Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

1 Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, 59030 Tekirdağ, Turkey 2Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey 3 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea 4Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

In the papers [13][14][15][16], the asymptotic formulas for the eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument and a spectral parameter in the boundary conditions were derived.In spite of their being already a long years, these subjects are still today enveloped in an aura of mystery within scientific community although they have penetrated numerous mathematical field.
The asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in [17][18][19][20].

Asymptotic Formulas for Eigenvalues and Eigenfunctions
Now we begin to study asymptotic properties of eigenvalues and eigenfunctions.In the following we will assume that is sufficiently big.From ( 14) and (26), we obtain By ( 15) and ( 27), this leads to By ( 16) and (28), this leads to The existence and continuity of the derivatives   1 (, ) for 0 ≤  ≤ ℎ Now let  ≥ 2 3 .Then   ≤ 2 0 and the validity of the asymptotic formula (45) follows.Formulas ( 43) and (44) may be proved analogically.
Theorem 6.Let  be a natural number.For each sufficiently big  there is exactly one eigenvalue of the problem (1)-( 7) near  2 .
Proof.We consider the expression which is denoted by (1) in (39).If formulas (40)-( 45) are taken into consideration, it can be shown by differentiation with respect to  that for big  this expression has bounded derivative.We will show that, for big , only one root (39) lies near to each .We consider the function () =  sin  +  (1).Its derivative, which has the form   () = sin  +  cos  + (1), does not vanish for  close to  for sufficiently big .Thus our assertion follows by Rolle's Theorem.