Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions

Abstract

In this study, we consider a new type boundary value problem consisting of a Sturm-Liouville equation on two disjoint intervals together with interaction conditions and with eigenvalue parameter in the boundary conditions. We suggest a special technique to reduce the considered problem into an integral equation by the use of which we define a new concept, the so-called weak eigenfunction for the considered problem. Then we construct some Hilbert spaces and define some self-adjoint compact operators in these spaces in such a way that the considered problem can be interpreted as a self-adjoint operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of weak eigenfunctions form a Riesz basis of the suitable Hilbert space.

Yeni Tipten Sturm-Liouville Problemlerinin Ürettiği Diferansiyel Operatörlerin Kendine Eşlenikliği ve Pozitivliği

“…Thus the integro-differential equation 12is transformed into the system of equalities (13)- (14) all terms of which are defined for the 3 (7), if this element satisfy the equations (13) and (14) for any Consequently the concept of generalized solution is an extension of a classical solution.…”

“…Oscillation and comparison results have been obtained in [7]. The completeness of the eigenfunctions and eigenfunction expansions in various function spaces for the Sturm-Liouville problems with spectral parameter in the boundary conditions have been considered in [10][11][12][13][14][15][16]. Problems with various singularities have been analyzed in [17][18][19][20][21][22][23][24][25][26][27][28][29][30].…”

“…The compactness of  is well-known fact (see, [34]). The proof of the compactness of 1  can be found by using the same arguments, as in [10,11] and [13].…”

The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions

“…Boundary value problems including transmission conditions appears in many fields of natural sciences. Recently, such type of transmission problems have been an important topic in theoretical and applied mathematics (see, [17][18][19][20][21][22][23][24][25][26][27][28]). In this study we will investigate some basic spectral properties of a new type periodic Sturm-Liouville problems.…”

Construction and Stability of Riesz Bases

“…Since then many results on the Riesz basis have been obtained [3][4][5]. Also the Riesz basis of the systems of sines and cosines in 2 [0, ] and Riesz basis associated with Sturm-Liouville problems have been studied in many papers [6][7][8][9][10][11][12]; moreover, on the problems of expansion of eigenfunctions, we refer to [13][14][15][16][17][18] and references cited therein.…”