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Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators

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1. Introduction

Consider the differential equation

(1)

where are complex valued functions and is a complex parameter.

Differential equations of type (1) often appear in connection with some spectral problems and nonlinear evolution equations (see [1] [2] [3] ). In the case the equation is the classical Sturm-Liouville equation and in this case there are a wide class of spectral problems and inverse spectral problems which were investigated by constructing integral representations for the independent solutions of the Sturm-Liouville equation (see [4] ). We studied in [5] , the solutions of the Equation (1) satisfying the initial conditions

and it is proved that in the sectors of complex plane

the solutions have the following integral representations:

(2)

where, and,

belong to and respectively. Moreover, if denotes Riemann-Liouville fractional derivative of order (see [6] ) with respect to t, i.e.

then for all the functions and belong to and respectively. Furthermore, the following equalities are valid:

(3)

(4)

where

(5)

In the present paper we use the above facts about special solutions of the Equation (1) to obtain some trace formulas for the boundary value problem generated by the Equation (1) in the segment with simple boundary conditions

2. Asymptotic Formulas and the Trace Formulas

Using (2), (3) and (4) it is easy to prove the following theorem where we seek two solutions which have special representations.

Theorem 1. If and then the Equation (1) has solutions

(6)

and

(7)

where

(8)

(9)

(10)

(11)

(12)

(13)

Since the solutions and are linearly independent for we have

for the solution of the Equation (1) with initial conditions

Then the Theorem 1 gives

(14)

where

(15)

(16)

(17)

Now let us connect the Equation (1) to the boundary conditions

(18)

In [2] it is obtained the asymptotic formulas for the eigenvalues of the boundary value problem (1)-(2). Let be a characteristic function of this boundary value problem. Then

(19)

where

(20)

Let us consider the circles where is sufficiently large integer. On circles the functions and are

bounded by the constants independent of. So we have that the module of the maximum of the function approaches to zero when. Hence, if are the series of eigenvalues of the problem (1), (18) we have

(21)

Using (19) and (20) we compute the integrals on the right hand side of the Equation (21) and prove the following theorem.

Theorem 2. If are the series of eigenvalues of the boundary value problem (1), (18) then

(22)

(23)

(24)

(25)

where are constants defined by the help of the functions . Here

(26)

in which the numbers are defined from the asymptotic equality

From Theorem 2 we have that if the Fourier series are conver-

gent and denoting their sums by we obtain the following regularized trace formulas for the eigenvalues of the boundary value problem (1), (18):

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Applied Mathematics*,

**7**, 2411-2417. doi: 10.4236/am.2016.718189.

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