Comparison Criteria For Two-İnterval Sturm-Liouville Equations Coupled With Transmission Conditions

Abstract

The purpose of this work is to establish some comparison properties of the Sturm-Liouville equations defined on two non-intersecting intervals with a common end, on which two conditions for the interaction of the left and right solutions are given, the so-called transmission conditions. We will call equations of this type as two-interval Sturm-Liouville equations, coupled with transmission conditions. It is not clear how to apply the classical methods of Sturm’s comparison theory to two-interval differential equations under given transmission conditions. We have derived some new criteria for comparison properties and have developed new approaches to obtain these criteria. The results obtained are an extension and generalization of the corresponding classical comparison properties for the Sturm-Liouville equations.

Keywords

• Non-classical Sturm-Liouville problem
• transmission conditions problems
• comparison theorems
• oscillation.

Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri

Abstract

Bu çalışmanın amacı ortak uç noktasına sahip olan iki ayrık aralıkta tanımlı olan Sturm-Liouville denklemleri için bazı karşılaştırma özelliklerini elde etmektir. Ortak uç noktada sağ ve sol çözümleri birbirine birleştiren ve geçiş şartları olarak adlandırılan iki şart verilmiştir. Bu çeşit denklemleri geçiş şartları ile birleştirilmiş iki aralıklı Sturm-Liouville denklemleri olarak adlandıracağız. Geçiş şartları ile birbirine birleştirilmiş iki aralıklı diferansiyel denklemlere Sturm karşılaştırma teorisinin klasik yöntemlerinin nasıl uygulanacağı açık değildir. Bu çalışmada karşılaştırma özellikleri için bazı yeni kriterler elde ettik ve bu kriterleri elde etmek için yeni yaklaşımlar geliştirdik. Elde edilen sonuçlar Sturm-Liouville denklemleri için klasik karşılaştırma özelliklerinin genişlemesi ve genelleştirilmesidir.

Keywords

• Klasik olmayan Sturm-Liouville problemi
• geçiş şartları
• karşılaştırma teoremleri
• salınım.

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