summary:Integral criteria are established for $\dim V_i(p)=0$ and $\dim V_i(p)=1, i\in \lbrace 0,1\rbrace $, where $V_i(p)$ is the space of solutions $u$ of the equation \[ u^{\prime \prime }+p(t)u=0 \] satisfying the condition \[ \int ^{+\infty }\frac{u^2(s)}{s^i}ds<+\infty \,. \

Chantladze, T.

Kandelaki, N.

Lomtatidze, A.

Czech digital mathematics library

On quadratically integrable solutions of the second order linear equation

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

Archivum MathematicumT. Chantladze; Nodar Kandelaki; Alexander LomtatidzeOn quadratically integrable solutions of the second order linear equationArchivum Mathematicum, Vol. 37 (2001), No. 1, 57--62Persistent URL: http://dml.cz/dmlcz/107786Terms of use:© Masaryk University, 2001Institute 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 37 (2001), 57 – 62ON QUADRATICALLY INTEGRABLE SOLUTIONS OF THESECOND ORDER LINEAR EQUATIONT. CHANTLADZE, N. KANDELAKI AND A. LOMTATIDZEAbstract. Integral criteria are established for dimV i(p) = 0 and dimVi(p) =1, i ∈ {0,1}, where Vi(p) is the space of solutions u of the equationu′′ + p(t)u = 0satisfying the condition+∞u2(s)sids < +∞ .1. Main ResultsConsider the equationu′′ + p(t)u = 0 ,(1)where p : [0,+∞[→]−∞,+∞[ is a locally integrable function.Under a solution of equation (1) is understood a locally absolutely continuoustogether with its first derivative function u : [0,+∞[→ ]−∞,+∞[ satisfying (1)almost everywhere.Denote by Vi(p) (i = 0, 1) the set of solutions u of equation (1) satisfying thecondition+∞∫u2(s)sids < +∞ ,(2)and denote the set of solutions u satisfyinglimt→+∞u(t) = 0by Z(p).Below we give some new results on the interlocation as well as on the dimen-sionality of the above sets.2000 Mathematics Subject Classification: 34C11.Key words and phrases: second order linear equation, quadratically integrable solutions, van-ishing at infinity solutions.Received October 11, 1999.58 T. CHANTLADZE, N. KANDELAKI AND A. LOMTATIDZETheorem 1. Let i ∈ {0, 1}, p(t) ≤ 0 in some neighbourhood of +∞, and+∞∫dssi[∫ s0η|p(η)| dη]2 < +∞ .(3)Then Vi(p) = Z(p) and dimVi(p) = 1.Suppose that there exists a finite limitlimt→+∞1tt∫1s∫1p(η) dη ds = cpand putp∗ = lim inft→+∞tcp − t∫1p(s) ds , p∗ = lim supt→+∞tcp − t∫1p(s) ds .Theorem 2. Let p∗ ≤ −34 andp∗ < p∗ − 1 +12√1− 4p∗ .(4)Then dimV0(p) = 1. If, moreover, p(t) ≤ 0 in some neighbourhood of +∞, thenV0(p) = Z(p).Theorem 3. Let p∗ < 0 andp∗ < p∗ −12+12√1− 4p∗ .Then dimV1(p) = 1. If, moreover, p(t) ≤ 0 in some neighbourhood of +∞, thenV1(p) = Z(p).It is proved in [1] that if p∗ > −12 and p∗ < 14, then V0(p) = {0}. The followingtheorem makes this result more complete.Theorem 4. Let p∗ < −12 andp∗ < −√p2∗ − p∗ −34.(5)Then V0(p) = {0}.2. ProofProof of Theorem 1. From (3) it follows that∫+∞s|p(s)|ds = +∞. Howeverthis condition is necessary and sufficient for Z(p) 6= {0} (see [2] and [3]). On theother hand, obviously dimZ(p) = 1 and Vi(p) ⊂ Z(p). Thus it is sufficient to showthat Z(p) ⊂ Vi(p).ON QUADRATICALLY INTEGRABLE SOLUTIONS 59Suppose u ∈ Z(p). Without loss of generality we can assume that for somet0 > 0,p(t) ≤ 0 for t > t0 ,(6)u(t) > 0, u′(t) < 0 for t > t0 .(7)Multiplying both sides of (1) by t and integrating from t0 to t, we obtaint∫t0sp(s)u(s) ds = tu′(t) − t0u′(t0)− u(t) + u(t0) for t > t0 .Hence on account of (6) and (7), we easily conclude that for some r > 0,u(t)t∫0s|p(s)| ds < M for t > t0 .Therefore, by virtue of (3) condition (2) holds. Thus the theorem is proved.Proof of Theorem 2. According to (4), we can find ε > 0 such that p∗ < −12−2εandp∗ < p∗ − 1 +12√1− 4(p∗ − ε) − 3ε .(8)SupposeQ(t) = tcp − t∫1p(s) ds for t > 0(9)and choose tε > 1 such thatp∗ − ε < Q(t) < p∗ + ε for t > tε .(10)Let α = −12 − p∗ − 2ε. It is evident that α > 0 andα−√α+ p∗ + ε < 0 .(11)Due to (8), it is easy to see that α+√α+ p∗ − ε < 0. If along with this we takeinto account (11), then from (10) we get−α−√α < Q(t) < −α+√α for t > tε ,and therefore, Q2(t) + 2αQ(t) +α(α− 1) < 0 for t > tε. In view of this, it is clearthat the function w(t) = tα satisfies the inequalityw′′(t) ≤ −Q2(t)t2w(t) − 2Q(t)tw′(t) for t > tε .Consequently, the equationv′′ = −Q2(t)t2v − 2Q(t)tv′(12)has a solution v satisfying the inequalities0 < v(t) < tα for t > tε(13)60 T. CHANTLADZE, N. KANDELAKI AND A. LOMTATIDZE(see, e.g., [4]).It can be directly checked that the functionu(t) = v(t) exp t∫1Q(s)sds for t > tε(14)is a solution of equation (1). By (10) and (13), there are M > 0 and t1 > tε suchthat0 < u(t) < Mtα+p∗+ε for t > t1 .Hence, taking into consideration how α is, we conclude that u ∈ V0(p). Thereforewe have proved that V0(p) 6= {0}.Since u(t) > 0 for t > t1, we have dimV0(p) ≤ 1 (see, e.g., [1]). HoweverdimV0(p) = 1, since V0(p) 6= {0}.Let us now suppose that p(t) ≤ 0 in some neighbourhood of +∞. Then it isobvious that dimZ(p) ≤ 1 and V0(p) ⊂ Z(p). Hence in view of the fact thatdimV0(p) = 1, we obtain V0(p) = Z(p). This completes the proof of the theorem.The proof of Theorem 3 is omitted, since it is analogous to that of Theorem 2with the only difference α = −p∗ − ε.Proof of Theorem 4. Assume the contrary. Let u be a nontrivial solution ofequation (1) and u ∈ V0(p). According to (5) and applying Theorem 1.6 from [5],equation (1) is nonoscillatory. Thus without loss of generality we can assume thatu(t) > 0 for t > t0. Choose ε ∈ ] 0, 12 [ and tε > t0 such that (10) holds andp∗ − ε < −12,(15)p∗ + ε < −√(p∗ − ε)2 − (p∗ − ε) −34.(16)It is evident that the function v defined by (14) and (9) is a solution of equation(12). According to our assumption,v(t) > 0 for t > t0 ,(17)+∞∫v2(s) exp2 s∫1Q(η)ηdη ds < +∞ .(18)Let us show that for some t1 > t0,v′(t) > 0 for t > t1 .(19)Indeed, if there exists t∗ > t1 such that v′(t∗) ≤ 0, then by virtue of the equalityv′(t) exp2 t∫1Q(η)ηdη′ = −Q2(t)t2exp2 t∫1Q(η)ηdη v(t) for t > 0 ,ON QUADRATICALLY INTEGRABLE SOLUTIONS 61we havev′(t) < 0 for t > t∗ .(20)Then in view of (15), from (12) we find v′′(t) < 0 for t > t∗. But this togetherwith (20) contradicts inequality (17).By (10), (17), and (19), from (12) we getv′′(t) ≤ −(p∗ + ε)2t2v(t) − 2(p∗ − ε)tv′(t) for t > t1 .Consequently, the equationw′′ = −(p∗ + ε)2t2w − 2(p∗ − ε)tw′(21)has a solution w satisfying the inequalities0 < w(t) < v(t) for t > t1 .Hence due to (10) and (18), we have+∞∫w2(s)s2(p∗−ε) ds < +∞ .(22)On the other hand, we can easily check that the functions wk(t) = tλk , k = 1, 2,whereλk =12[1− 2(p∗ − ε) − (−1)k√(1− 2(p∗ − ε))2 − 4(p∗ + ε)2], k = 1, 2,are linearly independent solutions of equation (21). By (16), 2λ1 + 2(p∗− ε) > −1and therefore,+∞∫w2k(s)s2(p∗−ε) ds = +∞ , k = 1, 2 .Thus neither of nontrivial solutions of equation (21) satisfies condition (22). Thisconcludes the proof of the theorem.References[1] Wintner, A., On the non-existence of conjugate points, Amer. J. Math. 73 (1951), 368–380.[2] Kneser, A., Untersuchung und asymptotische Darstellung der Integrale gewisser Differential-gleichungen bei grossen reelen Werten des Arguments, J. Reine Angew. Math. 116 (1896),178–212.[3] Kiguradze, I.T. and Chanturia, T.A., Asymptotic properties of solutions of nanautonomousordinary differential equations, Kluwer Academic Publishers, Dordrecht–Boston–London,1992.62 T. CHANTLADZE, N. KANDELAKI AND A. LOMTATIDZE[4] Kiguradze, I.T. and Shekhter, B.L., Singular boundary value problems for second order differ-ential equations, in ”Curent Problems in Mathematics: Newest Results,” vol. 30, pp. 105–201,Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyzn. Inst. Nauchn. i Tekhn. Inform., Moscow,1987.[5] Chantladze, T., Kandelaki, N. and Lomtatidze, A., Oscillation and nonoscillation criteriafor second order linear equations, Georgian Math. J. 6 (1999), No 5, 401–414.T. Chantladze and N. KandelakiN. Muskhelishvili Institute of Computational MathematicsGeorgian Academy of Sciences8, Akuri St., 380093 Tbilisi, GEORGIAA. LomtatidzeDepartment of Mathematical Analysis, Masaryk UniversityJanáčkovo nám. 2a, 662 95 Brno, CZECH REPUBLICE-mail: bacho@math.muni.cz

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On quadratically integrable solutions of the second order linear equation

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