Eigenvalues and normalized eigenfunctionsof discontinuous Sturm-Liouville problem with transmission conditions

Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

“…Lemma 3.3. Assume that (7) and (10) to (13) ) , and Φ( , ) = Φ( − 0, ) denotes the left limit at point .…”

“…Theorem 3.1. Assume 2 ≠ 0, and let (7), (10) to (13) hold. Let H( ) be defined as that in lemma 2.1.…”

“…. Define a partition of J by (10). Next, we construct piecewise constant functions̄,q andw on interval J satisfying (7), (11) to (13).…”

“…As we all know, Sturm-Liouville problem with interior discontinuous points inside an interval has been one of the important research topics in mathematical physics, such as heat and mass transfer, vibrating string problems when the string loads with additional point masses, diffraction problems, and so on (see previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references cited therein). It is well-known that the spectrum of the standard self-adjoint Sturm-Liouville problem is infinite (see Zettl 16 ), and lots of researchers studied the properties of self-adjoint or non-self-adjoint differential operator, such as the deficiency indices, the dependence of eigenvalues, inverse problems, the self-adjoint realization, the oscillation of eigenfunctions, and so on (see other works 8,14,15,[17][18][19][20].…”

Eigenvalue problems of second order impulsive differential equations

“…tending to +∞. Recently, a few authors discussed boundary value problems for impulsive differential equations (see [5][6][7][8][9][10][11][12][13] and the references therein). Faydaoglu and Guseinov [14] had considered the following eigenvalue problems of differential equations with impulsive perturbation…”