Öz
Banach daralma koşulunu sağlamayan ve bir tek sabit noktası ya da birden fazla sabit noktası olan fonksiyon örnekleri mevcuttur. Bu durumda, metrik sabit-nokta teorisi bazı teknikler kullanılarak kapsamlı olarak genelleştirilmektedir. Bu tekniklerden biri Jaggi tipinde daralma koşulu, Dass-Gupta tipinde daralma koşulu gibi kullanılan daralma koşulunun genelleştirilmesidir. Diğer bir teknik ise b-metrik uzay, S-metrik uzay gibi kullanılan metrik uzayın genelleştirilmesidir. Son teknik ise sabit çember, sabit disk gibi verilen bir fonksiyonun sabit nokta kümesinin geometrik özelliklerinin incelenmesidir. Bu amaç için, “sabit-çember problemi” metrik sabit-nokta teorisinin geometrik bir genellemesi olarak çeşitli tekniklerle çalışılmaktadır. Bu problem ayrıca “sabit-figür problemi” olarak da düşünülebilir. Bu son problemlere bazı çözümler hem metrik uzaylar üzerinde hem de genelleştirilmiş metrik uzaylar üzerinde farklı daralmalar kullanılarak elde edilmiştir. Bu makalenin ana amacı S-metrik uzaylar üzerinde bazı sabit-disk teoremleri ispatlamaktır. Bunun için, Bunun için bilinen bazı daralma koşullarını modifiye edeceğiz. Ayrıca elde edilen bu yeni teoremleri bazı gerçekleyici örnekler ile destekleyeceğiz.
Anahtar Kelimeler
Sabit disk sabit çember ikili tipinde daralma S-metrik uzay sabit çember problemi
Kaynakça
- Banach, S., Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fundamenta Mathematicae, 2, 133–181, (1922).
- Sedghi, S., Shobe, N. and Aliouche, A., A generalization of fixed point theorems in S-metric spaces, Matematički Vesnik, 64(3), 258–266, (2012).
- Bakhtin, I. A., The contraction principle in quasimetric spaces, Func. An. Ulian. Gos. Ped. Ins., 30, 26–37, (1989).
- Sedghi, S. and Dung, N. V., Fixed point theorems on S-metric spaces, Matematički Vesnik, 66(1), 113–124, (2014).
- Özgür, N. Y. and Taş, N., Some fixed-circle theorems on metric spaces, Bulletin of the Malaysian Mathematical Sciences Society, 42(4), 1433–1449, (2019).
- Mlaiki, N., Çelik, U., Taş, N., Özgür, N. Y. and Mukheimer, A., Wardowski type contractions and the fixed-circle problem on S-metric spaces, Journal of Mathematics, Art. ID 9127486, 9 pp, (2018).
- Özgür, N. Y., Taş, N. and Çelik, U., New fixed-circle results on S-metric spaces. Bulletin of Mathematical Analysis and Applications, 9(2), 10–23, (2017).
- Özgür, N. Y. and Taş, N., Fixed-circle problem on S-metric spaces with a geometric viewpoint, Facta Universitatis. Series: Mathematics and Informatics, 34(3), 459–472, (2019).
- Taş, N. and Özgür, N., On the geometry of fixed points for self-mappings on S-metric spaces, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69(2), 190–198, (2020).
- Özgür, N. Y. and Taş, N., Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25), Mathematical Sciences, 11(1), 7–16, (2017).
- Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019).
- Chen, C. M., Joonaghany, G. H., Karapınar, E. and Khojasteh, F., On bilateral contractions, Mathematics, 7, 38, (2019).
- Taş, N., Bilateral-type solutions to the fixed-circle problem with rectified linear units application, Turkish Journal of Mathematics, 44(4), 1330–1344, (2020).
- Özgür, N. Y. and Taş, N., Geometric properties of fixed points and simulation functions, arXiv:2102.05417, (2021).
Öz
There are some examples of self-mappings which does not satisfy the Banach contractive condition and have a unique fixed point or more than one fixed point. In this case, metric fixed-point theory has been extensively generalized using some techniques. One of these techniques is to generalize the used contractive conditions such as the Jaggi type contractive condition, the Dass-Gupta type contractive condition etc. Another technique is to generalize the used metric spaces such as a b-metric space, an S-metric space etc. The last technique is to investigate geometric properties of the fixed-point set of a given self-mapping such as fixed circle, fixed disc etc. For this purpose, “fixed-circle problem” has been studied with various techniques as a geometrical generalization of the metric fixed-point theory. This problem was also considered as “fixed-figure problem”. Some solutions to these recent problems were obtained using different contractions both a metric space and a generalized metric space. The main purpose of this paper is to prove some fixed-disc theorems on an S-metric space. To do this, we modify the known contractive conditions. Also, the obtained new theorems are supported by some illustrative examples.
Anahtar Kelimeler
Fixed disc fixed circle bilateral type contraction S-metric space fixed-circle problem
Kaynakça
- Banach, S., Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fundamenta Mathematicae, 2, 133–181, (1922).
- Sedghi, S., Shobe, N. and Aliouche, A., A generalization of fixed point theorems in S-metric spaces, Matematički Vesnik, 64(3), 258–266, (2012).
- Bakhtin, I. A., The contraction principle in quasimetric spaces, Func. An. Ulian. Gos. Ped. Ins., 30, 26–37, (1989).
- Sedghi, S. and Dung, N. V., Fixed point theorems on S-metric spaces, Matematički Vesnik, 66(1), 113–124, (2014).
- Özgür, N. Y. and Taş, N., Some fixed-circle theorems on metric spaces, Bulletin of the Malaysian Mathematical Sciences Society, 42(4), 1433–1449, (2019).
- Mlaiki, N., Çelik, U., Taş, N., Özgür, N. Y. and Mukheimer, A., Wardowski type contractions and the fixed-circle problem on S-metric spaces, Journal of Mathematics, Art. ID 9127486, 9 pp, (2018).
- Özgür, N. Y., Taş, N. and Çelik, U., New fixed-circle results on S-metric spaces. Bulletin of Mathematical Analysis and Applications, 9(2), 10–23, (2017).
- Özgür, N. Y. and Taş, N., Fixed-circle problem on S-metric spaces with a geometric viewpoint, Facta Universitatis. Series: Mathematics and Informatics, 34(3), 459–472, (2019).
- Taş, N. and Özgür, N., On the geometry of fixed points for self-mappings on S-metric spaces, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 69(2), 190–198, (2020).
- Özgür, N. Y. and Taş, N., Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25), Mathematical Sciences, 11(1), 7–16, (2017).
- Wolfram Research, Inc., Mathematica, Version 12.0, Champaign, IL (2019).
- Chen, C. M., Joonaghany, G. H., Karapınar, E. and Khojasteh, F., On bilateral contractions, Mathematics, 7, 38, (2019).
- Taş, N., Bilateral-type solutions to the fixed-circle problem with rectified linear units application, Turkish Journal of Mathematics, 44(4), 1330–1344, (2020).
- Özgür, N. Y. and Taş, N., Geometric properties of fixed points and simulation functions, arXiv:2102.05417, (2021).
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 5 Ocak 2022 |
| Gönderilme Tarihi | 14 Eylül 2021 |
| Yayımlandığı Sayı | Yıl 2022 Cilt: 24 Sayı: 1 |