МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2006 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2006 Proceeding of the Thirty Fifth Spring Conference of the Union of Bulgarian Mathematician Borovet, April 5 8, 2006 OSCILLATIONS OF A CLASS OF EQUATIONS AND INEQUALITIES OF FOURTH ORDER * Zornitza A. Petrova We etablih ufficient condition for ocillation of a wide cla of ordinary differential equation of fourth order. So that, we begin with ufficient condition for the abence of eventually poitive olution of proper ordinary differential inequalitie. We extend a rezult of Kuano and Yohida concerned with uch an inequality. There i extenive literature devoted to the ocillation behaviour of linear and nonlinear ordinary differential equation and inequalitie with or without delay. The excellent monographie of Györi and Lada [1], Ladde, Lakhmikantham and Zhang [3], Shevelo [5] and Mihev and Bainov [4] conit large lit of article. We obtain ufficient condition for ocillation of the equation (1) z iv (t) + mz (t) + g(z(t), z (t), z (t), z (t)) = f(t), where m = cont R, f(t) C([T, ), R), T 0 i a large enough contant and g(z, ξ, η, ζ) C(R 4, R) atifie the condition: (2) g(z, ξ, η, ζ) qz, (z, ξ, η, ζ) R + R 3, g(z, ξ, η, ζ) qz, (z, ξ, η, ζ) R R 3, where q = cont > 0. We prove the preent theorem via our concluion for the abence of eventually poitive olution of the inequality: (3) z iv (t) + mz (t) + g(z(t), z (t), z (t), z (t)) f(t) under the ame aumption. We mention that we could rewrite our reult for (3) a repective reult for the abence of eventually negative olution of (4) z iv (t) + mz (t) + g(z(t), z (t), z (t), z (t)) f(t). Then we apply them to find ufficient condition for ocillation of the equation (1). We point out that (2) guarantee that every poitive olution of (3) in a et M R i alo a poitive olution of (5) z iv (t) + mz (t) + qz(t) f(t) in the ame et. Simillarly, every negative olution of (4) in M R i alo a negative olution of (6) z iv (t) + mz (t) + qz(t) f(t) * Key word: Ocillation, eventually poitive olution, eventually negative olution. 2000 Mathematic Subject Claification: 34A30, 34A40, 34C10. 195
in M. The conideration for the equation: (7) z iv (t) + mz (t) + qz(t) = f(t) make u ure that all the reult are harp. We hall finih thi paper with: (8) (9) and z iv (t) + mz (t) + qz(t) + λz(t τ) f(t), z iv (t) + mz (t) + qz(t) + λz(t τ) f(t) (10) z iv (t) + mz (t) + qz(t) + λz(t τ) = f(t), where we uppoe additionaly that λ R and τ 0 are contant. All the ufficient condition for ocillation here are baed on the following lemma of Yohida [6]. We uppoe that L = cont > 0 and ρ = cont 0 everywhere. (11) Lemma 1. ([6]) If there i a number ρ uch that +π/l then the ordinary differential inequality F (t) in L(t )dt 0, (12) z (t) + L 2 z(t) F (t) ha no poitive olution in (, + π/l]. Definition 1. We ay that the function ϕ(t) C([c, ), R) i ocillating when t, if there exit a equence {t n } n=1 uch that lim t n = and ϕ(t n ) = 0. n Definition 2. We ay that the function ϕ(t) C([c, ), R) i eventually poitive (eventually negative), if there exit c = cont c uch that ϕ(t) > 0 (ϕ(t) < 0), t [ c, ). The next corollary wa not formulated in [6] but it wa applied there. Let define (13) Φ() = +π/l F (t) in L(t )dt. Corollary 1. If the function Φ() i eventually nonpoitive, then the ordinary differential inequality (12) ha no eventually poitive olution. Corollary 2. The ordinary differential inequality (14) z (t) + L 2 z(t) 0 ha no eventually poitive olution. The reult of Corollary 2 wa extracted in [8] a a particular cae of the paper [2]. It gurantee that all the above reult are harp. 196
Let u explain the connection between the inequalitie of fourth order at the begining and thee one of econd order here. Yohida and Kuano conidered Timohenko beam equation: 4 u(x, t) (15) t 4 +αβγ 2 u(x, t) t 2 (β +γ) 4 u(x, t) x 2 t 2 +βγ 4 u(x, t) x 4 +c(x, t, u)=f(x, t) in [8], where α, β and γ are poitive contant. They followed the claical method in the ocillation theory of partial differential equation there. In particular, it mean that they etablihed a pair of fourth order ordinary differential inequalitie correponding to the average of poitive olution a well a of the negative one in [8]. After a proper integration by part they found uch a relation for thee average, which helped them to apply Corollary 2. Obviouly, the equation (15) i a non-homogeneou one and the inequality (14) i a homogeneou one. Thi fact make the proof complicated. Later, Yohida [6] invetigated hyperbolic equation via Lemma 1, which wa proved there. Since the inequality (12) i a non-homogeneou one then the ame author applied it for (15) in the mentioned way in [7]. There were not any common reult for fourth order ordinary differential equation nor in [7] neither in [8]. We apply (16) Φ 4 (θ) = θ+π/ L +π/l θ f(t) in L(t ) in L( θ)dtd, L = m L 2 intead of Φ() in all the bellow. We point out that Yohida [7] obtained an unique function of the above type. In fact, we find a family of function Φ 4 (θ), which depend on the contant L. Thi i very important ince thee function give information for the ditribution of the zero of (1) and a whole family allow u to apply computer. Moreover, there wa a retriction on the number m and q in [7], which i omitted here. Theorem 1. Aume that (2) hold and that the poitive contant L and q atify the condition: (17) L (0, m) and q + (L 2 m)l 2 > 0. If there i a number ρ uch that (18) Φ 4 ( + π/l) 0, then the ordinary differential inequalitie (3) and (5) have no poitive olution in (, + π/l + π/ L]. Further, if the function Φ 4 (θ) i eventually nonpoitive, then the ordinary differential inequalitie (3) and (5) have no eventually poitive olution. Proof. It i enough to conider (5) only. We uppoe to the contrary, i. e. (19) z(t) > 0, t (, + π/l + π/ L]. Then we obtain a contradiction with Lemma 1. More exactly, from one hand we have that the function (20) ζ() = z( + π/l) + z() i poitive olution of the inequality (21) z () + L 2 z() F (), in ( + π/l, + π/l + π/ L], where (22) F () = 1 L +π/l f(t) in L(t )dt 197
but from the other hand, all the condition of Lemma 1 are fulfilled for (21) ince (18) i the preent particular cae of (11). Obviouly, we obtain that (23) ζ() > 0, t (, + π/l + π/ L] directly from (19) and (20). Now we hall how that ζ() i olution of (21). Since z(t) atifie (5) in (, + π/l + π/ L] then (24) +π/l ( z iv (t) + mz (t) + qz(t) ) +π/l in L(t )dt f(t) in L(t )dt and we replace the following integration by part: +π/l +π/l +π/l z (t) in L(t )dt = L(z( + π/l) + z()) L 2 z(t) in L(t )dt, +π/l z iv (t) in L(t )dt = L(z ( + π/l) + z ()) L 2 z (t) in L(t )dt there. Alo, (17) guarantee that (25) ( q + (L 2 m)l 2) z(t)inl(t )dt 0. Then the contrary aumption lead to the fact that ζ() i uch that: (26) L ( ζ () + (m L 2 )ζ() ) +π/l f(t)inl(t )dt. We apply Corollary 1 to finih the econd part of the theorem. Theorem 2. Let (2) and (17) be atified. If the function Φ 4 () i ocillating, then every olution of the ordinary differential equation (1) and (7) ocillate. Proof. Everything i baed on Theorem 1 a well a on the inequalitie (5) and (6). More exactly, every olution of the equation (7) i alo olution of (5) and (6) ince if z iv (t) + mz (t) + qz(t) = f(t), then obviouly z iv (t) + mz (t) + qz(t) f(t) and z iv (t) + mz (t) + qz(t) f(t). Moreover, every olution of the equation (1) i alo olution of (5) and (6) ince if (1) hold then alo if z(t) > 0, z iv (t) + mz (t) + qz(t) z iv (t) + mz (t) + g(z(t), z (t), z (t), z (t)) = f(t), z iv (t) + mz (t) + qz(t) z iv (t) + mz (t) + g(z(t), z (t), z (t), z (t)) = f(t), if z(t) < 0. Here we prove much more. In fact, we etablih a ufficient condition for the ditribution of the zero of (1) and (7) in the cae, where the function Φ 4 () i ocillating. It mean that there exit a equence { n } n=1: lim n n = and Φ( n + π/l) = 0, i. e. Φ( n + π/l) 0 and Φ( n + π/l) 0. We apply Theorem 1 to conclude that thee equation have nor poitive olution in ( n, n + π/l + π/ L] neither negative olution in ( n, n + π/l + π/ L], n N. It mean 198
jut that (1) and (7) ha at leat one zero in ( n, n + π/l + π/ L], n N and we finih the proof. Theorem 3. Let (2) and (17) be fufilled and let there i a number ρ uch that (18) hold. If λ 0 then (8) ha no poitive olution in ( τ, + π/l + π/ L]. If λ [ ( q + (L 2 m)l 2), 0] then (8) ha no poitive and monotonically increaing olution in ( τ, + π/l + π/ L]. Further, let the function Φ 4 (θ) be eventually nonpoitive. If λ 0 then (8) ha no eventually poitive olution. If λ [ ( q + (L 2 m)l 2), 0] then the ordinary differential inequality (8) ha no eventually poitive and monotonically increaing olution. Proof. It i enough to invetigate the cae, where t ( τ, + π/l + π/ L]. We uppoe to the contrary again. Our aim i to explain that (26) i fufilled in both ituation becaue it allow u to apply Lemma 1. Since here we have +π/l [ z iv (t) + mz (t) + qz(t) + λz(t τ) ] in L(t )dt +π/l f(t)inl(t )dt intead of (24) then the unique difficult moment i to ee that if λ [ ( q + (L 2 m)l 2), 0] then the contrary aumption give u that Hence, z(t) z(t τ) and q + (L 2 m)l 2 λ 0. ( q + (L 2 m)l 2) z(t) λz(t) λz(t τ). Theorem 4. Let (2) and (17) be fulfilled and let the function Φ 4 () be ocillating. If λ 0 then every olution of the equation (10) ocillate. If λ [ ( q + (L 2 m)l 2), 0] then the equation (10) ha nor eventually poitive and monotonically increaing olution neither eventually negative and monotonically decreaing olution. Proof. Everything follow from Theorem 3. REFERENCES [1] I. Györi, G. Lada. Theory of Delay Differential Equation with Application, Clerendon Pre, Oxford, 1991. [2] K. Kreith, T. Kuano, N. Yohida. Ocillation Propertie of Nonlinear Hyperbolic Equation, SIAM J. Math. Anal., 15 (1984), 570 578. [3] G. S. Ladde, V. Lakhmikantham, B. Zhang. Ocillation Theory of Differential Equation with Deviating Argument, Marcel Dekker, New York, 1987. [4] D. P. Mihev, D. D. Bainov. Ocillation Theory for Neutral Differential Equation with Delay, Adam Hilger, 1991. [5] V. N. Shevelo. Ocillation of the Solution of Differential Equation with Deviating Argument, Naukova Dumka, 1978. [6] N. Yohida. On the Zero of Solution to Nonlinear Hyperbolic Equation, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), 121 129. [7] N. Yohida. On the Zero of Solution of Beam Equation, Annali di Math. pura ed appl. (IV) CLL (1987), 389 398. [8] N. Yohida, T. Kuano. Forced Ocillation of Timohenko Beam, Quart. Appl. Math., 43 (1985), 167 177. 199
Zornitza A. Petrova Faculty of Applied Mathematic and Informatic Technical Univerity of Sofia Sofia, Bulgaria e-mail: zap@tu-ofia.bg ОСЦИЛАЦИИ НА КЛАС ОТ УРАВНЕНИЯ И НЕРАВЕНСТВА ОТ ЧЕТВЪРТИ РЕД Зорница А. Петрова Установяваме достатъчни условия за осцилация на широк клас от обикновени диференциални уравнения от четвърти ред. Започваме с достатъчни условия за отсъствието на финално положителни решения на подходящи обикновени диференциални неравенства. Разширяваме резултат на Йошида и Кусано, свързан с това неравенство. 200