Örtüşen Derece Teorisi Üzerine Bazı Düzeltme ve İzahlar

Year 2019, Volume: 2 Issue: 1, 17 - 34, 30.03.2019

Ali Sırma

Abstract

Gaines-Mawhin
örtüşen derece teorisi olarak da bilinen örtüşen derece teorisi, özellikle
doğrusal olmayan denklemlerdeki çözümün varlığı probleminde güçlü bir
tekniktir. Özellikle doğrusal olmayan differensiyel denklemlerin periyodik
çözümlerinin varlığının gösterilmesinde çok geniş bir uygulaması olduğundan pek
çok araştırması çalışmalarında bu metodu kullanmışlardır. Örtüşen derece
teorisinde, bir Banach uzayındaki  açık ve sınırlı kümesinde tanımlı  formundaki bir operatör denkleminin
çözümlerinin varlığı araştırılır. Burada  bir doğrusal operatör ve  doğrusal olmayan bir operatör olmak üzere  ve  bazı özel koşulları sağlayan operatörlerdir.
Bu çalışmada esas olarak Gaines ve Mawhin’in çalışmaları takip edilmiş, örtüşen teorinin
sürdürülebilirlik teoreminin ifadesindeki ikinci sonuç düzeltilmiş ve gerekçesi
belirtilmiş. Her ne kadar uygulamalarda birinci sonuç kullanılsa da bu
ikincisinin düzeltilmesi de önemli bir çalışmadır. Bu sürdürülebilirlik
teoreminin farklı bir şekilde ifadesi verilmiş. Gaines ve Mawhin’in çok az
izahla verdiği ispat ilerideki çalışmalara bir yardımcı olmak amacıyla
yeterince detaylı bir şekilde izah edilmeye çalışılmıştır.

Keywords

Coincidence Degree Theory, L-compact operator, Homotopi Teorisi

On Coincidence Degree Theory Some Corrections and Explanations

Abstract

Coincidence degree theory, also known Mawhin’s coincidence theory is very powerful technique especially in existence of solutions problems in nonlinear equations. It has especially so broad applications in the existence of periodic solutions of nonlinear differential equations so that many researchers have used it for their investigations. In coincidence degree, mainly existence of solutions of the operator equation in the form in an open and bounded set in some Banach space was researched. Here, is a linear operator and is a nonlinear operator satisfying some special properties. In this study mainly the studies of Gaines and Mahwin are followed, the statement of continuation theory in a coincidence degree theory was corrected and the reason is expressed. A continuation theorem was expressed in different manner. In order to help the researchers with their studies on this subject, the proof that was provided by Gaines and Mawhin has now been presented with more detailed explanation.

Keywords

Coincidence Degree Theory, -compact operator, Homotopy Theory

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