Second-order differential operators with interior singularity

Abstract

The purpose of this study is to investigate a new class of boundary value transmission problems (BVTPs) for a Sturm-Liouville equation on two separate intervals. We introduce a modified inner product in the direct sum space L 2 [a, c) ⊕ L 2 (c, b] ⊕ C 2 and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then, by suggesting own approaches, we construct the Green's function for the BVTP under consideration and find the resolvent function for the corresponding inhomogeneous problem.

Unconditional basicity of eigenfunctions’ system of Sturm-Liouville operator with an involutional perturbation

“…A specific example of a boundary-value problem for the second-order differential operator with involution that produces an infinite number of associated functions is given in [11,12]. We also note valuable results on the Green's function for the boundary value problems related to functional-differential operators with involution (see Cabada and Tojo [13,14]) and new types of non-classical Sturm-Liouville problems (see Aidemir, Mukhtarov et al [15,16]).…”

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

“…15)-( 18) and ( 19) we have 22) is a direct consequence of the polar identity. Now, making use the Lemma 2, Theorem 1 and inequality (22) we have the following important result.…”

“…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].…”

Solvability and coerciveness of multi-point Sturm–Liouville problems with abstract linear functionals

“…Note that an abstract theory of the boundary value problems with continuous coefficients and an eigenvalue parameter in the boundary conditions have been constructed by Yakubov and Yakubov (see [4] and corresponding bibliography). Many authors have been devoted to the study of discontinuous problems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. To deal with the discontinuity of the problem, transmission conditions are imposed on the discontinuous points.…”