Abstract
In this paper the operatortheoretical method to investigate a new type boundary value problems consisting of a twointerval SturmLiouville equation together with boundary and transmission conditions dependent on eigenparameter is developed. By suggesting our own approach, we construct modified Hilbert spaces and a linear operator in them in such a way that the considered problem can be interpreted as a spectral problem for this operator. Then we introduce socalled left and rightdefinite solutions and give a representation of solution of the corresponding nonhomogeneous problem in terms of these onehand solutions. Finally, we construct Green’s vectorfunction and investigate some important properties of the resolvent operator by using this Green’s vectorfunction.
Keywords:
SturmLiouville problems; eigenparameterdependent boundary and transmission conditions; Green’s function; resolvent operator1 Introduction
Many important special equations which appear in physics, such as Airy’s equation, Bessel’s equation, wave equation, heat equation, Schrödinger’s equation, Heun’s equation, advectiondispersion equation, etc., are associated with SturmLiouville type operators. For instance, the onedimensional form of the advectiondispersion equation for a nonreactive dissolved solute in a saturated, homogeneous, isotropic porous medium under steady, uniform flow is
where
This example makes it clear that the SturmLiouville problems are of broad interest. There is a welldeveloped theory for classical SturmLiouville problems (see, e.g., [15] and the references therein). Details of the derivation of the theory and of related background results can be found in the cited references. Although the subject of SturmLiouville problems is over 160 years old, these problems are an intensely active field of research today. The main tool for solvability analysis of such problems is the concept of Green’s function. Green’s functions have played an important role as a theoretical tool in the field of physics, since the possibility of a transition from the problems in mathematical physics to integral equations is based on the fundamental concept of Green’s function. Therefore, the powerful and unifying formalism of Green’s functions finds applications not only in standard physics subjects such as perturbation and scattering theory, boundstate formation, etc., but also at the forefront of current and, most likely, future developments (see [6]). Green’s function transforms the differential equation into the integral equation, which, at times, is more informative. In terms of Green’s function, the BVP with arbitrary data can be solved in a form that shows clearly the dependence of the solution on the data. Namely, Green’s function approach would allow us to have an integral representation of the solution instead of an infinite series. Determination of Green’s functions is also possible using SturmLiouville theory. This leads to series representation of Green’s functions (see, e.g., the monograph [1] as well as the recent results in [7] and the references therein).
SturmLiouville type problems with transmission conditions have become an important area of research in recent years because of the needs of modern technology, engineering and physics. Many of the mathematical problems encountered in the study of boundaryvaluetransmission problem cannot be treated with the usual techniques within the standard framework of boundary value problem (see [812]). In this study we shall consider a new type of SturmLiouville problems consisting of the twointerval SturmLiouville equation
together with eigenparameterdependent boundary conditions of the form
and eigenparameterdependent transmission conditions at one interaction point
where
2 Hilbert space formulation of the problem
In certain cases the boundary value problem can be characterized by means of a uniquely determined unbounded selfadjoint operator. In these cases the eigenvalues and eigenfunctions of the boundary value problem are determined by the eigenvalues and eigenvectors of the corresponding operator; these will be called a selfadjoint case of the boundary value problem. In some cases such a characterization is not possible and these will be referred to as ‘symmetric’ cases in general. In classical point of view, our problem cannot be characterized as ‘selfadjoint case’. For ‘selfadjoint characterization’ of the considered problem (1)(5), we shall define a new Hilbert space as follows.
Denote the determinant of the ith and jth columns of the matrix
by
hold. Define a new innerproduct space ℋ as a direct sum space
for
Lemma 1 ℋ is a Hilbert space.
Proof Let
Let us now define the boundary and transmission functionals
and action low
Then problem (1)(5) can be written in the operator equation form as
Theorem 1The linear operator ℜ is symmetric in the Hilbert space ℋ.
Proof By applying the method of [22] it is not difficult to prove that the operator ℜ is densely defined in ℋ, i.e.,
where, as usual,
Further, taking in view the definition of ℜ and initial conditions (14)(19) we can derive that
Finally, substituting (8), (9) and (10) in (7) we obtain that
The proof is complete. □
Theorem 2The linear operator ℜ is selfadjoint in ℋ.
Proof Since ℜ is symmetric and densely defined on ℋ, it is sufficient to show that if
for all
On the other hand, by two partial integrations we get
Thus,
From this equality, by applying the technique of Theorem 2.5 in our previous work
[11], it can be derived easily that
Theorem 3The operator ℜ has only point spectrum, i.e.,
Proof It suffices to prove that if
3 Leftdefinite and rightdefinite solutions
In this section we shall define two basic solutions
and
respectively. By using these solutions we shall define the other solutions
and
respectively. The existence of these solutions follows from the wellknown CauchyPicard
theorem of ordinary differential equation theory. Moreover, by applying the method
of [20], we can prove that each of these solutions are entire functions of the parameter
4 Construction of Green’s function
In this section we develop the idea of a resolvent operator to solve nonhomogeneous boundaryvalue transmission problems (BVTP) as follows. Consider the operator equation
for arbitrary
Let us define the Wronskians
where the functions
and the functions
for
where
By differentiating we have
By using (26), (27) and conditions (22) we can derive that
and
Putting in (26) gives
Thus we find the needed resolvent function
from (28) and (29) we have that the considered problem (21)(22) has a unique solution given by
5 Representations of the resolvent operator in terms of Green’s vectorfunction
We now shall define Green’s vectorfunction as follows:
Consequently, for the solution
Using this, the resolvent function (30) can be written in the form
where
Theorem 4For the resolvent operator
where
Theorem 5The estimation
holds for all regular value
Proof Let
Using the wellknown CauchySchwarz inequality, we conclude that
Consequently,
The proof is complete. □
Theorem 6The resolvent operator
Proof Let
Similarly to [22] we can easily show that
Competing interests
The author declares that she has no competing interests.
Acknowledgements
The author is grateful to anonymous referees for their constructive comments and suggestions, which led to the improvement of the original manuscript.
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