Aydemirand Mukhtarov Boundary Value Problems (2016) 2016:82 DOI 10.1186/s13661-016-0589-4

0 Boundary Value Problems

a SpringerOpen Journal

RESEARCH

Open Access

Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem

CrossMark

Kadriye Aydemir1* and Oktay S Mukhtarov2,3

Correspondence: kadriyeaydemr@g mail.com 1 Faculty of Education, Giresun University, Giresun, 28100, Turkey Fulllist of author information is available at the end of the article

Abstract

The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).

Keywords: Sturm-Liouville problems; boundary-transmission conditions; eigenvalues; Fourier series of eigenfunctions; minimax principle

ft Spri

ringer

1 Introduction

Sturm-Liouville eigenvalue problems appear frequently in solving several classes of partial differential equations, particularly in solving the heat equation or a wave equation by separation of variables. Other examples of Sturm-Liouville boundary value problems are Hermite equations, Airy equations, Legendre equations etc. Also, many physical processes, such as the vibration of strings, the interaction of atomic particles, electrodynamics of complex medium, aerodynamics, polymer rheology or the earth's free oscillations, yield Sturm-Liouville eigenvalue problems (see, for example, [1-6] and references therein).

In different areas of applied mathematics and physics many problems arise in the form of boundary value problems involving transmission conditions at the interior singular points. Such problems are called boundary value-transmission problems (BVTPs). For example, in electrostatics and magnetostatics the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem (see [7] and references therein). Another completely different field is that of'hydraulic fracturing' (see [8]) used in order to increase the flow of oil from a reservoir into a producing oil well. Some problems with transmission conditions arise in thermal conduction problems for a thin laminated plate (i.e. a plate composed by materials with different characteristics piled in the thickness; see [9]). Some aspects of spectral problems for differential equations having singularities with classical boundary conditions at the endpoints were studied among others in [10-23] and references therein.

© 2016 Aydemir and Mukhtarov. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tionalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution,and reproduction in any medium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

In this paper we shall investigate some qualitative properties of the eigenvalues and the corresponding eigenfunctions of one boundary value problem which consists of the Sturm-Liouville equation,

t(u) := -u" + q(x)u = Xu(x), x e (-n, 0) U (0, n), (1)

together with end-point conditions given by

i1u := cosau(-n) +sinau'(-n) = 0, (2)

l2u := cos ¡3u(n) +sin¡3u'(n) = 0, (3)

and with transmission conditions at the interior singular point x = 0 given by

tiu := nu(0-) - 5iu(0+) = 0, (4)

t2U := Y2u'(0-) - S2u'(0+) =0, (5)

where q(x) is a real-valued function; Si, Yi (i = 1,2) are real numbers; a, fi e [0, n); X is a complex spectral parameter. Throughout we shall assume that q(x) is continuous in ^ := [-n, 0) and := (0, n] with finite one-hand limits q(0±); Yi Y2 > 0, and SiS2 > 0.

It is the aim of this study to investigate such important spectral properties as the eigenfunction expansion, Parseval's equality, the Rayleigh-Ritz formula (minimization principle), the minimax principle, and monotonicity of the eigenvalues for the Sturm-Liouville problem (1)-(5). The 'Rayleigh quotient' is the basis of an important approximation method that is used in solid state physics as well as in quantum mechanics. In the latter, it is used in the estimation of energy eigenvalues of nonsolvable quantum systems.

Often in physical problems, the sign of the eigenvalue X is quite important. For example, the equation ^ + Xh = 0 occurs in certain heat flow problems. Here, positive X corresponds to exponential decay in time, while negative X corresponds to exponential growth. In the vibration problems dh + Xh = 0 only positive X corresponds to the 'usual' expected oscillations.

The Rayleigh quotient cannot be used to explicitly determine the eigenvalue since the eigenfunction is unknown. However, interesting and significant results can be obtained from the Rayleigh quotient without solving the differential equation. Particularly, it can be quite useful in estimating the eigenvalues.

2 Preliminary results about eigenvalues and eigenfunctions

Inthe direct sum ofthe Lebesgue spaces H := i2(^1) ® ¿2(^2) we shall define anew inner product in terms of the coefficients of the considered transmission conditions as follows:

/0- pn

f (x)g(x) dx + S1S2 f (x)g(x) dx. (6)

Remark 2.1 It is easy to see that the space H is also a Hilbert space with respect to the modified inner product (6).

Lemma 2.2 Let u and u be eigenfunctions ofBVTP (l)-(5) corresponding to distinct eigenvalues X and ¡x, respectively. If X = x then u and u are orthogonal in the Hilbert space H, i.e.

/0- pn

uu dx + S1S2 I uu dx = 0.

Proof Since t(u) = Xu and t(u) = ¡xu,

(X - ¡) (u, u)H = (Xu, u)H - (u, ¡xu)H

= (t(u), u)H - (u, t(u)H. (7)

By using the Lagrange identity we have

(t (u), u )H - (u, t (u))H = Y1Y2 W (u, u ; x)^ + M2 W (u, u; x)|£+, (8)

where W(u, u; x) denotes the Wronskians of u and u. The boundary conditions (2) and (3) implies W(u, u; -n ) = W (u, u ; n ) = 0. Further the transmission conditions (4) and (5) imply

YiY2 W(u, u; 0°) = S1S2.W(u, u;0+). (9)

By using these equations we get (X - /x)(u, u)H. Thus, X = ¡x implies <u, u)H = 0, which completes the proof. □

Theorem 2.3 All eigenvalues of the BVTP (l)-(5) are real.

Proof Let (X0,u0(x)) be any eigen-pair of the problem (l)-(5). Taking the complex-conjugate of the BVTP (l)-(5) we see that the pair (X0, u0(x)) is also an eigen-pair of this problem. From the boundary-transmission conditions (2)-(5) it follows easily that

W(u0, u>;-n) = W(u0, M0; n) = 0 (10)

Y1Y1 W(u0,M0;0-)- 5i52W(u0,M0;0+) =0. (11)

Putting these equalities in the equality (7) we have (X - X0)||u0||2 = 0. This implies that X0 - X0 = 0, i.e. X0 is real. □

Remark 2.4 Let X0 be an eigenvalue of (1)-(5) with corresponding eigenfunction u0(x) = u0(x) + i«0(x), where u0(x) and «0(x) are real-valued. Then both u0(x) and «0(x) are also eigenfunctions corresponding to the same eigenvalue X0. Indeed, putting u = u0 = u0 + iw0 and X = X0 in (1)-(5) and in view of X0 being real, we have

t (U0) + it («0) = (XU0) + i(X«0),

li(u0) + ili((o<0) = 0 and ti(u0) + iti («0) = 0, i = 1,2,

from which it follows that both v0(x) and rno(x) are eigenfunctions corresponding to the same eigenvalue Xo.

Theorem 2.5 There exists only one independent eigenfunction corresponding to each eigenvalue of the BVTP (1)-(5), i.e. each of eigenvalues of this problem is geometrically simple.

Proof By way of contradiction suppose that there exist two linearly independent eigenfunctions uo(x) and uo(x) corresponding to the same eigenvalue Xo. The boundary conditions (l)-(3) imply that W(uo, uo; n) = o and consequently W(uo, uo;x) = o for all x e [-n,o). Since uo(x) and uo(x) satisfy equation (l), uo(x) and uo(x) are linearly dependent on ^ by the well-known theorem of ordinary differential equation theory, i.e. there exists a constant cl =o such that uo(x) = cluo(x) for all x e Similarly, from the second boundary condition it follows that there exists a constant c2 =o such that uo(x) = c2uo(x) for all x e Œ2. Hence

Iciuo(x) for x e ^i, uo(x) = \ . . _ ^ (12)

I c2uo(x) for x e Œ2.

Substituting the transmission conditions (4)-(5) we have

(ci- c2)yiu^o ) = (ci- c2)5iu^o^ = o

(ci - c2)Y2Uo(o-) = (ci - c2)52uo(o+) = o.

From these equalities we get c1 - c2 = o. Consequently uo(x) and uo(x) are linearly dependent on the whole ^ = ^ U Œ2. Hence we have obtained a contradiction, which completes the proof. □

Remark 2.6 By virtue of Theorem 2.5 the eigenfunctions of a BVTP (1)-(5) can be chosen to be real-valued. Indeed, let X0 be an eigenvalue with the eigenfunction u0(x) = u0(x) + i«0(x). By Remark 2.4 both u0(x) and w0(x) are also eigenfunctions corresponding to the same eigenvalue. By Theorem 2.5 there is a complex number C0 =0 such that &>0(x) = C0u0(x). Hence u0(x) = u0(x) + irn0(x) = (1 + iC0)u0(x), i.e. here is only one real-valued eigenfunction, except for a constant factor, corresponding to each eigenvalue. In view of this fact, from now on we can assume that all eigenfunctions of the BVTP (1)-(3) are real-valued.

Now from Lemma 2.2, Theorem 2.3, and Remark 2.6 we have the next corollary.

Corollary 2.7 Let u1 and u2 be eigenfunctions of BVTP (1)-(5) corresponding to distinct eigenvalues X1 and X2. Then u1 and u2 are orthogonal in the sense of the following equality:

/o- pn

u1(x)u2(x)dx + S1S2 I u1(x)u2(x)dx = o. (13)

3 Reduction of (1 )-(5) to the integral equation with the Green kernel

Let u1(x, X) be the solution of equation (1) on the left interval ^ (the so-called left-hand solution) satisfying u1(-n) = sin a, u[(-n) = -cos a. Next we proceed from u1(x, X) to define the right-hand solution u2(x, X) of equation (1) on the right-hand interval by the initial conditions

u(0) = — u^0-, X), u'(0) = — u1(0-, X). Y1 Y2

Now, let v2(x, X) be the solution of equation (1) on the right-hand interval satisfying the initial conditions v1(-n) = sin j, v[(-n, X) = - cos j. Similarly we proceed from v2(x, X) to define the left-hand solution v1(x, X) of equation (1) on the left-hand interval by the initial conditions

8 8 v(0) = - v2(0+, X), v2(0) = — v^0+, X). Y1 Y2

The existence of the solutions ui and vi (i = 1,2) is obvious. Moreover, by using totally similar arguments as in [24] we can prove that each of these solutions is an entire function of the parameter X e C for each fixed x. Since the Wronskian W[ui(x, X), vi(x, X)] is independent of the variable x e (i = 1,2), we can denote «i(X) := W[ui(-, X), vi(-, X)] (i = 1,2). Using the transmission conditions (4)-(5) it is easy to see that y1Y2«1(X) = 8182«2(X). Both sides of this equality we shall denote by «(X). Now consider the following nonhomoge-neous BVTP:

t (u)-Xu = f, li (u) = ti (u) = 0, i = 1,2. (14)

Let us define a Banach space ®Ckas

0Ck(^):= if = if(1)(x)) for x e ^ : fa)(x) e Ck [-n,0],f(2)(x) e Ck [0, n] 1 [ [f(2)(x) for x e ^2 J

(k = 0,1,2,...) with the norm |f ||eckw := maxf Hck,0], |f(2) Hak[0,n]}. Below instead of 0C°(^) we shall write eC(ft).

Theorem 3.1 Letf e ©C(^). Then for X not an eigenvalue, the nonhomogeneous BVTP (14) has a unique solution uffor which the following formula holds:

Y1Y2{v1(x,X) j:n ux(%,X)f (H) dH + u1(x,X) V1(H,X)f (H) dH}

+ 8^u1(x,X)/0+ V2(H,X)f (H)df forx e [-n,0), 8182{V2(x, X) /;+ u2(H, X)f (H) df + ua(x, X) £ V2(H, X)f (H) df} ( ) + Y1Y2V2(x, X) f°n u1 (H, X)f(H) dH forx e (0, n ].

Proof By differentiating equation (15) twice we can easily see that t (u) = Xu + f, li (uf) = ti (uf) = 0 (i = 1,2) so the function uf given by (15) is the solution of the problem. We shall prove the uniqueness by way of contradiction. Suppose that there are two different solutions u0 and u0 of the system (14) corresponding to the same X0, which is not an eigenvalue. Denoting «0 := u0 - u0 we get t(«0) = X0«0, li(«0) = ti(«0) = 0 for i = 1,2, i.e.

uf (x, X) =

X0 is an eigenvalue with the corresponding eigenfunction «0. So we get a contradiction, which completes the proof. □

Let us introduce to the consideration the function G(x, f, X) given by

u\(x, X)vi(f, X) for- n < f < x < 0,

, X)v1(x, X) for- n < x < f < 0,

u2(f, X)v1(x, X) for- n < x <0<f < n,

u2(x, X)v1(f, X) for- n < f <0<x < n,

u2(x, X)v2(f, X) for0< f < x < n,

u2(f, X)v2(x, X) for0< x < f < n.

G(x, f; X) = 1

Then equation (15) can be written in the following form:

/0- pn

G(x, f; X)f (f) df + S1S2 G(x, f; X)f (f) df, (17)

i.e. uf (x, X) = (G(x, •, X),f (-)>H. Consequently the function G(x, f, X) given by (16) is the Green's function for the considered BVTP. Now suppose that X = 0 is not an eigenvalue and letf e ©C(^) be an arbitrary function. Denoting G(x, f) = G(x, f ;0) we have

r (u)=f, li(u) = ti(u) = 0, i = 1,2, (18)

has an unique solution u = u(x) given by

/0- pn

G(x, f)f (f) df + S1S2 G(x, f )f(f) df. (19)

Puttingf = Xu in equation (19) we have the following integral equation with Green's kernel: u(x) = X^y1y2 J G(x, f )u(f) df + G(x, f )u(f) df^j. (20)

4 Uniform and mean-square convergence of the eigenfunction expansions

Let us define the integral operator F by

/0- pn

G(x, f )u(f) df + S1S2 G(x, f )u(f) df. (21)

Then the BVTP (1)-(5) converts to the spectral problem for the integral operator F given by

(I - XF)u = 0,

where I is the identity operator. Since the kernel G(x, f) of the integral operator F is symmetric and continuous we can apply the well-known extremal principle (see, for example, [25]). Let {X„j be a sequence of eigenvalues of the integral operator F determined by the extremal principles and {^„(x)j be the corresponding sequence of orthonormal eigenfunc-tions.

Lemma 4.1 Let g e фС(^). Then

Jim^YiYij ($g - j^Ct(Fg)<p)j dx + SiSij^Ug - j^cdSg^A dxj = 0,

where ci(Fg) = {Fg, pi)H denote the Fourier coefficients ofFg with respect to the orthonormal set (pi).

Proof Denote gm(x) =g(x) - ^m=i{g,PÙhPî. Since {pn} is the orthonormal system in H, {gm, pi)H = 0 for i = 1,..., m. From the fact that the eigenvalues Xn are determined by the extremal principle with the corresponding sequence of orthonormal eigenfunctions {pn} we have ||Fgm IIh < iXm+i ilIgm IIh. Since Xm+i ^ 0, ||Fgm IIh ^ 0. Then we have

Fg = Fgm + (g,F$i = Fgm + £ Xi(g, $i>H

i=1 i=1 mm

= Fgm + J2 (g, F$i>H$i = Fgm + J2 (Fg,&>H$i (23)

i=1 i=1

for arbitrary m = 1,2,____Letting m ^^ we get

Fg = £ (Fg, <&>h (24)

where the convergence is in the Hilbert space H, i.e. the equality (22) holds. □

Corollary 4.2 Ifg e ©C(^) then the Parseval equality

= £ c2(Fg)

holds.

Corollary 4.3 The set of orthonormal eigenfunction of the integral operator F is complete in the range of the integral operator F given by

R(F) = {h e фС(^)| there exists g e еС(^) such that h = Fg}.

Theorem 4.4 Let the hypotheses and notation of Lemma 4.1 hold. Then, for any h e R(F),

hфi dx + S1SW hфi dx ) Ф1 (x),

TO / „0- pn

h = £ \Y1Y2J hфi dx + S1S2 J

where the series converges with respect to the norm ©C(^), i.e. uniformly on ^ = ^ U

Proof Let h = Fg. Then for any n, p we have

J^Xiig, &)h & = F

J2<g, & )h &

In view of the fact that the integral operator F is a bounded linear operator in the Banach space ©C(^) we get from (25)

£Xig &)H &

£l<g, &

for some constant C independent of n. By Bessel's inequality, the right-hand side of this inequality tends to zero as n uniformly. Thus the series

£<fg, &)h &i(x)

converges in the Banach space ©C(^). Let h (x) be the sum of the last series. Consequently h e ©C(^) and

h (x) = £ <Fg,&)h &i(x).

From (23) and (28) it follows that ||Fg - h||H = 0, i.e. h(x) = h(x) almost everywhere. Since h is also continuous in ^ we have h(x) = h(x) for all x e Thus

h = £ <h,&)h &,

where the series converges with respect to the norm of ©C(^), i.e. uniformly on □

Theorem 4.5 The set of all nonzero eigenvalues of the integral operator F coincide with the set of the eigenvalues (Xn) which are obtained from the extremal principle.

Proof By way of contradiction, suppose there is a nonzero eigenvalue X* distinct from all eigenvalues (Xn). Let u* be the eigenfunction corresponding to the eigenvalue X*. Then from Theorem 4.4 we get

X*u* = Fu* = £ (Fu*, &)H & = X* £ (u*, &)H & = 0

since <u*, &i)H = 0 for all i = 1,2,... by Theorem 2.7. Thus we get a contradiction. □

Theorem 4.6 Letf e ®C2(^) and satisfy the boundary-transmission conditions (2)-(5). Then the Fourier series off with respect to {&} converges uniformly on U i.e.

lim {sup

n / p 0- p n \2

f (x)-£\y1y2j f&idx + 8182 J f&idxj &

Proof Let f e ©C2(fi) satisfy the boundary-transmission conditions (2)-(5) and denote g = tf). Theng e ©C(fi). By virtue of (17) and (21) we havef = Fg. From Lemma 4.1,

f = Fg = £ <Fg, &i)h & = Y, f, &)h&>

where the series is convergent in the Banach space ©C(fi).

Theorem 4.7 The set of eigenfunctions {&i(x)} is a complete orthonormal set in the Hilbert space H.

Proof Denote by ©C^ (fi) the set of all functionsf e Ck (fi) which vanishes at some neighborhoods of the points x = -n, x = 0, and x = n. Letf e H and e > 0 be given. Then there exists a functiong e ©C^fi) such that |f -g|H < § since the set ©C^fi) is dense in the Hilbert space H, i.e. ©C2(fi) = H (see, for example, [26]). It is clear that

f -E <f,&)h &

< If -g\\H +

g -J2 <g, &i)H&

E((g -f),&)h & i=1

for arbitrary m. By Bessel's inequality we have

X> -f), &)h &

m i \ 2

= £K (g-f), &)h I2 <ff - gI\H

and, by Theorem 4.6, there exists an integer n0 = n0(e) such that, for m > n0,

g -£ <g, &i)H&i i=1

Finally, from (32) and (33) we get

f -£ f, &i)H &

for m > n0. The proof is complete. □

Now we are ready to prove the next important result.

Theorem 4.8 The set of eigenfunctions (&i(x)) of the problem (1)-(5) form an orthonormal basis in the Hilbert space H and for anyf e H the Parseval equality

/ f 0- f n

^2\Y1Y2 J f&idx + 8x82 J f&idx

holds.

Proof Without loss of generality we shall assume that k = 0 is not an eigenvalue. Otherwise, we can select a real k0 = 0 such that the problem tu = k0u, ii(u) = ti(u) = 0, i = 1,2 has no nontrivial solutions. Then denoting k = k - k0 and q(x) = q(x) - k0 we see that the problem

-u" + q (x)u = k u, ii(u) = ti(u) = 0, i = 1,2, (34)

has the same properties for the eigenfunctions and eigenvalues as the considered problem (1)-(5). Namely, the pair (k, u(x)) is the eigen-pair of the problem (34) if and only if the pair (k, u(x)) is an eigen-pair of (1)-(5). Clearly, k = 0 is not an eigenvalue of the problem (34). Hence, without loss of generality we can assume that k = 0 is not an eigenvalue of the considered BVTP (1)-(5). Moreover, if k =0, then the pair (k, u(x)) is the eigen-pair of the BVTP (1)-(5) if and only if the pair (¿, u(x)) is the eigen-pair of the integral operator F. Consequently the set fai form an orthonormal set of eigenfunctions either for F and (1)-(5). Moreover, this set is complete by Theorem 4.7. It is well known that any complete orthonormal set in a Hilbert space forms an orthonormal basis. Consequently, every function f e H may be expanded in a Fourier series with respect to the orthonormal set of eigenfunctions (fai), i.e. the equality

f = £ f, fai)H fa

holds, where the series converges with respect to the norm of the Hilbert space H. Further, the Parseval equality follows immediately from the last equality. □

5 The Rayleigh-Ritz principle for the BVTP (1 )-(5)

In the last sections of this study we will investigate some extremal properties of the eigenvalues and corresponding eigenfunctions of the considered BVTP (1)-(5) by using some variational methods.

Lemma 5.1 Let q(x) > 0 for allx e Then all eigenvalues of the problem (1)-(5) are nonnegative.

Proof Let (k, u(x)) be any eigen-pair of the problem (1)-(5). Multiplying (1) by u(x) and integrating by parts from x = -n to x = 0, and from x = 0 to x = n ,we have

u[u" - qu + ku] dx + S1S2 I u[u" - qu + ku] dx

/0- pn

[-u'2 - qu2 + ku2] dx + SiS2 I [-u'2 - qu2 + ku2] dx

+ y\y2uu!\_n + 5i52uu'in+ = 0. (35)

By using the equalities (4)-(5), we get Y1Y2uu'\07r + 5152uu'\n+ = 0. Hence

/0- pn

[-u'2 - qu2 + ku2] dx + 5152a I [-u'2 - qu2 + ku2] dx = 0. (36)

Consequently

Y1Y2 1°- [u'2 + qu2] dx + S1S2 /°+ [u'2 + qu2] dx

k = --n- — ° (37)

yiy2 f°n u2 dx + 5i52 fQ + u2 dx

since q — ° on U by assumption. □

Theorem 5.2 Let q(x) — ° for all x e Then, all the eigenvalues of the problem (1)-(5) are positive if any one of the following conditions holds:

(1) q ^ i.e. there exists at least one x° e ^ such that q(x°) >

(2) cos2 a + cos2 fi =

Proof Let Xl be the first eigenvalue with the corresponding eigenfunction ul(x). Show that k >°.

(1) Since q(x) is continuous in ^ there are S > ° and q° > ° such that q(x) — q° for all x e [x° - S,x° + S] c Then from (37) it follows immediately that kl > °.

(2) Suppose that it possible that kl = °. Then from (37), ul(x) = ° for all x e i.e. ul(x) is a constant function in each of and Putting in (2) and (3) we have cosau1(-n) = cos fiul (n) = °. Consequently at least one of ul(-jr) and ul(jr) is equal to zero and therefore ul is identically zero or Then by applying the transmission conditions (3) and (4) we see that ul is identically zero on the whole ^ = U We have a contradiction, which completes the proof. □

Theorem 5.3 Suppose that any one of the following conditions holds:

(1) q ^ ° and q(x) — °;

(2) q(x) — ° and cos2 a + cos2 fi =

Let kl < k2 < ••• be the sequence of eigenvalues with the corresponding normalized eigen-functions ^l(x), <2(x),... and let

Sn = {u(x)|u e©C2(^); u ^ l (u) = ti(u) = ° for i = l,2; (u, <k )H = °fork = l, 2, ...,n -l}, n = l, 2,...

(naturally by Sl we mean Sl = {u(x)|u e ®C2(^); u ^ li(u) = ti(u) = °for i = l, 2}). Then for alln = l, 2,... we have

kn = min{I(u) |u e S^,

where the functional I (u) is given by

I(u) = —ly| y1 Y2 f [u'2 + qu2] dx + SlS2 i [u'2 + qu2] dxl. (38)

llullH [ J-TC J°+ J

Moreover, the minimizingfunction is <n, i.e. kn = I(<n).

Proof Let <p(-) e ®C2(^) with liy = tiy = °, i = 1,2. Then by Theorem 4.6 we have

y(x) = J2(y(^), <n()H<n(x),

where the convergence is uniform on Then, by integration by parts, we get

(t(P), fan)H = t(fan))H = kn[fan V(-))H. ()

Since {fan(x)} is a complete orthonormal set, by Parseval's equation

(v, <P)h = £ c», (40)

where Cn(v) = (fan(0, v())h. By using (35) we get

[v'2 + qv2] dx + 515W [v'2 + qv2] dx

= V, T (<p))H = -/ I £ Cnfan ), T (vn = -£ Cn( fan, T \\n=1 ) Ih n=1

= £ c„Xn<p, fan>^ = £ ^ — MM^. (1)

n=1 n=1

Consequently

[yiy2j + qv2]+ sis2f0 W1 + qq)2]. (42)

Putting p = fa1 in equation (41) we have cn(p) = <fa^fan> = 0 for n = 1,3,... and 1

Y1Y1 f fa + #1] dx + S1S1 f [fa? + ¿4. (43)

J-n Jo+ J

From (42)-(43) it follows immediately that k1 = min{I(p)\p e 51}

and the minimizing function is v = fa1(x), i.e. k1 = I(fa1). Next, let v(x) e ®C2(^) with iiv = tiv = 0, i = 1,2, and (v, fan) = 0 for n = 1,...,//. Then

/0° pn

\_p'2 + qv2] dx + S1S2 l [v'2 + qv2] dx

= knCn(v)2 > k/+1 £ CnW)2 = k/+1HvHH. (44)

'■n^nv^/ — /v-k+1 / unV

n=k+1 n=k+1

Hence, by the same arguments as before, we have

A.k+1 = min{I(p)\p e Sk+1} for k = 1,1,... and the minimizing function is fak+1, i.e. Xk+1 = I(fak+1). □

Remark 5.4 By applying the minimization principle directly, it is not possible to determine explicitly the eigenvalues and corresponding eigenfunctions, since we do not know how to minimize over all 'admissible' functions. Nevertheless, using the Rayleigh functional (38) with appropriate test functions one can obtain useful approximations for the eigenvalues.

6 The minimax property of eigenvalues

According to the minimization principle which is given by the preceding Theorem 5.3 we can find the nth eigen-pair (kn, <n) only after the previous eigenfunctions <1(x), <2(x),..., <n-1(x) are known. But in many applications it is important to have a characterization of any eigen-pair (kk, <k) that makes no reference to other eigen-pairs. By applying the following theorem we can determine the nth eigen-pair (kn, <n) without using the preceding eigenfunctions <1(x), <2(x),..., <n_1(x).

Theorem 6.1 Let u1(x), u2(x),..., un _1(x) e ©C(^) be arbitrary functions. Denote

Dn-i(ui,u2,...,un-i) = {u e ®C1(^)|ly(u) = tj(u) = °,j = 1,2;

(u, ui) = °for i = 1,2,..., n -1;}, n = 1,2,...

(naturally by D°(u1, u2,..., un_1) we mean the linear manifold

{u e 0C1(^)|lj(u) = tj(u) = °,j = 1,2}).

Then the nth eigenvalue of the BVTP (1)-(5) is

kn = max{min{I(u)|u eDn_1(u1,u2,...,un-1)}|< e ©C(^)

fori = 1,2,..., n-1}. (5)

Proof Let ° < k1 < k2 < ••• be the sequence of eigenvalues determined by the extremal principles and <1(x),<2(x),... be the corresponding sequence of orthonormal eigenfunctions and let u1, u2,..., un-1 e ©C(^) be arbitrary functions. Define

Fn(ui, u2,...,un-i) = inf{ I (u)|u e Dn-i(ui, u2,...,un_i)}, n = 1,2,....

Now let f^x),f2(x), ...,fn-1(x) e ®C1(^) be any given functions, such that li(^j) = ti(fj) = ° (i = 1,2; j = 1,2,...,n - 1). Denoting aj = < fi)H for i,j = 1,2,...,n - 1, consider a system of n - 1 homogeneous linear equations in n unknowns z1,z2,...,zn given by

Y^aijZi = °, j = 1,2,..., n-1. (46)

Obviously this system of n -1 homogeneous linear equations in n unknowns has a nontrivial solution. Let a1, a2,..., an-1 be any nontrivial solution of system (46). Define a function

Vn(x) by

vn(x) = a1<1(x) + ••• + an<n(x).

It is easy to see that

(v«, fj)H = 0 for; = 1,2,...,« -1. (48)

Consequently vn e D«°1(^1, ..., ^n-1) and ||v||H = a2 + «2 + ' ' ' + a2. Then we have

/0 pir

[v'« + qv«] dx + S1S2 [v« + qv«\ dx

= «ia^ Y1Y2 j < < + q<<j\dx + sis2 « + qfatjdXj. Integrating by parts we get

/0 pTT

« + q«]dx + 5i5W « + q«]dx

= i°n + < & + (</, r (</i))H = Xi(</, Ufai)H = ^i^i/, where Si; is the Kronecker delta. Then we have

7(vn) = "¡j-¡¡r £ — ^n.

||Un||H i=i

Consequently F«(u^u2,...,un-1) — 7(vn) — Xn for all u1,u2,...,un-1 e ©C(^). Furthermore, by virtue of the preceding theorem

F«(<h, <2,...,<n°l) = V

X« = max{F(ui,u2,...,u«°i) e ©C(^),; = 1,2,...,« -1, which completes the proof. □

Remark 6.2 In many problems of mathematical physics, the smallest eigenvalue (the so-called principal eigenvalue) plays an important role. For example, the principal eigenvalue of the simple boundary value problem

°u" = Xp(x)u, x e [—n, n], u(°jr) = u(n) = 0,

where p (x) > 0 is the given function, can be interpreted as the square of the lowest frequency of vibration of a rod of nonuniform cross section given by p(x). Therefore it is significant to determine explicitly the principal eigenvalue, or at least a 'good' estimation of it. Note that useful approximation values for the principal eigenvalue can be drawn from the minimax property (45) by using certain principles of the theory of the calculus of variations. In particular we can find an upper bound for the lowest eigenvalue X1. In fact let

0C(Œ) be any nontrivial function satisfying the boundary-transmission conditions (2)-(5) called the trial function. Then by virtue of the minimax principle we have the inequality

which gives us an upper bound for the principal eigenvalue. By taking the trial function u(x) as close as possible to the corresponding eigenfunction we can expect to get the 'good' estimation for principal eigenvalue X1. In many special cases the useful trial function can be found by applying some principles of variational analysis.

7 Dependence of eigenvalues on the potential

The minimax principle of the eigenvalues, i.e. equation (45) for the eigenvalues makes it possible to study the dependence of the eigenvalues on the coefficients of the differential equation. In this section we shall establish the monotonicity of the eigenvalues with respect to the potential q(x) for fixed boundary-transmission conditions.

Theorem 7.1 Let Xn (q) be the nth eigenvalue of the BVTP (1)-(5). Then Xn (q) is a mono-tonically increasing function with respect to the variable q = q(x), i.e. if q1(x) < q2(x) for all x e Q then Xn(q1) < Xn(q2).

Proof Define

Let the notation of the preceding theorem hold and let u1(x), u2(x),..., un-1(x) e 0C(Q) be any arbitrary functions.

Since 0 < q1(x) < q2(x) for all x e Q it is obvious that I1(u) < I2(u) for all u e Dn_1(u1,u2,...,un-1). Then by virtue of Theorem 6.1, we find the required inequality

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The authors read and approved the final manuscript. Author details

1 Faculty of Education, Giresun University, Giresun, 28100, Turkey. 2Department of Mathematics, Faculty of Arts and Science, Gaziosmanpa§a University, Tokat, 60250, Turkey. 3Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan.

Acknowledgements

The authors would like to thank the referees for their valuable comments.

Received: 1 January 2016 Accepted: 8 April 2016 Published online: 15 April 2016

References

1. Gesztesy, F, Macedo, C, Streit, L: An exactly solvable periodic Schrödinger operator. J. Phys. A, Math. Gen. 18, 503-507 (1985)

2. Kong, Q, Wu, H, Zettl, A: Geometric Aspects of Sturm-Liouville Problems. Preprint

3. Petrovsky, IG: Lectures on Partial Differential Equations, 1 st edn. Interscience, New York (1954) (translated from Russian by A Shenitzer)

4. Pryce, JD: Numerical Solution of Sturm-Liouville Problems. Oxford University Press, London (1993)

Xn (qi) < Xn (q2). The proof is complete.

5. Rotenberg, M: Studies of the quadratic Zeeman effect. I. Application of the sturmian functions. Adv. At. Mol. Phys. 6 (1970)

6. Sherstyuk, AI: Problems of TheoreticalPhysics, vol. 3. Leningrad. Gos. Univ., Leningrad (1988)

7. Huy, HP, Sanchez-Palencia, E: Phénomènes des transmission à travers des couches minces de conductivité élevée. J. Math. Anal. Appl. 47, 284-309 (1974)

8. Cannon, JR, Meyer, GH: On a diffusion in a fractured medium. SIAM J. Appl. Math. 3,434-448 (1971)

9. Titeux, I, Yakubov, Y: Completeness of root functions for thermalconduction in a strip with piecewise continuous coefficients Math. Models Methods Appl. Sci. 7(7), 1035-1050 (1997)

10. Allahverdiev, BP, Bairamov, E, Ugurlu, E: Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions. J. Math. Anal. Appl. 401 (1), 388-396 (2013)

11. Allahverdiev, BP, Ugurlu, E: On dilation, scattering and spectraltheory for two-intervalsingular differentialoperators. Bull. Math. Soc. Sci. Math. Roum. 58(106)(4), 383-392 (2015)

12. Aydemir, K: Boundary value problems with eigenvalue depending boundary and transmission conditions. Bound. Value Probl. 2014,131 (2014)

13. Aydemir, K, Mukhtarov, OS: Second-order differentialoperators with interior singularity. Adv. Differ. Equ. 2015, 26 (2015). doi:10.1186/s13662-015-0360-7

14. Bairamov, E, Sertbas, M, Ismailov, ZI: Self-adjoint extensions of singular third-order differentialoperator and applications. AIPConf. Proc. 1611 (1), 177-182 (2014)

15. Hira, F, Altinisik, N: Sampling theorems for Sturm-Liouville problem with moving discontinuity points. Bound. Value Probl. 2015,237(2015)

16. Kadakal, M, Mukhtarov, OS: Sturm-Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367-1379 (2007)

17. Kandemir, M: Irregular boundary value problems for elliptic differential-operator equations with discontinuous coefficients and transmission conditions. Kuwait J. Sci. Eng. 39(1A), 71-97 (2010)

18. Kandemir, M, Mukhtarov, OS, Yakubov, S: Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter. Mediterr. J. Math. 6, 317-338 (2009)

19. Mukhtarov, OS, Olcjar, H, Aydemir, K: Resolvent operator and spectrum of new type boundary value problems. Filomat 29(7), 1671-1680 (2015)

20. Mukhtarov, OS, Kandemir, M: Asymptotic behavior of eigenvalues for the discontinuous boundary-value problem with functional-transmission conditions. Acta Math. Sci. Ser. B 22(3), 335-345 (2002)

21. Ugurlu, E, Allahverdiev, BP: On selfadjoint dilation of the dissipative extension of a direct sum differentialoperator. Banach J. Math. Anal. 7(2), 194-207 (2013)

22. Aliyev, ZS, Kerimov, NB: Spectralproperties of the differentialoperators of the fourth-order with eigenvalue parameter dependent boundary condition. Int. J. Math. Math. Sci. 2012, Article ID 456517 (2012)

23. Mamedov, KR, Cetinkaya, FA: Inverse problem for a class of Sturm-Liouville operator with spectralparameter in boundary condition. Bound. Value Probl. 2013,183 (2013)

24. Akdocjan, Z, Demirci, M, Mukhtarov, OS: Green function of discontinuous boundary-value problem with transmission conditions. Math. Methods Appl. Sci. 30,1719-1738 (2007)

25. Courant, R, Hilbert, D: Methods of Mathematical Physics, vol. 1. Interscience, New York (1953)

26. Indritz, J: Methods in Analysis. Macmillan Co., New York (1963)

Submit your manuscript to a SpringerOpen journal and benefit from:

► Convenient online submission

► Rigorous peer review

► Immediate publication on acceptance

► Open access: articles freely available online

► High visibility within the field

► Retaining the copyright to your article

Submit your next manuscript at ► springeropen.com