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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 306917, 8 pages
Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument
1Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, 59030 Tekirda####^~^~^~^~^~^####x11f;, Turkey
2Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
3Department 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
Received 18 November 2012; Accepted 1 March 2013
Academic Editor: Suh-Yuh Yang
Copyright © 2013 Erdo####^~^~^~^~^~^####x11f;an ####^~^~^~^~^~^####x15e;en et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.
Delay differential equations arise in many areas of mathematical modelling, for example, population dynamics (taking into account the gestation times), infectious diseases (accounting for the incubation periods), physiological and pharmaceutical kinetics (modelling, for example, the body####^~^~^~^~^~^####x2019;s reaction to CO2, and so forth, in circulating blood) and chemical kinetics (such as mixing reactants), the navigational control of ships and aircraft, and more general control problems. Also, differential equations and nonlinear differential equations have been studied by many mathematicians in several ways for a long time cf. [1####^~^~^~^~^~^####x2013;20].
Boundary value problems for differential equations of the second order with retarded argument were studied in [5####^~^~^~^~^~^####x2013;10, 13####^~^~^~^~^~^####x2013;16], and various physical applications of such problems can be found in .
In the papers [13####^~^~^~^~^~^####x2013;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.
In this paper we study the eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument. Namely, we consider the boundary value problem for the differential equation on , with boundary conditions and transmission conditions where the real-valued function is continuous in and has finite limits the real-valued function continuous in and has finite limits if then ; if then ; if then ; is a real spectral parameter; , , , , , are arbitrary real numbers such that and .
It must be noted that some problems with transmission conditions which arise in mechanics (thermal condition problem for a thin laminated plate) were studied in .
Consequently, the function is defined on by the equality is a solution of (1) on , which satisfies one of the boundary conditions and four transmission conditions.
Lemma 1. Let be a solution of (1) and . Then the following integral equations hold:
Proof. Let be an eigenvalue of the problem (1)####^~^~^~^~^~^####x2013;(7) and
be a corresponding eigenfunction. Then, from (2) and (10), it follows that the determinant
and, by Theorem####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;2.2.2, in  the functions and are linearly dependent on . We can also prove that the functions and are linearly dependent on and and are linearly dependent on . Hence
for some , ####^~^~^~^~^~^####x2009;, and . We must show that and . Suppose that . From the equalities (6) and (19), we have
Since it follows that
By the same procedure from equality (7) we can derive that
From the fact that is a solution of the differential (1) on and satisfies the initial conditions (21) and (22), it follows that identically on .
By using this method, we may also find From the latter discussions of , it follows that and identically on and , but this contradicts (10), thus completing the proof.
2. An Existence Theorem
Lemma 3. (1) Let . Then for the solution of (14), the following inequality holds:
(2) Let . Then for the solution of (15), the following inequality holds:
(3) Let . Then for the solution of (16), the following inequality holds:
Proof. Let . Then from (14), it follows that, for every , the following inequality holds: If we get (26). Differentiating (14) with respect to , we have From expressions of (30) and (26), it follows that, for , the following inequality holds: Let . Then from (11), (26), and (31) it follows that, for and , the following inequality holds: Hence, if , it reduces to (27). Differentiating (15) with respect to, we get From (26) and (33), it follows that, for and , the following inequality holds: Let . Then from (16), (27), and (34) it follows that, for ,####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; and , the following inequality holds: Hence, if we procure (28).
Proof. Differentiating (16) with respect to , we readily see that With the helps of (14), (15), (16), (24), (30), and (36), we have the following: Let be sufficiently big. Then, by (26), (27), and (28), (37) may be rewritten in the following form: Obviously, for big (39) has an infinite set of roots. Thus, the proof of the theorem is completed.
3. 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 for , , for , and for , follows from Theorem####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;1.4.1 in .
Lemma 5. The following holds true:
Proof. By differentiating (16) with respect to , we get, by (43) and (44) the following: Let . Then the existence of follows from continuity of derivation for . From (46) Now let . Then and the validity of the asymptotic formula (45) follows. Formulas (43) and (44) may be proved analogically.
Proof. We consider the expression which is denoted by in (39). If formulas (40)####^~^~^~^~^~^####x2013;(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 . Its derivative, which has the form , does not vanish for close to for sufficiently big . Thus our assertion follows by Rolle####^~^~^~^~^~^####x2019;s Theorem.
Let be sufficiently big. In what follows we will denote by the eigenvalue of the problem (1)####^~^~^~^~^~^####x2013;(7) situated near . We set . Then from (39) it follows that . Consequently Formula (48) makes it possible to obtain asymptotic expressions for eigenfunction of the problem (1)####^~^~^~^~^~^####x2013;(7). From (14), (30), and (40), we get From expressions of (15), (31), (39), and (41), we easily see that By substituting (48) into (50), (52), we find that Hence the eigenfunctions have the following asymptotic representation: Under some additional conditions the more exact asymptotic formulas which depend upon the retardation may be obtained. Let us assume that the following conditions are fulfilled.
(a) The derivatives and exist and are bounded in and have finite limits , and ,####^~^~^~^~^~^####x2009; , respectively.
(b) in , , , and .
It is easy to see that, using (b) are obtained.
Under the conditions (a) and (b) the following formulas: can be proved by the same technique in Lemma####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;3.3.3 in .
Putting the expressions (56), (57), and (58) into (37), and then using (59), after long operations we have Hence Again, if we take , then from (48) Hence for big , and finally Thus, we have proven the following theorem.
Now, we may obtain sharper asymptotic formulas for the eigenfunctions. From (14), (56), and (59), we have Now, replacing by and using (64), we have From (15), (57), (59), and (65) we have Now, replacing by and using (64), we have
Thus, we have proven the following theorem.
Theorem 8. If conditions (a) and (b) are satisfied, then the eigenfunctions of the problem (1)####^~^~^~^~^~^####x2013;(7) have the following asymptotic formula for : where , and are determined as in (49), (68), and (70), respectively.
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