On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

The boundary value problem with a spectral parameter in the boundary conditions for a polynomial pencil of the Sturm-Liouville operator is investigated. Using the properties of the transformation operators for such operators, the asymptotic formulas for eigenvalues of the boundary value problem are obtained.


Introduction
In this paper the boundary value problem, generated on the finite interval 0 π x ≤ ≤ by equation ( ) ( ) ( ) ( ) and the boundary conditions It is known that the Sturm-Liouville problems play an important role in solving many problems in mathematical physics.There has been a growing interest in Sturm-Liouville problems with spectral parameter in boundary conditions in recent years and there are a lot of articles on this subject in the literature.For more detailed analysis we refer to the papers [1]- [9] and the references therein.In the case 1 n > the simple boundary value problem for the Equation (1) with conditions ( ) ( ) is investigated in [10] (also see [11]).
Note that many of these investigations are based on some integral representations for the fundamental solutions of the Sturm-Liouville equation called transformation operators.The transformation operators for Sturm-Liouville equation and quadratic pencil of the Sturm-Liouville equation are constructed and studied in [12] [13] and [14] [15] respectively, while the corresponding operators for the pencil (1) are investigated in [10] [16].
In this paper using the properties of transformation operators, the considering boundary value problem is investigated and asymptotic formula for the eigenvalues is obtained.
We studied in [10], the solutions and it is proved that in the sectors of complex plane ( ) the solutions ( ) , j y x λ have the following integral representations: where ( ) ( ) ,. , ,. ,.
x m D K x belong to ( ) ( ) denotes Riemann-Liouville fractional derivative of order ( ) (see [17]) with respect to t, i.e. where

Asymptotic Formulas for the Solutions and Eigenvalues
By ( ) , s x λ and ( ) , c x λ we denote the solutions of the Equation ( 1) with initial con- ditions Using integral representations (3) and formulae (4), (5), it is easy to show that for 1 2 e , Let us consider the boundary problem ( 1), (2).Denote by ( ) Zeros of the function ( ) , ; w x h λ be the solution of the Equation ( It is clear that From formulae ( 8)-( 11) we find that ( ( ) ( ) ( ) ( ) Then for ( ) where k θ and k h are constants.From this we conclude that there exists the constant for all λ , where ( ) From (20) we have that for sufficiently large positive integer k there are a finite number of zeros of ( ) . In other words, the total number of zeros of ( ) O is equal to the total number of zeros of the func- tion sin π. n λ Moreover, there exists a positive number N such that on the circle satisfies.Hence, from (28), (30) and the equality according to Rouche's theorem we conclude that ( )

∑
with the given constants kj β .
in the circle k O for sufficiently large k .Using a simple asymptotic estimations (see[2]), we obtain that zeros having sufficiently large module of the funceigenvalues of the problem (1),(2) consist of 2n series.Solving the equation ( ) 0 λ ∆ = asymptotically we find the following asymptotic formula for th m series of eigenvalues of the problem (1), (2):

Theorem 2 .
Boundary value problem (1), (2) has a countable number of eigenvalues.The eigenvalues having sufficiently large module are placed near the rays th m series of these satisfy the asymptotic formula (23).