Hölder uzaylarında bazı integral denklemlerin çözülebilirliği
Yükleniyor...
Dosyalar
Tarih
2022
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
İnönü Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Dört bölümden oluşan bu tezin ilk bölümünde, integral denklemlerin tarihsel gelişimi ve kullanım alanları hakkında genel bilgiler ve bu denklemlere ilişkin bazı tanımlar verilerek, tez çalışmasının literatürdeki yeri ve önemi vurgulanmıştır. Tezin ikinci bölümünde, ileriki bölümlerde ele alacağımız (3.0.1) ve (4.0.1) denklemleriyle ilgili olan α üslü Hα[a,b] Hölder uzayı ve ω süreklilik modülünün doğurduğu Cω[a,b] uzayları ile ilgili tanımlar ve bu uzaylarla ilgili bazı özellikler verildi. Diğer bölümlerin anlaşılmasını sağlayacak bazı temel tanımlar ve teoremler verildi. Ayrıca bazı teoremler ispatlarıyla verildi. Çalışmanın orijinal kısmını üçüncü ve dördüncü bölüm oluşturmaktadır. Tezin üçüncü bölümünde, son zamanlarda yapılan çalışmalar ve ilerlemeler araştırılarak; Hölder uzaylarında önceden çalışılan (1.1.15)-(1.1.17) denklemlerinden daha genel olan (3.0.1) Fredholm tipi kuadratik integral denkleminin çözümü için bir varlık teoremi verilmiş ve bu teoremin uygulanabilirliği ile ilgili iki örnek sunulmuştur. Tezin dördüncü bölümünde ise tezin üçüncü bölümünde ele aldığımız denklemin kısmen daha geneli olan (4.0.1) Fredholm tipi kuadratik integral denkleminin çözümü, tezin üçüncü bölümündeki α üslü Hα[0,1] Hölder uzayından farklı olarak ω süreklilik modülünün doğurduğu Cω[a,b] uzayında araştırıldı ve bir varlık teoremi elde edildi. Bu sonuca varabilmek için Cω[a,b] uzayında rölatif kompaktlık ve Schauder sabit nokta teoreminden yararlanıldı. Akabinde bu teoremin uygulanabilirliği ile ilgili bir örnek sunuldu.
In the first part of this thesis, which consists of four chapters, general information about the historical development and usage areas of integral equations and some definitions about these equations were given, and the place and importance of the thesis study in the literature was emphasized. In the second part of the thesis, some definitions and some properties were given about both the Hα[a,b] Hölder space and the Cω[a,b] spaces formed by the modulus of continuity ω, which are related to the (3.0.1) and (4.0.1) equations. Some basic definitions and theorems were given to help understand the other chapters. In addition, some theorems were given with their proofs. The third and fourth chapters constitute the original part of the study. In the third part of the thesis, by researching the recent studies and developments; An existence theorem was given for the solution of the (3.0.1) Fredholm type quadratic integral equation, which is more general than the previously studied (1.1.15)-(1.1.17) equations in the Hölder spaces, and two examples of the applicability of this theorem was presented. In the fourth part of the thesis, the solution of the (4.0.1) Fredholm type quadratic integral equation, which is partly more general of the equation we discussed in the third part of the thesis, was investigated in Cω[a,b] space, which is different from the Hα[a,b] Hölder space in the third part of the thesis, and an existence theorem was obtained. Relative compactness in Cω[a,b] space and Schauder fixed point theorem were used to reach this conclusion. Afterwards, an example of the applicability of this theorem was presented.
In the first part of this thesis, which consists of four chapters, general information about the historical development and usage areas of integral equations and some definitions about these equations were given, and the place and importance of the thesis study in the literature was emphasized. In the second part of the thesis, some definitions and some properties were given about both the Hα[a,b] Hölder space and the Cω[a,b] spaces formed by the modulus of continuity ω, which are related to the (3.0.1) and (4.0.1) equations. Some basic definitions and theorems were given to help understand the other chapters. In addition, some theorems were given with their proofs. The third and fourth chapters constitute the original part of the study. In the third part of the thesis, by researching the recent studies and developments; An existence theorem was given for the solution of the (3.0.1) Fredholm type quadratic integral equation, which is more general than the previously studied (1.1.15)-(1.1.17) equations in the Hölder spaces, and two examples of the applicability of this theorem was presented. In the fourth part of the thesis, the solution of the (4.0.1) Fredholm type quadratic integral equation, which is partly more general of the equation we discussed in the third part of the thesis, was investigated in Cω[a,b] space, which is different from the Hα[a,b] Hölder space in the third part of the thesis, and an existence theorem was obtained. Relative compactness in Cω[a,b] space and Schauder fixed point theorem were used to reach this conclusion. Afterwards, an example of the applicability of this theorem was presented.
Açıklama
Anahtar Kelimeler
Hölder eşitsizliği, Sabit nokta teoremleri, Schauder bazları
Kaynak
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Dal, İ. (2022). Hölder uzaylarında bazı integral denklemlerin çözülebilirliği. İnönü Üniversitesi, Malatya.
