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We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

Various modifications of classical Sturm-Liouville problems have attracted a lot of attention in the recent past because of the appearance of new important applications in mathematics, mechanics, physics, electronics, geophysics, meteorology, and other branches of natural sciences (see [

In this paper we will examine a new type of Sturm-Liouville equation involving an abstract linear operator

Some special cases of this problem arise after an application of the method of separation of variables to the varied assortment of physical problems. For example, some boundary value problems with transmission conditions arise in heat and mass transfer problems [

For operator-theoretic interpretation we will introduce some modified Hilbert spaces according to boundary-transmission conditions. For this throughout in the below we will assume that

It is readily seen that the modified inner product (

Denoting

Consequently we can reformulate the boundary value-transmission problem (BVTP) (

The linear operator

It is enough to prove that, if

Now let

The operator

Let

The eigenvalues of

For shortening denote

To establish the topological isomorphism and coerciveness we need to introduce a new inner product space

Let

If the operator T is compact from

It is obvious that the linear operator

Let

From the coercive estimate (

Let

If the operator

At first show that the embedding

Define a linear operator

The eigenvalues of the problems (

Let

Let

The following theorem can be deduced from Theorem 3.2 in [

Let

The operator

Let

Let the operator

the spectrum of

for any arbitrary small

for the sequence of eigenvalues

holds.

From asymptotic formula (

Under conditions of previous lemma the spectrum

Taking in view that for all small

The main result of this section is the following theorem.

Let the operator

By virtue of Theorem

Let us give some examples of abstract linear operator

Consequently, the results of this study can be applied to the wide variety class of boundary value problems.

All results in this study are derived under condition

The authors declare that there is no conflict of interests regarding the publication of this paper.