summary:In this paper, we introduce the concept of upper and lower solutions for third order periodic boundary value problems. We show that the monotone iterative technique is valid and obtain the extremal solutions as limits of monotone sequences. We first present a new maximum principle for ordinary differential inequalities of third order that is interesting by itself

Nieto, Juan J.

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Commentationes Mathematicae Universitatis CarolinaeJuan J. NietoPeriodic solutions for third order ordinary differential equationsCommentationes Mathematicae Universitatis Carolinae, Vol. 32 (1991), No. 3, 495--499Persistent URL: http://dml.cz/dmlcz/118430Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1991Institute 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 mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 32,3 (1991)495–499 495Periodic solutions for third orderordinary differential equations*Juan J. NietoAbstract. In this paper, we introduce the concept of upper and lower solutions for thirdorder periodic boundary value problems. We show that the monotone iterative techniqueis valid and obtain the extremal solutions as limits of monotone sequences. We firstpresent a new maximum principle for ordinary differential inequalities of third order thatis interesting by itself.Keywords: periodic solution, maximum principle, upper and lower solutions, monotonemethodClassification: 34B15, 34C251. Introduction.The existence of solutions for third order ordinary differential equations has beenwidely studied in the last years and applications of third order differential equationsare encountered in physics, engineering and mathematical biology. See, for instance,[1]–[6], [9] and the references therein. Recently has been considered the existenceof periodic solutions of third order ordinary differential equations [2], [3], [4], [8].In this paper, we study the existence of solutions for the following periodic bound-ary value problem (PBVP) for a third order ordinary differential equation(1.1) u′′′ = f(t, u), u(0) = u(2π), u′(0) = u′(2π), u′′(0) = u′′(2π).We first present a new maximum principle for third order ordinary differentialinequalities. As it is well known, the maximum principles have applications to thequestion of uniqueness and continuous dependence on the boundary values for linearequations and, also, to the question of existence for nonlinear equations by meansof the monotone iterative method. We apply the maximum principle to the studyof the existence of periodic solutions for the periodic boundary value problem (1.1).To this purpose, we introduce a new concept of upper and lower solutions as limitsof monotone iterates.Finally, we mention some open problems and questions relative to the periodicboundary value problem and questions relative to the periodic boundary value prob-lem (1.1) for further research.*Research partially supported by DGICYT, project PS88–0054.496 J. J. Nieto2. Maximum principle.Let m > 0, M = m2, and for u ∈ C1(I), v ∈ C2(I), define the operators L1u =u′ − mu, and L2v = v′′ − mv, respectively. Relative to L1 and L2, we have thefollowing comparison results.Lemma 2.1. Suppose that L1u ≥ 0 on I and u(0) ≥ u(2π), then u ≤ 0 on I.Lemma 2.2. If L2u ≥ 0 on I, u(0) = u(2π), and u′(0) ≥ u′(2π), then u ≤ 0 on I.For the proofs of these results see Lemma 1.2.2 and Lemma 2.1.1 in [10], respec-tively. They are utilized to generate monotone sequences that converge to extremalsolutions of first and second order PBVP, respectively.Now, for u ∈ C3(I), let L = L1◦L2, that is, Lu = u′′′−mu′′−mu′+Mu. We arein a position to prove the following maximum principle for third order differentialinequalities.Theorem 2.3. Suppose that Lu ≥ 0 on I and u(0) = u(2π), u′(0) ≤ u′(2π) andu′′(0) ≥ u′′(2π). Then we have that u ≤ 0 on I.Proof: Let v = L2u. Thus, L1v ≥ 0 and v(0) ≥ v(2π). In consequence, we getthat v ≤ 0 on I, that is, L2(−u) ≥ 0. Now, by Lemma 2.2, we conclude that −u ≤ 0on I. 3. Upper and lower solutions.We say that α ∈ C3(I) is a lower solution of (1.1) if there exists m > 0 such thatα′′′(t)− mα′′(t)− mα′(t) ≤ f(t, α(t)), t ∈ I,(3.1)α(0) = α(2π), α′(0) ≥ α′(2π), α′′(0) ≤ α′′(2π).(3.2)Analogously, we say that β ∈ C3(I) is an upper solution of (1.1) if there existsn > 0 withβ′′′(t)− nβ′′(t)− nβ′(t) ≥ f(t, β(t)), t ∈ I,(3.3)β(0) = β(2π), β′(0) ≤ β′(2π), β′′(0) ≥ β′′(2π).(3.4)Conditions (3.1) and (3.3) may seem very artificial, but in practice we have somegeneral situations, where they are satisfied. Indeed, we have the following two casesfor α ∈ C3(I). Obviously, similar situations are valid for β.Case I. Suppose that α ∈ C3(I) and α′′′(t) < f(t, α(t)), t ∈ I. Then we can choosek, m > 0, such that α′′′(t)+k ≤ f(t, α(t)), t ∈ I and m|α′′(t)|+m|α′(t)| ≤ k, t ∈ I.Therefore, Condition (3.1) holds.Case II. Suppose that a ∈ R is such that 0 ≤ f(t, a), t ∈ I. Thus, defining α(t) = a,t ∈ I, we have that (3.1) is satisfied for any m > 0.Now, let us assume that there exist α, β lower and upper solutions of (1.1),respectively, such that(3.5) α(t) ≤ β(t), t ∈ I.Periodic solutions for third order ordinary differential equations 497Suppose that m = n and assume that f satisfies the following condition forα(t) ≤ v ≤ u ≤ β(t), t ∈ I,(3.6) f(t, u)− f(t, v) ≥ −M(u − v).Let E = {u ∈ C3(I) : u(0) = u(2π), u′(0) = u′(2π), u′′(0) = u′′(2π)}, F = C(I),and consider the linear operator L : E → F . The analysis of this differentialoperator [3] shows that KerL = {0} sincem > 0. Thus, L−1 : F → E is continuous.For η ∈ F , let us solve the following PBVP(3.7) Lv = f(t, η) +Mη, v ∈ E.The unique solution of this PBVP is denoted by Kη, and we can define theoperator K : F → E. For u ∈ F , we say that u ∈ [α, β] if α ≤ u ≤ β on I.Theorem 3.1. The operatorK has the following two properties in the sector [α, β].(P1) If η ∈ [α, β], then Kη ∈ [α, β], and(P2) if η1, η2 ∈ [α, β] and η1 ≤ η2 on I, then Kη1 ≤ Kη2 on I.Proof: To show (P1), let v = Kη and u = v − α. Thus, on using (3.1) and (3.6),we getLu = f(t, η) +Mη − α′′′ +mα′′ +mα′ − Mα ≥ f(t, η) +Mη − f(t, α)− Mα ≥ 0.Now, taking into account (3.2) and Theorem 2.3, we obtain that v ≥ 0 on I, thatis, v ≥ α on I. Similarly we obtain that v ≤ β on I holds. In order to prove (P2),note thatL(Kη2 − Kη1) = f(t, η2) +Mη2 − f(t, η1)− Mη1 ≥ 0.This allows us to conclude that u ≥ 0 on I. Next, we present the monotone iterative technique which yields monotone se-quences that converge to the extremal solutions between the lower and upper solu-tions.Theorem 3.2. Suppose that there exist α and β lower and upper solutions of (1.1),respectively, satisfying (3.5) and (3.6). Then there exist monotone sequences {αn} ↑r and {βn} ↓ s such thatα = α0 ≤ α1 ≤ · · · ≤ αn ≤ βm ≤ · · · ≤ β1 ≤ β0 = β, for every n, m ∈ N.Here, r, s are the minimal and maximal solutions of (1.1) in the sector [α, β], re-spectively, that is, any solution u ∈ [α, β] of (1.1) is such that u ∈ [r, s].Proof: Let α0 = α and α1 = Kα0. Thus α1 ∈ [α, β] by the property (P1). Now,by induction, it is easy to prove that the sequence {αn} defined by αn+1 = Kαn,is such that α ≤ αn ≤ αn+1 ≤ β, n ∈ N. Moreover, {αn} is bounded and, for everyt ∈ I, we have that {αn(t)} ↑ r(t). On the other hand, we have that there exists498 J. J. Nietoa constant C > 0 with |f(t, αn(t)) +Mαn(t)| ≤ C for every n ∈ N and t ∈ I since{αn} is bounded in F and f is continuous. In consequence, ‖αn+1‖E ≤ ‖L−1‖ · Cand {αn} is bounded in C3(I). This implies that {αn} converges to r uniformlyon I. Using standard arguments, we obtain that r is actually a solution of (1.1).Analogously, we obtain a sequence {βn} ↓ s, where s is a solution of (1.1).To show that r ≤ s are the minimal and maximal solutions of (1.1) in [α, β], letu be a solution of (1.1) in [α, β]. Thus, α1 = Kα ≤ Ku = u ⊂ Kβ = β1, and byinduction we get that αn ≤ u ≤ βn for every n ∈ N. Passing to the limit, we seethat r ≤ u ≤ s on I. As an important and practical case we obtain the following result.Corollary 3.3. Suppose that there exist constants a ≤ b and M > 0 such thatf(t, a) ≤ 0 ≤ f(t, b), t ∈ I andf(t, u)− f(t, v) ≥ −M(u − v), for a ≤ v ≤ u ≤ b, t ∈ I.Then there exist r and s minimal and maximal solutions of (1.1), respectively, witha ≤ r(t) ≤ s(t) ≤ b, t ∈ I.4. Open problems.It is interesting to investigate the existence of solutions of the PBVP when someor all of the inequalities that appear in the definition of lower and upper solutionsare not satisfied. Some results in this direction for second order PBVP are givenin [7], [10], [11].Suppose that we have that α ≥ β on I instead of (3.5). Is it possible to ensure theexistence of solution for the PBVP (1.1)? For second order differential equationswith periodic boundary conditions, this question is solved in [12].If r = s in Theorem 3.2, then the problem (1.1) has a unique solution in thesector [α, β]. On the other hand, if r 6= s, then it would be interesting to studythe structure of the set of solutions of (1.1) between α and β. Following the ideasof [12] and [13], we conjecture that under the conditions of Theorem 3.2, the setof solutions of (1.1) is compact and connected provided that f is monotone in u(either increasing or decreasing) for every t ∈ I and there exists a sufficiently smallconstant k > 0 with |f(t, u)− f(t, v)| ≤ k|u − v| for every t ∈ I, u, v ∈ R.For η ∈ [α, β], we can define η1 = Kη, ηn+1 = Kηn, n ≥ 1. Thus, we havea discrete dynamical system and obviously r and s are fixed points for K. For somegeneral properties on dynamical systems see, for instance, [14]. What is the globalattractor J for this discrete dynamical system? Is J stable in any sense? Thesequestions are considered for second order PBVP in [7]. It is easy to see that anysolution u ∈ [α, β] of (1.1) is such that u ∈ J .References[1] Aftabizadeh A.R., Gupta C.P., Xu J.M., Existence and uniqueness theorems for three-pointboundary value problems, SIAM J. Math. Anal. 20 (1989), 716–726.Periodic solutions for third order ordinary differential equations 499[2] Aftabizadeh A.R., Gupta C.P., Xu J.M., Periodic boundary value problems for third orderordinary differential equations, Nonlinear Anal. 14 (1990), 1–10.[3] Afuwape A.U., Omari P., Zanolin F., Nonlinear perturbations of differential operators withnontrivial kernel and applications to third-order periodic boundary value problems, J. Math.Anal. Appl. 143 (1989), 35–56.[4] Afuwape A.U., Zanolin F., An existence theorem for periodic solutions and applications tosome third order nonlinear differential equations, preprint.[5] Agarwal R.P., Boundary Value Problems for Higher Order Differential Equations, WorldScientific, Singapore, 1986.[6] Agarwal R.P., Existence-uniqueness and iterative methods for third-order boundary valueproblems, J. Comp. Appl. Math. 17 (1987), 271–289.[7] Cabada A., Nieto J.J., A generalization of the monotone iterative technique for nonlinearsecond order periodic boundary value problems, J. Math. Anal. Appl. 151 (1990), 181–189.[8] Ezeilo J.O.C., Nkashama M.N., Resonant and nonresonant oscillations for some third ordernonlinear ordinary differential equations, Nonlinear Anal. 12 (1988), 1029–1046.[9] Gregus M., Third Order Linear Differential Equations, D. Reidel, Dordrecht, 1987.[10] Ladde G.S., Lakshmikantham V., Vatsala A.S., Monotone Iterative Techniques for NonlinearDifferential Equations, Pitman, Boston, 1985.[11] Lakshmikantham V., Nieto J.J., Sun Y., An existence result about periodic boundary valueproblems of second order differential equations, Appl. Anal., to appear.[12] Nieto J.J., Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl.130 (1988), 22–29.[13] Rudolf B., Kubacek Z., Remarks on J. J. Nieto’s paper: Nonlinear second order periodicboundary value problems, J. Math. Anal. Appl. 146 (1990), 203–206.[14] Temam R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1988.Departamento de Análisis Matemático, Facultad de Matemáticas,Universidad de Santiago de Compostela, Spain(Received May 13, 1991)

Periodic solutions for third order ordinary differential equations

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Periodic solutions for third order ordinary differential equations

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