Some Qualitative Properties of a Periodic Sturm-Liouville Problem With an Inner Singular Point

Öz

In this paper we study a new type of boundary value problem consisting of a self-adjoint second-order differential equation (the so-called Sturm-Liouville equation) defined on two non-intersecting intervals with a common end, periodic boundary conditions and two additional transmission conditions specified at the common endpoint of the considered intervals. We proved some spectral properties of the boundary value problem consideration. In particular, we obtained an estimate of the principal eigenvalue using a modified Rayleigh quotient. In the special case where γ=δ=1, the obtained results are reduced to the corresponding classical results, so our results, generalize the classical results

Anahtar Kelimeler

• Sturm-Liouville problem
• periodic boundary condition
• transmission condition
• Rayleigh quotient

İç Tekil Noktası Bulunan Bir Periyodik Sturm-Liouville Probleminin Bazı Nitel Özellikleri

Öz

Bu makalede ortak uç noktası olan iki ayrık aralıkta tanımlı olan kendine eşlenik ikinci mertebeden diferansiyel denklemden (Sturm-Liouville denklemi olarak adlandırılan diferansiyel denklemden), periyodik sınır şartlarından ve verilmiş aralıkların ortak uç noktasında verilmiş iki tane ek geçiş şartlarından oluşmuş yeni tip bir sınır değer problemini inceledik. İncelediğimiz sınır değer probleminin bazı spektral özelliklerini ispat ettik. Ayrıca modifiye edilmiş (biçimi değiştirilmiş) Rayleigh oranından yararlanarak esas özdeğer için bir tahmin elde ettik. \gamma=\delta=1 olduğu özel durumda, elde edilen sonuçlar uygun gelen klasik sonuçlara indirgeniyor. Bu nedenle elde edilen sonuçlar klasik sonuçları genelleştiriyor.

Anahtar Kelimeler

• Sturm-Liouville problemi
• periyodik sınır şartı
• geçiş şartı
• Rayleigh oranı.

Kaynakça

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