The spectral problem is investigated for the fourth order ordinary differential
operator with spectral parameter in the boundary conditions. The oscillation
properties of the eigenfunctions of this problem are established

Aliyev, Ziyatkhan S.

Kerimov, Nazim B.

English

Khazar University Institutional Repository

Transactions of NAS of Azerbaijan 61Nazim B. KERIMOV, Ziyatkhan S. ALIYEVTHE OSCILLATION PROPERTIES OF THEBOUNDARY VALUE PROBLEM WITH SPECTRALPARAMETER IN THE BOUNDARY CONDITIONAbstractThe spectral problem is investigated for the fourth order ordinary differentialoperator with spectral parameter in the boundary conditions. The oscillationproperties of the eigenfunctions of this problem are established.Research of boundary value problems with spectral parameter in the equationand in the boundary conditions form the important part of the spectral theory oflinear differential operators. Many concrete problems of mathematical physics (seefor example, [1, pp.149-152] and [2-5]) are led to such type problems.The main purpose of this paper is to study the ocillation properties of the eigen-functions of the boundary value problem(p(x)y′′)′′ − (q (x) y′)′ = λρ (x) y, 0 < x < l, (1)y′ (0) cosα− (py′′) (0) sinα = 0, (2.a)y (0) cosβ + Ty (0) sinβ = 0, (2.b)y′ (l) cos γ +(py′′)(l) sin γ = 0, (2.c)(aλ+ b) y (l)− (cλ+ d)Ty (l) = 0, (2.d)whereTy ≡ (py′′)′ − qy′,and λ is a spectral parameter, the functions p (x) , q (x) , ρ (x) are strictly positive andcontinuous on [0, l], p (x) has an absolutely continuous derivative, q (x) is absolutelycontinuous on [0, l], α, β, γ, a, b, c, d are real constants, moreover 0 ≤ α, β, γ ≤ pi2andσ = bc− ad > 0. (3)As in [6-8] to study the oscillation properties of eigenfunctions and its derivativesthe Pru˝fer type transformation is used:y (x) = r (x) sinψ (x) cos θ (x) , (4.a)y′ (x) = r (x) cosψ (x) sinϕ (x) , (4.b)(py′′)(x) = r (x) cosψ (x) cosϕ (x) , (4.c)62[N.B.Kerimov, Z.S.Aliyev]Transactions of NAS of AzerbaijanTy (x) = r (x) sinψ (x) sin θ (x) . (4.d)We first write equation (1) as the equivalent first order systemU ′ =MU, (5)whereU =yy′py′′Ty , M =000λρ10q001/p000010 .Assuming w (x) = ctgψ (x) and using transformation (4) in (5) we’ll obtain thesystem of a first order differential equations for the functions r, w, θ, ϕ of the followingform:r′ =[sin 2ψ sin (θ + ϕ) +(q +1p)cos2 ψ sin 2ϕ++λρ sin2 ψ sin 2θ] r2, (6.a)w′ = −w2 cos θ sinϕ+ 12(q +1p)w sin 2ϕ++sin θ cosϕ− λρ2w sin 2θ, (6.b)θ′ = −w sinϕ sin θ + λρ cos2 θ, (6.c)ϕ′ =1pcos2 ϕ− q sin2 ϕ− 1wsin θ sinϕ. (6.d)The following statements hold.Lemma 1 (see lemma 3 from [8]). For each fixed λ ∈ C there exists a unique(to within constant multiplier) nontrivial solution y (x, λ) of problem (1), (2.a), (2.b),(2.c).Remark 1 (see remark 1 from [8]). Without loss of generality, for everyfixed x ∈ [0, l] the function y (x, λ) can be taken as entire function of λ.It holds the followingLemma 2. The eigenvalues of boundary value problem (1)-(2) are real.Proof. Let y (x, λ) be a nontrivial solution of problem (1), (2.a), (2.b), (2.c).Then eigenvalues of problem (1)-(2) are the roots of the equation(aλ+ b) y (l, λ)− (cλ+ d)Ty (l, λ) = 0. (7)If λ is a non-real eigenvalue of problem (1)-(2), then λ will also be eigenvalue ofthis problem. Really, as the coefficients p (x) , q (x) , ρ (x) , α, β, γ, a, b, c, d, are real,Transactions of NAS of Azerbaijan[The oscillation properties of the boundary ... ]63then y(x, λ)= y (x, λ). Consequently, if the relation (7) holds for λ, then it will beas well as true also for λ.By virtue of (1) we have(Ty (x, µ))′ y (x, λ)− (Ty (x, λ))′ y (x, µ) = (µ− λ) ρ (x) (x, µ) y (x, λ) .Integrating this identity in the range from 0 to l, using formula of integration byparts and taking into account (2.a), (2.b), (2.c), we gety (l, λ)Ty (l, µ)− y (l, µ)Ty (l, λ) = (µ− λ)l∫0ρ (x) y (x, µ) y (x, λ) dx. (8)Assuming µ = λ in (8), we’ll obtainy (l, λ)Ty (l, λ)− y (l, λ)Ty (l, λ) = (λ− λ) l∫0ρ (x) |y (x, λ)|2 dx. (9)We’ll rewrite condition (2.d) in the formTy (l, λ) =aλ+ bcλ+ dy (l, λ) .Taking into account last relation from (9) we get(λ− λ) (ad− bc) |y (l, λ)|2|cλ+ d|2 =(λ− λ) l∫0ρ (x) |y (x, λ)|2 dx.As λ 6= λ, then hence it follows the equality−σ |y (l, λ)|2|cλ+ d|2 =1∫0ρ (x) |y (x, λ)|2 dx.The last equality contradicts to the conditions σ > 0,l∫0ρ (x) |y (x, λ)|2 dx > 0.Hence, λ must be real. Lemma 2 is proved.Lemma 3. The eigenvalues of boundary value problem (1)-(2) form no morethan countable set, which has no finite limit point. All eigenvalues of boundary valueproblem (1)-(2) are simple.Proof. The eigenvalues are the zeros of the entire function standing at the left-hand side of equation (7). We show (lemma 2) that this function doesn’t vanish fornon-real λ. Hence, it is not equal to zero identically. Therefore, its zeros form nomore than countable set, which has no finite limit point.Let’s prove, that equation (7) has only simple roots. Really, if λ = λ∗ is a multipleroot of equation (7), then it holds the equalities64[N.B.Kerimov, Z.S.Aliyev]Transactions of NAS of Azerbaijan(aλ∗ + b) y (l, λ∗)− (cλ∗ + d)Ty (l, λ∗) = 0, (10)ay (l, λ∗) + (aλ∗ + b)∂∂λy (l, λ∗)− cTy (l, λ∗)−− (cλ∗ + d) ∂∂λTy (l, λ∗) = 0. (11)Dividing the both parts of (8) by (µ− λ) and passing to the limit as µ→ λ we’llobtainy (l, λ)∂∂λTy (l, λ)− Ty (l, λ) ∂∂λy (l, λ) =l∫0ρ (x) y2 (x, λ) dx. (12)Supposing in equality (12) λ = λ∗ we’ll havey (l, λ∗)∂∂λTy (l, λ∗)− Ty (l, λ∗) ∂∂λy (l, λ∗) =l∫0ρ (x) y2 (x, λ∗) dx. (13)As σ 6= 0, then (a, λ∗ + b)2 + (cλ∗ + d)2 6= 0. Let cλ∗ + d 6= 0. From (10) and(11) it follows respectivelyTy (l, λ∗) =aλ∗ + bcλ∗ + dy (l, λ∗) ,∂∂λTy (l, λ∗) =aλ∗ + bcλ∗ + d∂∂λy (l, λ∗)− σ(cλ∗ + d)2y (l, λ∗) .Using the last two equalities in (13), we get− σ(cλ∗ + d)2y2 (l, λ∗) =l∫0ρ (x) y2 (l, λ∗) dx,that is not impossible by virtue of condition (3).The case aλ∗ + b 6= 0 is considered similarly. Lemma 3 is proved.The following statements are true.Lemma 4 (see theorem 3.1 from [7]). Let y (x, λ) be a nontrivial solution ofproblem (1), (2.a), (2.b), (2.c) for λ > 0. Then the Jacobian J [y] = r3 cosψ sinψof transformation (4) does not vanish in (0, l).Lemma 5 (see lemma 5 from [8]). Let y (x, λ) be a nontrivial solutionof problem (1), (2.a), (2.b), (2.c) and one of the following conditions be fulfilled:(i) λ < 0; (ii) , λ = 0, β ∈[0,pi2). Then the Jacobian J [y] = r3 cosψ sinψ oftransformation (4) does not vanish in (0, l).Let y (x, λ) be a nontrivial solution of problem (1), (2.a), (2.b), (2.c), and eitherλ ∈ R\{0}, or λ = 0, β ∈[0,pi2). Suppose that θ (x, λ) and ϕ (x, λ) are correspondingTransactions of NAS of Azerbaijan[The oscillation properties of the boundary ... ]65functions from (4). Without loss of generality, we’ll define the initial values of thesefunctions in the following wayθ (0, λ) = β − pi2, ϕ (0, λ) = α, (14)where α = 0, when ψ (0) =pi2.The proof of this fact is led completely by the scheme of the proof of theorem 3.1from [6]. (see also [7, theorem 3.3]).Lemma 6 (see theorem 4.2. from [7]). Let y (x, λ) be a nontrivial solutionof problem (1), (2.a), (2.b), (2.c) for λ > 0. Then θ (l, λ) is a strictly increasingcontinuous function of λ.Let’s introduce the following boundary conditiony (l) cos δ − Ty (l) sin δ = 0, δ ∈[0,pi2]. (2.d′)Theorem 1 (see theorems 5.4 and 5.5 from [7]). The eigenvalues of bound-ary value problem (1), (2.a), (2.b), (2.c), (2.d’) (except the case β = δ =pi2) forman infinitely increasing sequence {λδn}∞1 such that0 < λδ1 < λδ2 < ... < λδn < ...,θ(l, λδn)= (2n− 1) pi2− δ, n ∈ N. (15)Besides, the eigenfunction υδn (x), corresponding to the eigenvalue λδn, has exactlyn−1 simple zeros in the interval (0, l), and the function Tυδn (x) has exactly n zeroson the segment [0, l].Remark 2 (see [8]). In the case β = δ =pi2the first eigenvalue of boundaryvalue problem (1), (2.a), (2.b), (2.c), (2.d’) is equal to zero and the correspondingeigenfunction is a constant. In this case the statement of theorem 1 is valid for n ≥ 2.It is obvious, that the eigenvalues µn = λ0n and νn = λpi/2n (n ∈ N) are zeros ofthe entere functions y (l, λ) and Ty (l, λ) respectively.Let’s consider the functionTy (l, λ)y (l, λ)for λ ∈ K ≡ ∞∪n=1(µn−1, µn), where µ0 = −∞.It holds the followingLemma 7 (see lemma 6 from [8]). The functionTy (l, λ)y (l, λ)in each of theinterval(µn−1, µn)(n = 1, 2, ...) is strictly increasing function of λ.Lemma 8 (see lemma 7 from [8]). Let y (x, λ) be a nontrivial solution ofproblem (1), (2.a), (2.b), (2.c). Then it holds the relationlimλ→−∞Ty (l, λ)y (l, λ)= −∞.Remark 3 (see [8]). It is easy to observe that if λ < 0 or λ = 0, β ∈[0,pi2),thenTy (l, λ)y (l, λ)< 0; if λ = 0 and β =pi2, then Ty (l, λ) = 0. Besides, the relationν1 < µ1 < ν2 < µ2 < ... is valid.66[N.B.Kerimov, Z.S.Aliyev]Transactions of NAS of AzerbaijanLemma 9 (see lemma 8 from [8]). Let y (x, λ) be a nontrivial solution ofproblem (1), (2.a), (2.b), (2.c). If λ ≤ 0, then y (x, λ) 6= 0 for x ∈ (0, l) .Let y (x, λ) be a nontrivial solution of problem (1), (2.a), (2.b), (2.c). Denote bym (λ) the quantity of zeros of function y (x, λ) in the interval (0, l).Lemma 10. If λ ∈ (µn−1, µn] , n ∈ N, then m (λ) = n− 1.Proof. It is easy to observe, that the statement of the lemma for λ ∈ (µ0, 0]follows from lemma 9.Let λ > 0 and θ (x, λ) be a corresponding angular function determined by trans-formation (4). From (14) it follows that θ (0, λ) = β − pi2∈[−pi2, 0]. From (15) wehave θ (l, µn) = (2n− 1)pi2(n ∈ N). It is known (see theorems 5.1 and 5.2 from [7])that if λ > 0, then the function θ (x, λ) takes the values of the formkpi2(k ≥ −1)only strictly increasing. Taking into account the statement of lemma 6 we get thatθ (x, λ) ∈(−pi2,pi2)for λ ∈ (0, µ1] , x ∈ (0, l) ; θ (x, λ) ∈(−pi2, (2n− 1) pi2)forλ ∈ (µn−1, µn] , x ∈ (0, l) , n ≥ 2. Hence, we obtain the truth of statement oflemma 10 for λ > 0. Lemma 10 is proved.In the case c 6= 0 we’ll define the natural number N0 from the inequalityµN0−1 < −dc≤ µN0 .The following theorem are the main results of the paper.Theorem 2. The eigenvalues of boundary value problem (1)-(2) form the infi-nitely increasing sequence {λn}∞1 , such that λ1 < λ2 < ... < λn < ... . Besides,(a) if c = 0, then the eigenfunction yn (x), corresponding to the eigenvalue λn,has exactly n− 1 simple zeros in the interval (0, l);(b) if c 6= 0, then the eigenfunction yn (x), corresponding to the eigenvalue λn,has exactly n − 1 simple zeros in (0, l) for n ≤ N0, and exactly n − 2 simple zerosin (0, l) for n > N0.Proof. Let y (x, λ) be a nontrivial solution of problem (1), (2.a), (2.b), (2.c).By virtue of lemma 7 Φ (λ) =Ty (l, λ)y (l, λ)is continuous strictly increasing function inthe interval(µn−1, µn)(n ∈ N). From lemma 8 and equality y (l, µn) = 0 (n ∈ N) itfollows thatlimλ→µn−1+0Φ (λ) = −∞, limλ→µn−0Φ (λ) = +∞,and besides this function takes each value from (−∞,+∞) only at a unique point ofthe interval(µn−1, µn).Let H (λ) =aλ+ bcλ+ d. We have H ′ (λ) =−σ(cλ+ d)2. As by supposition (3) σ > 0,then the function H (λ) strictly decreases in the interval (−∞,+∞) for c = 0, andthe function H (λ) c 6= 0 strictly decreases in each of the intervals(−∞,−dc)and(−dc,+∞)for c 6= 0; moreoverTransactions of NAS of Azerbaijan[The oscillation properties of the boundary ... ]67limλ→− dc−0h (λ) = −∞, limλ→− dc+0H (λ) = +∞. (16)Let c = 0 or c 6= 0,−d/c /∈ (µn−1, µn]. From the above-stated it follows, that inthe interval(µn−1, µn)there will be found a unique value λ = λ∗n, for whichΦ (λ) = H (λ) , (17)i.e. condition (2.d) is fulfilled. Consequently, λ∗n is the eigenvalue of boundary valueproblem (1)-(2) and y (x, λ∗n) is a corresponding eigen function. From lemma 10 itfollows that m (λ∗n) = n − 1. It is easy to observe, that if c = 0 or c 6= 0, n < N0then λ∗n is the n−th eigenvalue of boundary value problem (1)-(2). Hence we obtainthe truth of statement (a) and the truth of a part of statement (b) concerning to thecase n < N0.Let c 6= 0 and −d/c ∈ (µN0−1, µN0). In a similar way, in each of the intervals(µN0−1,−d/c)and(−d/c, µN0) there can be found a unique value (λN0 and λN0+1,respectively) which (17) is satisfied i.e. condition (2.d) is fulfilled. From lemma 10it follows that m (λN0) = m (λN0+1) = N0 − 1.The case c 6= 0,−d/c = µN0 is considered in a completely similar way. At that itis used the fact, that µN0 in this case is also eigenvalue of boundary value problem(1)-(2). In the given case λN0 ∈(µN0−1, µN0), λN0+1 = µN0 . By virtue of the lemma10 we have m (λN0) = m (λN0+1) = N0 − 1.It is easy to see that for c 6= 0 and n > N0 a unique solution λ∗n of equation (17)in the interval(µn−1, µn]is the (n+ 1) -th eigenvalue of boundary value problem(1)-(2), i.e. λ∗n = λn+1. As m (λ∗n) = n− 1, then m (λn+1) = n− 1 = (n+ 1)− 2.From the above-stated discussions we obtain the truth of estimation (b).Theorem 2 is proved.References[1]. Tichonov A. N., Samarskii A.A. Equation of mathematical physics. M.:“Nauka”, 1972, 735 p. (Russian)[2]. Fulton C.T. Two-point boundary value problems with eigenvalue parameterin the boundary conditions. Math. Z., 1973, Bd. 133, No4, pp.301-312.[3]. Handeiman G.H., Keller J.B. Small vibrations of a slightly stiff pendulum.Proc. Fourth U.S. Natl. Congress of Appl. Mech., 1962, pp.195-202.[4]. Ahn H.J. Vibrations of a pendulum consisting of a bob suspended from a wire.Quart. Appl. Math., Bd.39, No1, pp.109-117.[5]. Meleshko S.V., Pokorny˘ı Yu.B. On one vibrational boundary value problem.1987, v.23, No8, pp.1466-1467. (Russian)[6]. Banks D.Q., Kurowski G.J. A Pru˝fer transformation for the equation of avibrating beam. Transaction of the American Mathematical Society, 1974, v.199,pp.203-222.[7]. Banks D.Q., Kurowski G.J. A Pru˝fer transformation for the equation of avibrating beam subject to axial forces. Journal of differential equations, 1977, v.24,No2, pp.57-74.68[N.B.Kerimov, Z.S.Aliyev]Transactions of NAS of Azerbaijan[8]. Kerimov N.B., Aliyev Z.S. On oscillation properties of eigenfunctions of thefourth order differential operator. Transactions of NAS of Azerb., ser. of phys.-tech.and math. sci., 2005, v.25, No4, pp. 63-76.Nazim B. KerimovInstitute of Mathematics and Mechanics of NAS of Azerbaijan9, F.Agayev str., AZ1141, Baku, AzerbaijanTel.: (99412) 439 47 20 (off.).E-mail: nazimkerimov@yahoo.com.Ziyatkhan S. AliyevBaku State University23, Z.I.Khalilov str., AZ1148, Baku, AzerbaijanTel.: (99412) 438 05 82 (off.)E-mail: z aliyev@mail.ruReceived September 01, 2005; Revised October 24, 2005.Translated by Mamedova Sh.N.

The Oscillation Properties Of The Boundary Value Problem With Spectral Parameter In The Boundary Condition

http://dspace.khazar.org/bitstream/20.500.12323/4643/1/The%20Oscillation%20Properties%20Of%20The%20Boundary%20Value%20Problem%20With%20Spectral%20Parameter%20In%20The%20Boundary%20Condition.pdf

The Oscillation Properties Of The Boundary Value Problem With Spectral Parameter In The Boundary Condition

Abstract

Similar works

Full text

Available Versions

Khazar University Institutional Repository