and Applied Mathematics, 43, 335. https://doi.org/10.1007/s40314-024-02853-w
19. Aseev, S. M. (1986). Quasilinear operators and their application in the theory of multivalued mappings. Proceeding of the Steklov Institute of Mathematics, 2, 23-52.
20. Debnath, L., & Mikusinski, P. (2005). Introduction to Hilbert spaces with applications, Elsevier Academic Press. USA.
21. Ameen, Z. A., & Alghamdi, O. F. (2025). Soft topologies induced by almost lower density soft operators. Engineering Letters, 33(3), 712-720.
22. Dalkılıc, O. (2025). Decision-making approaches focusing on best parameterobject pair for soft set. Neural Computing and Applications, 37, 14521-14539. https://doi.org/10.1080/00207721.2021.1949644
23. Stojanovi´c, N., Vucicevic, N., & Dalkılıc, O. (2025). Decision-making algorithm based on the scored-energy of neutrosophic soft sets. Afrika Matematika, 36, 144. https://doi.org/10.1007/s13370-025-01366-x30
24. Dalkılıc, O. (2022). Determining the membership degrees in the range (0,1) for hypersoft sets independently of the decision-maker. International Journal of Systems Science, 53(8), 1733-1743. https://doi.org/10.1080/00207721.2021.2023686
25. Dalkılıc, O. (2021). A novel approach to soft set theory in decisionmaking under uncertainty. International Journal of Computer Mathematics, 98(10), 1935-1945. https://doi.org/10.1080/00207160.2020.1868445
26. Dalkılıc, O., & Demirta，S, N. (2023). Similarity measures of neutrosophic fuzzy soft set and its application to decision making. Journal of Experimental and Theoretical Artificial Intelligence, 37 (3), 513–529. https://doi.org/10.1080/0952813X.2023.2222720
27. Dalkılıc, O. (2025). Unifying relationships in uncertain environments: Examining relations in binary soft sets for expressing inter-object correspondence. The Journal of Supercomputing, 81, 1529. https://doi.org/10.1007/s11227-025-08036-6
28. Dalkılıc, O. (2021). Karar vericilere odaklanan yeni bir matematiksel model: Sanal sezgisel bulanık parametreli esnek kume. El-Cezeri, 8(2), 1565-1575. https://doi.org/ 10.31202/ecjse.940871
29. Dalkılıc, O., & Demirta，S, N. (2021). Bipolar fuzzy soft D-metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 64–73. https://doi.org/10.31801/cfsuasmas.774658
30. Dalkılıc, O. (2021). (α, β)-cuts and inverse (α, β)-cuts in bipolar fuzzy soft sets. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(2), 582–599. https://doi.org/10.31801/cfsuasmas.770623
31. Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to interval analysis. Society for Industrial and Applied Mathematics, Philadelphia.
32. Faried, N., Ali, M. S., & Sakr, H. H. (2020). On fuzzy soft linear operators in Fuzzy soft Hilbert spaces. Abstract nnd Applied Analysis, 1, 1-13. https://doi.org/10.1155/2020/5804957
33. Mohsen, M. S. D. (2023). Some generalizations of fuzzy soft (k*A)-quasinormal operators in fuzzy soft Hilbert spaces. Journal of Interdisciplinary Mathematics, 26(6), 1133–1143. https://doi.org/10.47974/JIM-161231
34. Abbas, M., Ali, M. I., & Romaguera, S. (2017). Generalized operations in soft set theory via relaxed conditions on parameters. Filomat, 31(19), 5955–5964. https://doi.org/10.2298/FIL1719955A
35. Kumawat, S., & Gupta, V. (2025). A new approach of soft positive operators with theorem and soft square roots based on soft set theory. International Journal of Scientific Research in Mathematical and Statistical Sciences, 12(2), 71–76.
36. Azzam, A., Ameen, Z. A., Al-shami, T. M, & El-Shafei, M. E, (2022). Generating soft topologies via soft set operators. Symmetry, 14(5), 914. https://doi.org/10.3390/sym1405091432