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Singular Dirac systems in the Sobolev space

In this paper we construct Weyl’s theory for the singular left-definite Dirac systems. In particular, we prove that there exists at least one solution of the system of equations that lies in the Sobolev space. Moreover, we describe the behavior of the solution belonging to the Sobolev space around the singular point. | Turk J Math (2017) 41: 933 – 939 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1606-70 Research Article Singular Dirac systems in the Sobolev space ∗ ˘ Ekin UGURLU Department of Mathematics, Faculty of Arts and Sciences, C ¸ ankaya University, Ankara, Turkey Received: 14.06.2016 • Accepted/Published Online: 04.10.2016 • Final Version: 25.07.2017 Abstract: In this paper we construct Weyl’s theory for the singular left-definite Dirac systems. In particular, we prove that there exists at least one solution of the system of equations that lies in the Sobolev space. Moreover, we describe the behavior of the solution belonging to the Sobolev space around the singular point. Key words: Dirac systems, left-definite operators, Weyl theory 1. Introduction Consider the equation y ′′ + y = 0. One can immediately obtain that the solution is described by y = c1 sin(x + c2 ), where c1 and c2 are arbitrary constants. However, a different approach to obtain the solution from the ordinary way is possible. In fact, multiplying the equation by y ′ one can find (y ′ )2 + y 2 = c2 and y′ = (1.1) √ c2 − y 2 . Considering y = c sin γ it is obtained that γ = x + d. Consequently, the solution is y = c sin(x + d) [5]. The main point of this construction of the solution is the equation (1.1), which can arise naturally in Sturm–Liouville equations, called left-definite Sturm–Liouville equations. The name left-definite comes from the left-side of the equation −(py ′ )′ + qy = λwy, x ∈ (a, b) ⊆ R, (1.2) in contrast to the standard right-definite case. That is, as is well known, in the right-definite case we impose the condition to the weight function w to be positive, which gives rise to the standard Lebesgue space L2w (a, b) with the inner product ∫b (y, z) = yzwdx. a ∗Correspondence: ekinugurlu@cankaya.edu.tr 2000 AMS Mathematics Subject Classification: Primary 35Q41; Secondary 46C05, 30E25. 933 ˘ UGURLU/Turk J .