On the Qualitative Behavior of Solutions of Third Order Differential Equations

Transcription

1 Ž. JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 97, ARTICLE NO On the Qualitative Behavior of Solutions of Third Order Differential Equations M. Cecchi Department of Electrical Engineering, Uniersity of Florence, Via di S. Marta, Firenze, Italy Z. Dosla* Department of Mathematics, Masaryk Uniersity, Janackoo nam 2a, Brno, Czech Republic M. Marini Department of Mathematics & Computer Science, Via delle Scienze, 206, 3300 Udine, Italy and Gab. Villari Department of Mathematics U. Dini, Uniersity of Florence, Viale Morgagni 67a, 5034 Firenze, Italy Submitted by Jack K. ale Received December 5, 994. INTRODUCTION The purpose of this paper is to examine the oscillatory and nonoscillatory behavior of solutions of the third order linear differential equation d x qž t. x0 Ž E. pž t. rž t. dt * This paper was written during the author s visit to Department of Applied Mathematics G. Sansone, University of Florence, Florence, Italy, and has been partially supported by a grant from the Commission of the European Economic Community for Cooperation in Science and Technology with Central and Eastern European Countries X96 $8.00 Copyright 996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2 750 CECCI ET AL. where p, r, q are twice differentiable functions with fixed sign on 0,., that is, pž t. 0, rž t. 0, qž t. 0 for t 0,.. Ž. We just note that the regularity assumptions on the functions p, r, q are considered for the sake of simplicity. Namely, if p, r, q are continuous on 0,., then Eq. Ž E. can be interpreted as a first order linear differential system. In this case a solution of Eq. Ž E. is a function x with continuous derivative and continuous second and third quasiderivative x 2, x 3, where 2 Ž. Ž Ž. Ž.. 3 x t x t rt,x Ž. t Žx 2 pt Ž... We recall that a nontrivial solution of Eq. Ž E. is said to be oscillatory or nonoscillatory according to whether it does or does not have arbitrarily large zeros. Equation Ž E. is said to be oscillatory if it has at least one nontrivial oscillatory solution, and nonoscillatory if all of its solutions are nonoscillatory. In addition an oscillatory equation is said to be either a partially oscillatory equation or a completely oscillatory equation according to whether it has at least one nonoscillatory solution or all of its nontrivial solutions are oscillatory. The investigation of the oscillatory or nonoscillatory behavior of solutions of Eq. Ž E. or, more generally, of the equation x až t. x bž t. xcž t. x0, Ž. where a, b, c are real continuous functions on 0,., is often accomplished by dividing the considered equations in particular classes. For instance, when ct Ž. 0 on 0,., some authors introduce for equations of type Ž. the notion of an equation with property A and an equation with property B Žsee, e.g., 24.. More precisely, Eq. Ž. is said to be an equation with property A if every solution x is either oscillatory or Ži. satisfies the condition x Ž. t 0ast,i0,, 2; Eq. Ž. is said to be an equation with property B if every solution x is either oscillatory or Ži. satisfies the condition x Ž. t as t, i 0,, 2. Sufficient conditions concerning the property A or the property B have been given by several authors. A comprehensive list of references on this subject may be found in, 24. The above classification has been extended in several directions to linear and nonlinear equations of nth order. Among the wide literature on this field, we refer the reader to, 3, 24, 26, and references therein. Another basic classification concerning third order linear differential equations has been employed, in an implicit form, by Sansone 28and

3 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 75 later on has been formalized by anan 2. We recall that Eq. Ž. is said to be of class I if every solution x for which xt Ž. xt Ž. 0, x Ž t , 0t, has the property that xt Ž. 0inŽ 0,t.; Eq. Ž. 0 0 is said to be of class II if every solution x for which xt Ž. xt Ž. 0, x Ž t , 0t, has the property that xt Ž. 0inŽ t, In the literature there are numerous papers dealing with such a classification: we refer the reader to 57, 23, 27 and to 8, 30 for an extensive bibliography. The interest on such equations is caused by their considerable properties, especially as regards oscillation or nonoscillation conditions. Examples of equations of class I or of class II are equations with constant coefficients and equations of type Ž E. with pt Ž. rt Ž. 3. In this paper, we prove, in particular, that if p, r, q are positive functions on 0,., then Eq. Ž E. is of class I; if p, r, q are positive functions on 0,., then Eq. Ž E. is of class II. Using this property and a previous result Ž Correspondence Principle. stated in 9 for equations of higher order, we derive some criteria on the structure of the space of the solutions of Eq. Ž E.. In addition, some conditions concerning oscillation and nonoscillation results are also given. All the conditions are presented as integral criteria and involve only the integral behavior of the functions p, r, q on 0,.. The obtained results generalize well known oscillation criteria 24, 29, 3 and extend to Eq. Ž E. some criteria stated for Eq. Ž. with at Ž. 0 2 and for Eq. Ž E. with pt Ž. rt Ž. 3. Relations and comparison with other known results will be made throughout the paper. Related results may be obtained from many papers concerning linear equations of nth order Žsee, e.g., 9, 67, 26 and from nonlinear equations Žsee, e.g. 5, The plan of the paper is the following: in Section we recall the quoted Correspondence Principle and some known results which will be useful in the following. In Section 2 we give sufficient conditions in order for Eq. Ž E. to be of class I or of class II, together with a first application to the study of the structure of the space of solutions of Eq. Ž E.. Finally, in Sections 3 and 4 we present some integral criteria concerning the nonoscillation and oscillation of Eq. Ž E., respectively. ²: In a recent paper 9, a Correspondence Principle for equations of higher order has been presented. For n 3 such a result states some

4 752 CECCI ET AL. relations between the solutions of Eq. Ž E. and the solutions of equations ½ 5 ½ rž t. ž qž t. / 5 y rž t. y0, Ž E2. qž t. pž t. z pž t. z0, Ž E. obtained by means of an ordered cyclic permutation of the coefficients p, r, q of Eq. Ž E.. Such a result plays an important role in the study of the classification of solutions as well as in the theory of disconjugacy. In order to recall the above-mentioned principle, we now introduce some notation. Denote by S i, i, 2, 3, the linear space of solutions of Eq. Ž E. i, and by Oi and Ni two subsets of Si given by the oscillatory solutions and nonoscillatory solutions of Eq. Ž E. i, respectively. In 9, the following definition is posed: DEFINITION. The spaces S i, S j, i, j, 2, 3, are said to be isomorphic with respect to the oscillation if there exists an isomorphism L ij: Si Sj which keeps the oscillatory properties of the solutions, that is g Oi LijŽ g. O j, g Ni LijŽ g. N j. In this case the operator Lij is said to be an isomorphism of oscillation. It is obvious to note that if the spaces S i, S j, i, j, 2, 3, are isomor- phic with respect to the oscillation, then Lij maps Oi in Oj and Ni in N j. ence the existence of an isomorphism of oscillation between the spaces S i, Sj enables us to describe the oscillatory or nonoscillatory behavior of the solutions of Eq. Ž E. i by the oscillatory or nonoscillatory behavior of the solutions of Eq. Ž E. j, and vice versa. The following lemma gives a relation between Eqs. Ž E. i. Such a relation has been used already in the study of stability of linear equations of higher order 25 : LEMMA. Assume condition Ž.. If x is a solution of Eq. Ž E., then y Ž r. x is a solution of Eq. Ž E., zž p.žž r. xis. 2 a solution of Eq. Ž E., and w Ž q.ž p.žž r. xis. 4 a solution of Eq. Ž E.. 3 Let L, L 2, L3 be the first order differential operators L : S S2 LŽ x.ž t. xž t. rž t. L 2: S2 S3 L2Ž y.ž t. yž t. pž t. L 3: S3 S L3Ž z.ž t. zž t.. qž t. 3

5 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 753 Ž. The following holds 9 : CORRESPONDENCE PRINCIPLE. phisms of oscillation. The operators L, i, 2, 3, are isomori As consequence of the Correspondence Principle, the linear spaces S i, i, 2, 3, have the same structure with respect to oscillation or nonoscillation. ence oscillation nonoscillation criteria for Eqs. Ž E. Ž E. 2 3 may immediately be transformed in oscillation nonoscillation criteria for Eq. Ž E.. This fact enables us to easily obtain some new results about oscillation or nonoscillation of Eq. Ž E. that may be hard to prove directly. We recall that the above Correspondence Principle extends an analogous principle, the Duality Principle stated in 8, for the self-adjoint second order linear differential equation. We also obtain, as a straightforward consequence, that Eq. Ž E. is completely oscillatory if and only if Eqs. Ž E. and Ž E. 2 3 are completely oscillatory, and Eq. Ž E. is partially oscillatory if and only if Eqs. Ž E. 2 and Ž E. 3 are partially oscillatory. Other applications of the Correspondence Principle to the study of oscillation of Eq. Ž E. will be given in the following. We conclude this section with the following properties of equations of class I or II and with some relations among Eq. Ž E. and the equation d u qž t. u0 Ž E. rž t. pž t. dt which is called the adjoint equation of Eq. Ž E.. We recall that, if x, x are two independent solutions of Eq. Ž E., then u Ž r.žx x x x is a solution of Eq. ŽE.. The following holds: PROPOSITION. Assume condition Ž.Then. Ž. a If Eq. Ž E. is oscillatory and of class I, then eery solution of Eq. Ž E. with a zero is oscillatory. Ž b. If Eq. Ž E. is of class II, then eery solution with a double zero is nonoscillatory. Ž c. If Eq. Ž E. is both of class I and of class II, then Eq. Ž E. is nonoscillatory. Ž d. If Eq. Ž E. is of class I, then Eq. Ž E. has nonoscillatory solutions. Proof. Claim Ž. a is proved in 2. Claim Ž b. follows immediately from the definition of an equation of class II. Claim Ž. c follows from Claim Ž. a and Claim Ž b.. For the proof of Claim Ž d. we refer the reader to 23.

6 754 CECCI ET AL. PROPOSITION 2 2. Assume condition Ž.Then. Eq. Ž E. is of class I if and only if equation ŽE. is of class II. Since the adjoint equation of Eq. ŽE. is Eq. Ž E., from Proposition 2 we obtain that Eq. Ž E. is of class II if and only if Eq. ŽE. is of class I. PROPOSITION 3 4. Assume condition Ž.Then. Eq. Ž E. is oscillatory if and only if Eq. ŽE. is oscillatory. We remark that an analogous result was proved in 2 by anan for the more general third order linear equation Ž., by assuming, in addition, that Eq. Ž. is of class I or II. ²: 2 In this section first we give sufficient conditions for Eq. Ž E. to be of class I or of class II. As already assumed in condition Ž., the coefficients p, r, q have fixed sign. Concerning the sign of every single coefficient, the possible situations are eight, but it is straightforward to see that with respect to Eq. Ž E. there are only two significant ones, namely p, r, q positive in 0,.; Ž. p, r, q positive in 0,.. Ž 2. Both cases are now considered. The following holds: TEOREM. If condition Ž. holds, then Eq. Ž E. is of class I. Proof. Assume that Eq. Ž E. does not belong to the class I. Then for some a 0 there exists a solution x of Eq. Ž E. with xaxa Ž. Ž. 0, x Ž a. 0 which vanishes in Ž 0, a.. Let b, 0ba, such that xb0, Ž. xt Ž. 0 for t Ž b, a.. Then there exists c, b c a, such that xž. c 0, xt Ž. 0 for t Ž c, a.. By integrating Eq. Ž E. in Ž t, a., t a, we obtain a xž t. qž s. xž s. ds xž t. pž a. rž t. t pž t. rž t. ta or a 2 p t x a p t q s x s ds x t Ž. Ž. Ž. Ž. Ž. Ž., Ž 2. t rž t.

7 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 755 Ž. 2 where x Ž a. Žpa Ž.. rž t. xž t. ta. By integrating Ž 2. in Ž t, a. we have For t c we get a a a 2 x Ž a. pž s. ds pž s. qž u. xž u. du ds t t s xž a. xž t.. rž a. rž t. a a a 2 x Ž a. pž s. ds pž s. qž u. xž u. du ds c c s xž a. xž c. 0. Ž 3. rž a. rž c. Ž. Ž. Since x a 0 and x a 0, we have rž a. 2 x Ž a. xž t. xž a. xž a 2. pž a. rž t. pž a. rž a. r Ž a. ta xž a. 0. pž a. rž a. Then, taking into account that x, p, q are positive functions in Ž c, a., from Ž. 3 we obtain a contradiction. Remark. It is actually possible to give a more general definition of an equation of class I and II, in the case when the functions p, r, q satisfy weaker assumptions than the ones quoted at the beginning of this paper. Theorem is still valid in this case, with minor changes. Remark 2. In 2 some classes of solutions of three differential inequalities are investigated. The previous Theorem may also be obtained, with a different argument, from Theorem and Remark 5 in 2. If the functions p, r, q are positive in 0,., then also Eqs. Ž E. and Ž E. 2 3 are of class I. An analogous result concerning equations of class II holds: TEOREM 2. If condition Ž. holds, then Eq. Ž E. 2 is of class II. Proof. From Theorem, the adjoint Eq. ŽE. is of class I. Because the adjoint equation of Eq. ŽE. is Eq. Ž E., from Proposition 2 we get the assertion.

8 756 CECCI ET AL. Theorems and 2 extend previous results stated for Eq. Ž E. with pt Ž. rt Ž. 3. We are now able to state some interesting results on the structure of the space S. From a result in 3, the space S always contains a twodimensional subspace either of oscillatory solutions or of nonoscillatory ones. Clearly, if Eq. Ž E. is nonoscillatory, then the structure of the space of solutions is completely determined. We are going to consider the remaining two cases, that is, the cases in which Eq. Ž E. is either completely oscillatory or partially oscillatory. The following holds: PROPOSITION 4. Assume condition Ž.Then. Eq. Ž E. is neer completely oscillatory. Proof. As already emphasized, there are only two significant possibilities concerning the sign of every single coefficient, namely, the cases Ž. and Ž.. If Ž. holds, then from Theorem Eq. Ž E. 2 is of class I, and if Ž. holds, then from Theorem 2 Eq. Ž E. 2 is of class II. In both cases, from Proposition it follows that Eq. Ž E. is nonoscillatory or partially oscillatory. From a result in 2, we can now state the following: TEOREM 3. Assume condition Ž. and let Eq. Ž E. be partially oscillatory. Then the space S contains a two-dimensional subspace W of oscillatory solutions. In addition, if Ž. holds, then there exists x O, x W. Proof. At first assume that condition Ž. holds. Then Eq. Ž E. is of class I and so Eq. ŽE. is of class II. From Proposition 3 we obtain that Eq. ŽE. is partially oscillatory. ence, from a result in 2, the linear space of solutions of its adjoint equation contains a two-dimensional subspace W of oscillatory solutions. ence, the assertion follows by taking into account that the adjoint equation of Eq. ŽE. is Eq. Ž E.. Now assume that condition Ž. holds. Then Eq. Ž E. 2 is of class II and its adjoint equation is of class I. ence, from Proposition 3, it is partially oscillatory. By the same argument as that above, the first assertion follows. In order to complete the proof, let us show that if Ž. holds, then there exists x O, x W. Reasoning as in 6, let x x Ž. t, x x Ž. 2 2 t be a basis for the subspace W, and let x xt Ž. be a nonoscillatory solution of Eq. Ž E., xt Ž. 0 for t Ž a,., a 0, and let b Ž a,. such that x Ž b. 0. Now consider the solution x of Eq. Ž E. given by 3 xž b. x3ž t. xž t. xž t.. Ž 4. xž b.

9 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 757 Since x Ž b. 0, from Proposition Ž a. 3 we get that x3 is oscillatory. Let us prove, by contradiction, that x3 W. If x3 W, then there exist 2 2,,0, such that x x x. From Ž. 4 we obtain xž b. Ž. x2x2 xž t. Ž 5. xž b. which is a contradiction because the right side of Ž. 5 is a nonoscillatory solution of Eq. Ž E. and the left one is an oscillatory solution. The proof is now complete. Remark 3. If condition Ž. 2 holds, then the set of oscillatory solutions O may be exactly a two-dimensional subspace of S, i.e., it may happen that there do not exist solutions x of Eq. Ž E. such that x O, x W,as equations with constant coefficient show. Concerning the nonoscillatory solutions, the following holds: TEOREM 4. Assume condition Ž. and let Eq. Ž E. be partially oscillatory. Then S cannot contain a two-dimensional subspace of nonoscillatory solutions. Proof. From Proposition 3, Eq. ŽE. is partially oscillatory. Assume that the space S contains a two-dimensional subspace of nonoscillatory solutions. From a result in 2, Eq. ŽE. is either nonoscillatory or completely oscillatory, which is a contradiction. Ž. Ž. It is obvious that the above results are still valid for Eqs. E and E. 2 3 ²: 3 We state now some nonoscillation criteria which extend previous results in 2, 3. The following holds: Ž. TEOREM 5. Assume condition. If t s then Eq. Ž E. is nonoscillatory. Proof. Let a 0 such that qž t. r Ž s. pž u. du ds dt, Ž 6. t s a a a qž t. r Ž s. pž u. du ds dt

10 758 CECCI ET AL. Ž. Ž. Ž. 2 and let x be a solution of Eq. E such that xax a 0, x Ž a., where 2 x Ž a. xž a., x Ž a. xž t.. rž a. pž a. rž t. ta Ž. Ž. Integrating Eq. E three times in a, t, with t a, we obtain t t s 2 xž t. xž a. x Ž a. rž s. dsx Ž a. r Ž s. pž u. du ds a a a Ž. Then from 7 we have t s u a a a Ž. Ž. Ž. Ž. Ž. r s p u q x d du ds. 7 t s t s u a a a a a Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. x t r s p u du ds r s p u q x d du ds. 8 Assume that Eq. Ž E. is oscillatory. Equation Ž E. being of class I, then, by virtue of Proposition Ž. a, x is oscillatory. Let b a such that xb0 Ž. and xt Ž. 0 for t Ž a, b.. ence from Ž 8. we obtain for t Ž a, b. that t s xž t. r Ž s. pž u. du ds. Ž 9. a Ž. Ž. Ž. Taking into account that xb0, again from 8 and 9 we get b s b s u a a a a a Ž. Ž. Ž. Ž. Ž. Ž. Ž. 0x b r s p u du ds r s p u q x d du ds b s u a a a Ž. Ž. Ž. Ž. r s p u q x d du ds a b s u a a a a a Ž. Ž. Ž. Ž. Ž. r s p u q r p d d d du ds, which is a contradiction since b t s t s qž t. r Ž s. pž u. du ds dt qž t. r Ž s. pž u. du ds dt. a a a a a a Ž. Ž. Ž. Remark 4. If pt r t, then condition 6 implies qt Ž. 0 dt and tqž. t dt. Thus Theorem 5 extends a previous result in 3 0. We Ž. Ž. note also that if pt rt, then the condition tqž. 0 t dt is sufficient for existence of at least one oscillatory solution 3.

11 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 759 If condition Ž. holds, then we have the following: 2 Ž. TEOREM 6. Assume condition. If 2 t s then Eq. Ž E. is nonoscillatory. qž t. pž s. r Ž u. du ds dt, Proof. Applying Theorem 5 to Eq. ŽE., we obtain that Eq. ŽE. is nonoscillatory. The assertion now follows from Proposition 3. Applying the Correspondence Principle to Eq. Ž E., we obtain the following: TEOREM 7. Ž. i Assume condition Ž.. Then Eq. Ž E. is nonoscillatory in the case any of the following conditions is satisfied: t s Ž a. pž t. qž s. r Ž u. du ds dt, t s Ž b. r Ž t. pž s. qž u. du ds dt. Ž ii. Assume condition Ž.. Then Eq. Ž E. 2 is nonoscillatory in the case any of the following conditions is satisfied: t s Ž a. pž t. r Ž s. qž u. du ds dt, t s Ž b. r Ž t. qž s. pž u. du ds dt. We conclude this section with an interesting consequence concerning the nonoscillatory solutions. We recall that a nonoscillatory solution x of Eq. Ž E. is said to be a Kneser solution Žsee, e.g., 24. or a completely monotone solution Žsee, e.g., 22. if there exists t0 0 such that for t t0 k k Ž. xž t. D Ž x.ž t. 0, k 0,,2 Ž 0. where D k Ž x. denotes the kth quasiderivative of x defined as d 0 0 D Ž x.ž t. xž t., D Ž x.ž t. D Ž x.ž t., rž t. dt d 2 D Ž x.ž t. D Ž x.ž t.. Ž. pž t. dt

12 760 CECCI ET AL. Several criteria for existence of Kneser solutions may be found in 22 for the linear case and in, 4, 24 for the nonlinear one. Assume conditions pž t. dt rž t. dt. Ž Then, a classical result of Kiguradze, extended later by Elias 24, p. 43 Žsee also 9, 26., states that Ž. a if Ž. holds, then every nonoscillatory solution of Eq. Ž E. is either a Kneser solution or satisfies for t t0 0 xž t. D k Ž x.ž t. 0 k0,,2; Ž 3. Ž. Ž. Ž. b if holds, then Eq. E does not have Kneser solutions. 2 The following holds: COROLLARY. Assume conditions Ž.. Then the set of Kneser solutions of Eq. Ž E. is a nonempty subspace of S in case any of the following conditions is satisfied: t s Ž i. pž t. dt rž t. dt, qž t. r Ž s. pž u. du ds dt t s Ž ii. pž t. dt qž t. dt, r Ž t. pž s. qž u. du ds dt t s Ž iii. qž t. dt rž t. dt, pž t. qž s. r Ž u. du ds dt Proof. A well-known result of artman and Wintner Žsee, e.g., 22, Chap. XIV, Theorem 2.. states that, if Ž. holds, then Eq. Ž E. has Kneser solutions. Assume conditions Ž. i. From Theorem 5, every nontrivial solution x of Eq. Ž E. is nonoscillatory. Since Ž 2. holds, then x satisfies, for t t0 0, either Ž 0. or Ž 3.. Let us show that solutions satisfying Ž 3. are unbounded.

13 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 76 Let x be a solution of Eq. Ž E. satisfying Ž 3.. Without loss of generality, we may assume that xt Ž. 0 from t t. From Ž 3. we have or xž t. xž t0. rž t. rž t. 0 0 t xž t. xž t0. xž t0. rž. d. rž t. t 0 0 ence, by Ž 2., we get that x is unbounded. Now the assertion easily follows, because solutions satisfying Ž 0. are bounded. Assume now conditions Ž ii.. Reasoning as above, we get that the set of Kneser solutions of Eq. Ž E. 2 is a nonempty subspace of S 2. From Corollary 3 in 9, we obtain the assertion. Finally, if conditions Ž iii. are satisfied, then the assertion follows using a similar argument. ²: 4 In this section we prove some oscillation theorems for Eq. Ž E. which generalize a well-known result Žsee, e.g., 2, 3.. The following holds: Ž. TEOREM 8. Assume condition. If t pž t. dt rž t. dt qž t. r Ž s. ds dt, then Eq. Ž E. is partially oscillatory. Proof. From Proposition 4, Eq. Ž E. is never completely oscillatory. Assume that Eq. Ž E. is nonoscillatory. According to a result in 0, Eq. Ž E. has nonoscillatory solutions x which satisfy Ž 3. for all large t. Without loss of generality, we may assume that xt Ž. 0, D Ž x.ž t. 0, D 2 Ž x.ž t. 0 for t t 0, where D k Ž x.ž t. is defined in Ž. 0. Integrating Eq. Ž E. on Ž t, t., t t, we obtain 0 0 ž / t xž t. xž t. qž s. xž s. ds0 pž t. rž t. pž t. rž t. t tt0 0

14 762 CECCI ET AL. 2 Ž.Ž. or, since D x t 0, t xž t. qž s. xž s. ds. Ž 4. pž t. rž t. t 0 Ž. Since D x is a positive increasing function, reasoning as in the proof of Corollary, we have tt0 t xž t. xž t0. xž t0. rž. d. rž t. t From Ž 4. we get x Ž t. ž pž t. ž rž t. / / tt t s Ž. ž Ž 0. Ž 0. Ž. t rž t. t / q s x t x t r d ds t s xž t0. qž s. rž. d ds. Ž 5. rž t. t t Ž. As t, 5 gives a contradiction. Such a result agrees with oscillation results stated for equations of type Ž E. with pt Ž. rt Ž. in 2, 3, and for nonlinear equations in 3, 7. Remark 5. A well-known criterion Žsee, e.g., 9, 24. states that Eq. Ž E. Ž. is oscillatory if qt dt pt Ž. dt rt Ž dt. Obviously, if Ž. qt dt, then qt Ž. t rž s ds dt. ence Theorem 8 extends such a result. If condition Ž. holds, then we have the following: 2 Ž. TEOREM 9. Assume condition. If 2 pž t. dt rž t. dt 0 0 t qž t. pž s. ds dt, 0 0 then Eq. Ž E. is partially oscillatory. Proof. Applying Theorem 8 to Eq. ŽE., we obtain that Eq. ŽE. is oscillatory. The assertion now follows from Propositions 3 and 4.

15 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 763 Applying the Correspondence Principle to Eq. Ž E., we obtain the following: TEOREM 0. Ž. i Assume condition Ž.. Then Eq. Ž E. is partially oscillatory in the case any of the following conditions is satisfied: t Ž a. qž t. dt pž t. dt r Ž t. pž s. ds dt, t Ž b. rž t. dt qž t. dt pž t. qž s. ds dt Ž ii. Assume condition Ž.. Then Eq. Ž E. 2 is partially oscillatory in the case any of the following conditions is satisfied: t Ž a. qž t. dt r Ž t. qž s. ds dt, pž t. dt t Ž b. rž t. dt pž t. r Ž s. ds dt, qž t. dt We conclude this section with some consequences of the above results. Assume condition Ž.. From Theorem, Eq. Ž E. is of class I, and from Proposition we get that Eq. Ž E. has nonoscillatory solutions. In addition, if Eq. Ž E. is oscillatory, then every solution with a zero is oscillatory. Some properties concerning the qualitative behavior of oscillatory and nonoscillatory solutions of Eq. Ž E. are given by the following: COROLLARY 2. Assume conditions Ž.. If one of the following conditions t Ž i. pž t. dt rž t. dt qž t. r Ž s. ds dt, t Ž ii. qž t. dt pž t. dt r Ž t. pž s. ds dt, t Ž iii. rž t. dt qž t. dt pž t. qž s. ds dt, is satisfied, then: Ž. Ž. Ž. a eery solution x of Eq. E, which erifies at some t 0 0 x t 0 6 Ž. Ž. 0

16 764 or or CECCI ET AL. xž t0. 0 Ž 62. xž t. 0, Ž 6 3. rž t. tt 0 is oscillatory; Ž b. eery nonoscillatory solution of Eq. Ž E. is a Kneser solution on the whole half-axis 0,.. Proof. Claim Ž. a. From Theorems 8 and 0, Eqs. Ž E. i, i, 2, 3, are oscilla- tory. Let x be a solution of Eq. Ž E. verifying Eq. Ž 6.. Since Eq. Ž E. is of class I, from Proposition Ž. a x is oscillatory. Assume now that x is a solution of Eq. Ž E. verifying Ž Then yt Ž. rt Ž Ž.. xt Ž. is a solution of Eq. Ž E. 2. Taking into account that y vanishes at t t and Eq. Ž E. 0 2 is of class I and oscillatory, we obtain that y is oscillatory. Applying the Correspondence Principle, we get again the assertion. Finally, if x is a solution of Eq. Ž E. verifying Ž 6. 3, then the assertion follows by using a similar argument. Claim Ž b.. Assume conditions Ž i. and let x be a nonoscillatory solution of Eq. Ž E.. Equation Ž E. being oscillatory, from the quoted result of Chanturia 0 there exists t0 0 such that x is a Kneser solution on t,. 0. If, by contradiction, t0 0, then there exists t, 0tt 0, such that xt Ž. 0. From the claim Ž. a, x is oscillatory, which is a contradiction. If conditions Ž ii. or Ž iii. hold, then the assertion follows from the Correspondence Principle and using an argument similar to that above. Assume condition Ž. and let x, i, 2, 3, be the solution of Eq. Ž E. i which satisfies conditions Ž 6.. If Eq. Ž E. is oscillatory Ž i i.e., for example, if assumptions of Theorems 6 or 7 are satisfied., then a basis for the space S is given by the functions x, i, 2, 3. ence Eq. Ž E. i has a basis composed of oscillatory solutions. Note that the same result may be obtained from Theorem 4. If condition Ž. holds, then from Theorem 2, Eq. Ž E. 2 is of class II and so every solution with a double zero is nonoscillatory Ž Proposition Ž b...

17 3RD ORDER DIFFERENTIAL EQUATION SOLUTIONS 765 ence it is straightforward to show that every solution x, which verifies Žat some t 0. one of the following conditions 0 ž r t / ž r t / xž t0. 0 xž t0. 0 xž t. Ž 7. rž t. tt 0 xž t0. xž t0. 0 xž t. 0 Ž 72. Ž. tt 0 xž t0. 0 xž t0. xž t. 0 Ž 73. Ž. tt 0 is nonoscillatory. Then Eq. Ž E. has a basis of nonoscillatory solutions. Remark 6. Assume conditions Ž. 2 and let x i, i, 2, 3, be the solution of Eq. Ž E. which satisfies conditions Ž 7.. If Eq. Ž E. i is oscillatory Ž i.e., for example, if assumptions of Theorems 6 or 7 are satisfied., then, taking into account Theorem 4, there exists nontrivial linear combinations of xi which are oscillatory. ACKNOWLEDGMENT The authors thank the referee for his useful comments. REFERENCES. M. Bartusek, Asymptotic Properties of Oscillatory Solutions of Differential Equations of the n-th Order, Folia Fac. Sci. Nat. Univ. Brunensis Masarykianae, M. Bartusek, On the structure of solutions of a system of three differential inequalities, Arch. Math. Brno 30, No. 2 Ž 994., M. Bartusek and Z. Dosla, On solutions of a third order nonlinear differential equation, Nonlinear Anal. Theory, Methods, Appl. 23 Ž 994., M. Bartusek and Z. Dosla, Oscillatory criteria for nonlinear third order differential equations with quasiderivatives, preprint. 5. M. Cecchi, Oscillation criteria for a class of third order linear differential equations, Boll. Un. Mat. Ital. VI, C 2 Ž 983., M. Cecchi, Sul comportamento delle soluzioni di una classe di equazioni differenziali lineari del terzo ordine in caso di oscillazione, Boll. Un. Mat. Ital. VI, C 4 Ž 985., M. Cecchi and M. Marini, On the oscillatory behavior of a third order nonlinear differential equation, Nonlinear Anal. Theory, Methods Appl. 5, No. 2 Ž 990., M. Cecchi, M. Marini and Gab. Villari, Integral criteria for a classification of solutions of linear differential equations, J. Differential Equations 99, No. 2 Ž 992.,

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