Orlicz Fonksiyon Dizileri İle Wijsman Toplanabilirlik

Öz

Wijsman yakınsaması, metrik uzaylarda kapalı küme dizileri için bir yakınsama türüdür ve bir noktanın bir kümeye olan uzaklığını kullanır. Bu çalışmada, Orlicz fonksiyonlarının dizileri kullanılarak, Wijsman anlamında kapalı kümeler dizileri için bir toplanabilirlik kavramı tanımlanarak yeni bir dizi uzayı önerilmiştir. Bu küme dizileri uzayının tanımlanmasında kullanılan parametrelerin farklılaşması durumunda veya Wijsman istatistiksel yakınsak küme dizilerinin uzayıyla ilişkili çeşitli kapsama teoremleri sunulmuştur. Ayrıca, elde edilen sonuçlarda, asimptotik yoğunluk yerine ağırlık fonksiyonları kullanılarak elde edilen bir yoğunluk kavramı kullanılmıştır.

Anahtar Kelimeler

• Yoğunluk
• Orlicz fonksiyonu
• İstatistiksel yakınsaklık
• Wijsman yakınsaklık

Wijsman summability through Orlicz Function Sequences

Öz

The Wijsman convergence is a type of convergence for sequences of closed sets in metric spaces, utilizing the distance from a point to a set. This study introduces a new sequence space by defining a summability concept for sequences of closed sets in the Wijsman sense, using sequences of Orlicz functions. Various inclusion theorems related to the space of Wijsman statistically convergent sequences of sets have been presented, considering different parameters used in the definition of this set sequence space. Additionally, in the obtained results, a concept of density has been employed using weight functions instead of asymptotic density.

Anahtar Kelimeler

• Density
• Orlicz function
• Statistical convergence
• Wijsman convergence

Kaynakça

1. [1] Wijsman R.A. Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc., 70 186-188, 1964.
2. [2] Wijsman R.A. Convergence of sequences of convex sets, cones and functions II, Transactions of the American Mathematical Society, 123(1) 32-45, 1966.
3. [3] Beer G. Wijsman convergence: A survey, Set-Valued Analysis, 2 77-94, 1994.
4. Nuray F., Rhoades B. Statistical convergence of sequences of sets, Fasc. Math., 49 87–99, 2012.
5. [4] Ulusu U., Nuray F. Lacunary statistical convergence of sequence of sets, Prog. Appl. Math., 4(2) 99–109, 2012. doi: 10.3968/j.pam.1925252820120402.2264.
6. [5] Nuray F., Ulusu U., Dündar E. Lacunary statistical convergence of double sequences of sets, Soft Computing, 20 2883-2888, 2016.
7. [6] Altınok M., İnan B., Küçükaslan M. On Deferred Statistical Convergence of Sequences of Sets in Metric Space, Turkish Journal of Mathematics and Computer Science, 3(1), 1-9, 2016.
8. [7] Altınok M., İnan B., Küçükaslan M. On Asymptotically Wijsman Deferred Statistical Equivalence of Sequence of Sets, Thai Journal of Mathematics, 18(2), 803–817, 2020.

9. [8] Ulusu U., Dündar E., Gülle E. I_2-Cesàro summability of double sequences of sets, Palestine Journal of Mathematics, 9 561-568, 2020.
10. [9] Nuray F., Dündar E., Ulusu U. Wijsman statistical convergence of double sequences of sets, Iran. J. Math. Sci. Inform., 16 55-64, 2021.
11. [10] Kandemir H.Ş., Et M. On I-lacunary statistical convergence of order α of sequences of sets, Filomat, 31 2403-2412, 2017.
12. [11] Aral N.D., Kandemir H.Ş., Et M. ρ-statistical convergence of sequences of sets. 3rd International E-Conference on Mathematical Advances and Applications (ICOMAA 2020), Istanbul, Turkey, 2020.
13. [12] Aral N.D., Kandemir H.Ş., Et M. On ρ-statistical convergence of order α of sequence of sets, Miscolc Mathematical Notes, 24(2) 569-578, 2023. doi: 10.18514/MMN.2023.3503
14. [13] Fast H. Sur la convergence statistique, Colloq. Math., 2 241-244, 1951.
15. [14] Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloq Math 2:73-74, 1951.
16. [15] Schoenberg IJ. The integrability of certain functions and related summability methods. Amer Math Monthly 66:361-375, 1959.
17. [16] Šalát T. On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) 139-150, 1980. Fridy j.A., On statistical convergence, Analysis, 5(4) 301-314, 1985.
18. [17] Connor, J. On strong matrix summability with respect to a modulus and statistical convergence, Can. Math. Bull., 32(2) 194-198, 1989.
19. [18] Mursaleen M. λ-statistical convergence, Mathematica Slovaca, 50(1) 111-115, 2000.
20. [19] Bektaş Ç.A. On some new generalized difference sequence spaces on seminormed spaces defined by a sequence of Orlicz functions, Math. Slovaca, 61 227–234, 2011.
21. [20] Çolak R., Bektaş Ç.A. λ-statistical convergence of order α, Acta Math. Sci., 31(3) 953-959, 2011.
22. [21] Et M., Şengül H., Some Cesàro-Type Summability of order and Lacunary Statistical Convergence of order α, Filomat, 28 1593-1602, 2014.
23. [22] Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals, Acta Math. Hung., 147 97-115, 2015.