S-metrik uzaylarda ikili tipinde daralmalar yardımıyla yeni sabit-disk sonuçları

Abstract

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.

Keywords

• Sabit disk
• sabit çember
• ikili tipinde daralma
• S-metrik uzay
• sabit çember problemi

New fixed-disc results via bilateral type contractions on S-metric spaces

Abstract

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.

Keywords

• Fixed disc
• fixed circle
• bilateral type contraction
• S-metric space
• fixed-circle problem

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