Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 17 (2017), No. 2, pp. 911–923 DOI: 10.18514/MMN.2017.1098 STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS AND EIGEN-PARAMETER IN BOUNDARY CONDITION ABDULLAH KABLAN AND MANAF DZH. MANAFOV Received 15 January, 2014 Abstract. This paper deals with the Sturm-Liouville equation with a finite number of point ıinteractions and eigenvalue parameter contained in the boundary condition. Sturm–Liouville problem with discontinuities at one or two points and its different variants have already been investigated. In this study we extend these results to a finite number of point ıinteractions case. The crucial part of this study is the using graph demonstration to obtain asymptotic representation of solutions. 2010 Mathematics Subject Classification: 34B05; 34L20 Keywords: discontinuous Sturm-Liouville problems, transmission conditions, ıinteractions 1. I NTRODUCTION We consider the boundary problem (BVP) for the differential equation `y WD y 00 C q.x/y D y (1.1) on Œa;x 1 / [ .x 1 ;x 2 / [[ .x n1 ;x n / [ .x n ;b and boundary condition at x D a L 1 .y/ WD ˛ 1 y.a/ C ˛ 2 y 0 .a/ D 0; (1.2) with transmission conditions at discontinuous points x i ;i D 1;n U i .y/ WD y.x i  0/ D y.x i C 0/ D y.x i /; (1.3) V i .y/ WD y 0 .x i C 0/  y 0 .x i  0/ D  i y.x i / (1.4) and the eigenparameter-dependent boundary condition at x D b L 2 .y/ WD Œˇ 0 1 y.b/ C ˇ 0 2 y 0 .b/ C Œˇ 1 y.b/  ˇ 2 y 0 .b/ D 0; (1.5) where q.x/ is real-valued function and continuous in L 1 Œa;b. We assume that ˛ i ,  i , ˇ i ;ˇ 0 i ;i D 1;2 are real numbers, satisfying j˛ 1 jCj˛ 2 j¤ 0;  is a complex spectral parameter. Throughout this paper, we assume that r WD ˇ 0 1 ˇ 2  ˇ 1 ˇ 0 2 > 0: (1.6) c  2017 Miskolc University Press