summary:In this paper a singular third order eigenvalue problem is studied. The results of the paper complete the results given in the papers [3], [5]

Greguš, Michal

Greguš, Michal, Jr.

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On certain singular third order eigenvalue problem

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Archivum MathematicumMichal Greguš; Michal Greguš, Jr.On certain singular third order eigenvalue problemArchivum Mathematicum, Vol. 39 (2003), No. 3, 173--177Persistent URL: http://dml.cz/dmlcz/107863Terms of use:© Masaryk University, 2003Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 39 (2003), 173 – 177ON CERTAIN SINGULAR THIRD ORDER EIGENVALUEPROBLEMMICHAL GREGUŠ AND MICHAL GREGUŠ, JR.Dedicated to Professor’s Šeda 70th birthdayAbstract. In this paper a singular third order eigenvalue problem is studied.The results of the paper complete the results given in the papers [3], [5].1. We consider the third order linear differential equation in the normal form(a) y′′′ + 2A(t)y′ + [A′(t) + λb(t)]y = 0 ,where A(t) ≥ 0, A′(t) and b(t) > 0 are continuous functions of t ∈ [a,∞), a > −∞and λ is a positive parameter. It is assumed that (a) is strongly nonoscillatory on[a,∞), that is (a) is nonoscillatory there for each real positive λ. By nonoscillationof (a) we mean that all of its nontrivial solutions are nonoscillatory on [a,∞).A nontrivial solution of (a) is called oscillatory on [a,∞) if ∞ is a limit pointof zeros of that solution. In the contrary case the solution is called nonoscillatoryon [a,∞).The equation (a) is said to be oscillatory on [a,∞) if it has at least one oscil-latory solution on [a,∞).The problem studied in Section 2, is to find a nontrivial (nonoscilatory) solutiony(t, λ) of (a) which satisfies either of the boundary conditions at finite pointsy(a, λ) = y′(a, λ) = y(b, λ) = 0 , a < b,(1)y(a, λ) = y(b, λ) = y(c, λ) = 0 , a < b < c(2)a, b and c being any given constants, as well as the boundary condition at infinityy(t, λ) = o(t[k1u1(t)u2(t) + k2u22(t)]) for t → ∞ ,(3)2000 Mathematics Subject Classification: 34B05, 34B24, 34C10.Key words and phrases: singular eigenvalue problem, normal form of third order differentialequation, zeros of nonoscillatory solutions.Received June 7, 2001.174 M. GREGUŠ AND M. GREGUŠ, JR.together with the requirement thaty(t, λ) 6= 0in a certain neighbourhood of infinity (t0,∞), where b ≤ t0 < ∞ (or c ≤ t0 < ∞ inthe conditions (2)), and u1, u2 form a fundamental set of solutions of the secondorder differential equationu′′ +12A(t)u = 0(4)with initial conditions u1(t0) = 1, u′1(t0) = 0, u2(t0) = 0, u′2(t0) = 1, k1, k2 aresuitable constants. The motivation for this paper was given by the paper [1] of A.Elbert, T. Kusano and M. Naito for linear second order nonoscillatory differentialequations.In the paper [3] the case A(t) < 0 on [a,∞) was studied.2. At the beginning of this section we introduce certain auxiliary statementson the linear third order differential equation, given in monograph [4].Consider equation (a) and the third order differential equation(a1) y′′′ + 2A(t)y′ + [A′(t) + b(t)] = 0 ,Lemma 1. [4, Theorem 2.41] Let the differential equationy′′ + 2A(t)y = 0(5)be disconjugate in (a,∞) and let the functions A(t), A′(t) + b(t), b(t) − A′(t) bepositive in [a,∞). If∞∫at2[b(t) − A′(t)] dt < ∞ ,(6)then the differential equation (a1) is non-oscillatory in [a,∞).Remark 1. If A(t) ≡ 0 on [a,∞) and (6) holds then [2, Theorem 4] equation (a1)is non-oscillatory in [a,∞).By Lemma 1 and Remark 1 the following lemma can be proved.Lemma 2. Let the differential equation (5) be disconjugated in [a,∞) and letA(t) ≥ 0, A′(t) ≤ 0, b(t) > 0 and A′(t) + b(t) > 0 in [a,∞). Let further λ̄ be anyfixed positive value of the parameter λ. If (6) holds then∞∫at2[λ̄b(t) − A′(t)] dt < ∞ ,(7)and the differential equation (a) is non-oscillatory for λ = λ̄ in [a,∞).ON CERTAIN SINGULAR THIRD ORDER EIGENVALUE PROBLEM 175Lemma 3. [4, Theorem 2.31] Let b(t) > 0, A′(t) be continuous in [a,∞). Theequation (a1) is oscillatory in [a,∞) if and only if its adjoint equation(b1) z′′′ + 2A(t) + [A′(t) − b(t)]z = 0 ,is oscillatory in [a,∞).If we apply Lemma 3 we can formulate Theorem 2.51 [4] as follows.Lemma 4. Let A(t) ≥ 0, A′(t)−b(t) ≤ 0, b(t) > 0 in [a,∞) and let∞∫ab(t) dt = ∞,then the differential equation (a1) is oscillatory in [a,∞).Corollary 1. Let the suppositions of Lemma 4 be fulfilled and A′(t) ≤ 0 in [a,∞).Then the differential equation (a) for λ = λ̄ > 0 is oscillatory in [a,∞).Lemma 5. Let A(t) ≥ 0 in [a,∞) and let the differential equation (4) be discon-jugated in [a,∞). Let u1, u2 be independent solutions of (4) and let u1(t0) = 1,u′1(t0) = 0, u2(t0), u2(t0) = 1, a < t0 < ∞. Then there is u2(t) > 0 for t > t0.And u1(t) has at most one zero to the right of t0.Remark 2 [6, Lemma 2.23]. Let the suppositions of Lemma 5 be fulfilled. If u isa solution of (4) and u(t) 6= 0 for t ≥ t1, then0 < (t + d)v(t) ≤ 1, t ≥ t1where v(t) = u′(t)u(t) , d = −t1 + 1/v(t1).Lemma 6. Let the suppositions of Lemma 5 be fulfilled and let b(t) > 0 for t ∈[a,∞) and λ > 0. Let further y be a solution of (a) and let for λ = λ̄ be y(t0, λ̄) =0, y′(t0, λ̄) 6= 0, y′′(t0, λ̄) 6= 0 for a ≤ t0 < ∞. Let, moreover y(t, λ̄) 6= 0 for t > t0.Theny(t, λ̄) =u2(t)[y′′(t0, λ̄)2u2(t) + y′(t0, λ̄)u1(t)]−12λ̄t∫t0b(τ)∣∣∣∣u1(t) u2(t)u1(τ) u2(τ)∣∣∣∣2y(τ, λ̄)dτ ,(8)where u1, u2 form a fundamental set of solutions of (4) with the properties givenin Lemma 5.The proof of Lemma 6 is given in [4] at the beginning of Section 3, Chap. I, §3.Corollary 2. If y(t, λ̄) > 0 [y(t, λ̄) < 0] for t > t0 in (8) then y′(t0, λ̄) >0 [y′(t0, λ̄) < 0], u2(t) > 0 and u(t) = y′(t0, λ̄)u1(t) +y′′(t0,λ̄)2 u2(t) > 0 [u(t) < 0]for t > t0.Corollary 3. Let the suppositions of Lemma 6 be fulfilled then there exist con-stants k1 = y′(t0, λ̄), k2 =y′′(t0,λ̄)2 such that |y(t, λ̄)| ≤ u2(t)|k1u1(t)+ k2u2(t)| fort > t0, or y(t, λ̄) = o(tu2(t)[k1u1(t) + k2u2(t)]) for t → ∞.176 M. GREGUŠ AND M. GREGUŠ, JR.Adaptation of Oscillation theorem [4, Theorem 4.5] to (a) in our case yields thefollowing lemma.Lemma 7. Suppose that A(t) ≥ 0, A′(t) ≤ 0 and b(t) ≥ k > 0 for t ∈ [a,∞). Letλ ∈ (0,∞) and let y(t, λ) be a nontrivial solution of (a) with y(a, λ) = 0. Then forany fixed b > a, the number of zeros of y on [a, b] increases to infinity as λ → ∞and the distance between any consecutive zeros of y converges to zero.The continuous dependence of zeros of solutions of (a) upon the parameter λis given in following lemma.Lemma 8. [4, Lemma 4.2] Let A′(t), b(t) > 0 be continuous functions in [a,∞).Let y be a nontrivial solution of (a) on [a,∞) such that y(α, λ) = 0, a ≤ α < ∞,for all λ ∈ (0,∞). Then the zeros of y on (α,∞) (if they exist) are continuousfunctions of the parameter λ ∈ (0,∞).With the help of results given in the preceding lemmas, remarks and corollariesone can prove the following theorem regarding the singular eigenvalue problemproblem (a), (1), (3) or (a), (2), (3).Theorem 1. Let A(t) ≥ 0, A′(t) ≤ 0, b(t) > 0 be continuous functions in [a,∞)and let A′(t)+ b(t) > 0 for t ∈ [a,∞). Let∞∫at2[b(t)−A′(t)]dt < ∞ and let the sec-ond order differential equation (5) be disconjugate in [a,∞). Let further a ≤ b < cbe fixed arbitrarily. Then there exists a natural number ν and a sequence of pa-rameters λ{λν+p}∞p=0(eigenvalues) such that λν+p < λν+p+1, p = 0, 1, 2, . . . andlimp→∞λν+p = ∞ and a corresponding sequence of functions{yν+p}∞p=0(eigenfunc-tions) such that yν+p = y(t, λν+p) is a solution of (a) for λ = λν+p, has a finitenumber of zeros on (a,∞) with the last zero at tν+p0 . This solution yν+p fulfillsthe boundary conditions (1), (3) or (2), (3) and has exactly ν + p zeros in (b, c).Proof. We prove the case a < b < c. In the case a = b, i.e. (1), (3) the proof issimilar.Let a < b < c < ∞. Let y = y(t, λ),λ > 0, be a nontrivial solution of (a) suchthat y(a, λ) = y(b, λ) = 0 for all λ > 0. By Lemma 2 solution y is nonoscillatoryfor each λ = λ̄ > 0. Now, construct the differential equation(A) Y ′′′ + 2A(t)Y ′ + [A′(t) + λB(t)]Y = 0 on [a,∞)whereB(t) ={b(t) for t ∈ [b, c]b(c) for t ≥ c .Let Y = Y (t, λ) be a solution of (A) on [a,∞) such that Y (a, λ) = Y (b, λ) = 0and Y (t, λ) = y(t, λ) for t ∈ [a, c] and λ ∈ (0,∞).By Lemma 4 and Corollary 1 the differential equation (A) is oscillatory on[a,∞) for each λ̄ ∈ (0,∞) and therefore the solution Y is oscillatory on [a,∞) foreach λ̄ > 0.ON CERTAIN SINGULAR THIRD ORDER EIGENVALUE PROBLEM 177Let λ = λ̄ be fixed. Let Y (t, λ̄) have exactly ν zeros in (b, c). Let tν(λ) be ν-thzero of Y (t, λ). Then there is tν(λ̄) < c ≤ tν+1(λ̄). By Lemma 7 there exist λ∗such that tν+1(λ∗) < c and by Lemma 8 (continuous dependence of zeros) thereexists λν , λ̄ ≤ λν < λ∗ such that tν+1(λν) = c and y(t, λν) has exactly ν zeros in(b, c). But, we know that Y (t, λν) = y(t, λν) on [a, c]. By Lemma 2 applied to λνthere exists tν0 ≥ c such that y(t, λν) has finite number of zeros to the right of cand tν0 is its last zero on [c,∞). Then by Corollary 3, when t0 = tν0 , the inequality(3) holds.Continuing in the same manner we can find a sequence of valuesλν , λν+1, · · · , λν+p, · · ·and the corresponding sequence of functions{yν+p}∞p=0(eigenfunctions) with theprescribed properties and the theorem is proved.References[1] Elbert, A., Takaši, K. and Naito, M., Singular eigenvalue problems for second order linearordinary differential equations, Arch. Math. (Brno), T. 34 (1998), 59–72.[2] Gera, M., Bedingungen der Nicht-oszillationsfähigkeit für die lineare Differentialgleichungdritter Ordnung, Mat. Čas. Slov. akad. vied, 21 (1971), 65–80.[3] Greguš, M., On certain third order eigenvalue problem, CDDE Brno 2000, Masaryk Univer-sity, Arch. Math. (Brno), T. 36 (2000), 461–464.[4] Greguš, M., Third Order Differential Equations, D. Riedel Publishing Company, Dordrecht,Boston, Lancaster, Tokyo (1987).[5] Greguš, M., Jr., On certain third order boundary value problems on infinite interval, CDDEBrno 2000, Masaryk University, Arch. Math. (Brno), T. 36 (2000), 465–468.[6] Swanson, C. A., Comparison and oscillation theory of linear differential equations, Aca-demic Press, New York and London 1968.Department of Information SystemsFaculty of Management, Comenius UniversityOdbojarov 10, P.O. Box 95820 05 Bratislava 25, SlovakiaE-mail: michal.gregus@fm.uniba.sk

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On certain singular third order eigenvalue problem

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Institute of Mathematics AS CR, v. v. i.