Non-monotone Positive Solutions of Second-order Linear Differential Equations: Existence, Nonexistence and Criteria

We study non-monotone positive solutions of the second-order linear differential equations: (p(t)x) + q(t)x = e(t), with positive p(t) and q(t). For the first time, some criteria as well as the existence and nonexistence of non-monotone positive solutions are proved in the framework of some properties of solutions θ(t) of the corresponding integrable linear equation: (p(t)θ) = e(t). The main results are illustrated by many examples dealing with equations which allow exact non-monotone positive solutions not necessarily periodic. Finally, we pose some open questions.


Introduction
In recent years, mathematical models which admit non-monotone positive solutions pay attention in various disciplines of the applied sciences.For instance, non-monotonic behaviour of: the amplitude of harmonic oscillator driven with chirped pulsed force [9], the three-flavour oscillation probability [1,10], the particle density in Bose-Einstein condensates with attractive atom-atom interaction [2,5,14], the several kinds of cardiogenic oscillations [6], the structural analysis of blood glucosa [4], the response function in a delayed chemostat model [19].
In the paper, we consider the second-order linear differential equation: (p(t)x ) + q(t)x = e(t), t ≥ t 0 , (1.1) where p, q, e ∈ C[t 0 , ∞), p(t) > 0, q(t) ≥ 0 for t ≥ t 0 , and x = x(t).By a solution of (1.1), we mean a function x ∈ C 1 [t 0 , ∞) which satisfies p(t)x (t) ∈ C 1 [t 0 , ∞) and (1.1) on [t 0 , ∞).We say Corresponding author.Email: tanaka@xmath.ous.ac.jp that a function x(t) is (eventually) positive if x(t) > 0 for all t > t 1 and some t 1 ≥ t 0 (where it is not necessary, the word eventually is avoided).Also, a smooth x(t) is a non-monotone function on [t 0 , ∞) (or shortly said, x(t) is non-monotonic on [t 0 , ∞)) if x (t) is a sign-changing function on [t 0 , ∞), that is, for each t > t 0 , there exist t + , t − ∈ [t, ∞) such that x (t + ) > 0 and x (t − ) < 0 (in the literature, such a function x(t) is also called weakly oscillatory, see for instance [3,7]).It is easy to show that: lim inf t→∞ x(t) < lim sup t→∞ x(t) implies x(t) is non-monotonic on [t 0 , ∞), which is used here as a criterion for the non-monotonic behaviour of continuous functions.
However, in our main problems of the paper, the non-homogeneous part e(t) is not a point of any construction, but e(t) is an arbitrary given function just as p(t) and q(t).
Main problems.1) Find sufficient and necessary conditions on arbitrary given p(t), q(t), and e(t), such that every positive solution of (1.1) is non-monotonic.2) Prove the existence of at least one non-monotone positive solution of (1.1).
Taking into account the preceding observation, we can positively answer to the main problem concerning the concrete Euler equation: The purpose of this paper is to give some answers to the main problem in the framework of non-monotonic behaviour of the function θ = θ(t), θ ∈ C 2 (t 0 , ∞), which is a solution of the next integrable second-order linear differential equation: (p(t)θ ) = e(t), t ≥ t 0 . (1.3) The most simple model for the linear equation (1.1) having p(t), q(t), e(t), and x(t) that satisfy all required assumptions and conclusions of this paper is: ( For some a, b and e(t), the equation (1. , where the amplitude α(t) is positive, the frequency ω(t) goes to infinity as t goes to infinity, and S(τ) is a continuous periodic function.
In Section 2, we give some relations for lower and upper limits of x(t) and θ(t) as the solutions of respectively (1.1) and (1.3), in two different cases: bounded and possible unbounded solutions.It will ensure some conditions on θ(t) which imply the non-monotonicity of positive solutions of x(t).In Sections 3 and 4, some conditions on θ(t) are involved such that the main equation (1.1) allows or not the positive non-monotone solutions.Finally in Section 5, we suggest some open problems for further study on this subject.
Our approach here to non-monotone positive solutions of second-order differential equations is quiet different than in [13], where (without limits inferior and superior of x(t)) the sign-changing property of x (t) of positive solutions x(t) of a class of nonlinear differential equations has been studied by means of a variational criterion.On the existence of positive periodic solutions as a particular case of non-monotonic behaviour of the second-order linear differential equations, see for instance [18,Section 2], [11,Lemma 2.2] and references cited therein.

Criteria for non-monotonicity of solutions
Since the right-hand side of both equations (1.1) and (1.3) are the same, we can derive the next relation between all their solutions.Proposition 2.1.Let x(t) and θ(t) be two smooth functions on [t 0 , ∞) that satisfy the following equality: In what follows, we consider two rather different cases: the bounded and not necessarily bounded solutions of equation (1.1).

Non-monotone positive bounded solutions
In this subsection, the main assumption on p(t) and q(t) is: where c 1 , c 2 ∈ R and the real constants C 1 , C 2 and C 3 only depend on γ.Next, we have: if γ < 0, then lim inf t→∞ θ(t) = lim sup t→∞ θ(t) = c 1 , and if γ = 0, then lim inf t→∞ θ(t) = c 1 + 1 < c 1 + 3 = lim sup t→∞ θ(t).Thus, if γ = 0, then condition (2.5) is fulfilled, and by Theorem 2.3, every positive bounded solution x(t) of equation (1.4) is non-monotonic on [t 0 , ∞).Next, since a = 2 and b = 1, we especially have and thus, the extra assumption (2.3) is also satisfied in this case.Finally, it is worth to mention that the function x(t) = t γ 2 + sin(ln t) is an exact non-monotone positive solution of equation (1.4) with such a, b and e(t).We leave to the reader to make a related example in which the solution If q(t) ≡ 0, then assumption (2.2) implies 1/p ∈ L 1 (t 0 , ∞).By direct integration of equation (1.1), we obtain for some c 1 , c 2 depending on t 0 .Since in the subsection we are working with positive bounded solutions x(t), from the previous equality and (2.2), we have: Hence, from (2.8) and (2.9), we can easily prove the next two simple results.Theorem 2.5.Let q(t) ≡ 0 and assume then every positive bounded solution x(t) of equation (1.1) satisfies (2.6), and so, x(t) is nonmonotonic on [t 0 , ∞).
As pointed out above, the coefficients p(t) = t a and q(t) = t −b , t ≥ t 0 > 0, satisfy condition (2.2) if a > 1 and a + b > 2.Moreover, we have q(t) ≡ 0 and so, we may use Theorem 2.5.
The previous example can be generalised to the case when e(t) is the first derivative of an oscillating (chirped) function with general frequency ω(t).

Non-monotone positive not necessarily bounded solutions
Since p(t) > 0, we can define the next function, and we suppose that: At the first, we prove the following technical result.
Proposition 2.11.Let x(t) be a continuous function such that 0 ≤ x(t) P(t) ≤ M for all t ≥ t 0 and some M > 0. If assumptions (2.14) and (2.15) hold, then there exists a constant L ∈ [0, ∞) such that (2.16) then (2.18) Proof.We introduce two auxiliary functions X p (t) and X q (t) defined by: If q(t) ≡ 0 or x(t) ≡ 0, then the conclusion of this proposition obviously holds.Thus, we may assume q(t) ≥ 0, q(t) ≡ 0 and x(t) ≥ 0, x(t) ≡ 0. Hence, the functions X p (t) and X q (t) are positive, X p (t) is increasing and X q (t) is nondecreasing.Moreover, with the help of assumptions x(r) P(r) ≤ M and (2.15), we have Therefore, there exists L q ∈ (0, ∞) such that L q = lim t→∞ X q (t).In particular, X q (t) ≥ L q /2 on [t 1 , ∞) for some t 1 ≥ t 0 , and hence which implies lim t→∞ X p (t) = ∞.Hence, the L'Hospital rule yields that: and thus, the desired statement (2.16) is shown.Finally, from previous equality we especially conclude that lim t→∞ X q (t) − X p (t) as t → ∞, where (2.17) is used.It proves (2.18).
A model equation for (1.1) with the coefficients p(t) and q(t) satisfying required assumptions (2.14), (2.15) and (2.17) is equation (1.4), which is shown in the next example.
Example 2.12.Let p(t) = t a and q(t) = t −b , where a ≤ 1 and a + b > 2. If a = 1 then (2.14) is satisfied for a ≤ 1.Since a ≤ 1 and b > 2 − a imply b > 1, in both cases of P(t), we have:
Proof of Lemma 2.13.Firstly, from (2.1) with C 1 = C 2 = 0, we have: (2.29) Then from (2.29), x(r) = x(r) P(r) P(r), and 0 ≤ x(t) P(t) ≤ M, we derive: as well as by Proposition 2.11, there exists L ∈ [0, ∞) such that Proof of Theorem 2.14.The first part of this theorem is very similar to Theorem 2.3 and so, its proof is leaved to the reader.Next, according to the assumptions of the second part of this theorem, we my apply Lemma 2.13 (ii) which together with assumption (2.21) ensure that every positive solution x(t) of equation (1.1) satisfies the required condition (2.22).Now, Lemma 2.16 proves that x(t) is non-monotonic on [t 0 , ∞).

Theorem 3.2 (Existence of solution)
. Assume (3.1) and (3.2), and let θ(t) be a solution of equation then the main equation (1.1) has a positive solution x(t) such that where ω > 0, − √ 3 3 ω < γ ≤ 0, a > 1 and a + b > 2 + γ, then by (2.7), where the real constants C 1 , C 2 and C 3 only depend on parameters ω, γ, a and b.It follows that (3.3) is satisfied.On the other hand, x(t) = t γ 2 + sin(ω ln t) is an exact non-monotone non-periodic positive bounded solution of the equation (1.4) with above e(t) such that x(t) satisfies (3.4).
We can observe now that the coefficients p(t) = t a and q(t) = t −b of equation (1.4) simultaneously satisfy the required assumptions (2.2), (3.1) and (3.2) provided a > 1 and b > 1.In fact, in Section 2 it is mentioned that (2.2) holds if a > 1 and a + b > 2, and in the previous example, it is mentioned that (3.1) and (3.2) hold if b > 1 and a + b > 2. These together imply a > 1 and b > 1.
Proof of Theorem 3.2.According to (3.3), there exist t 1 ≥ t 0 , δ 1 > 0 and δ 2 > 0 such that Because of (3.1), we can take t 2 ≥ t 1 so large that Hence, by (3.5), we find that Then it is easy to check that x * is a solution of ( We take t 2 ≥ t 1 so large that Hence we have (F y)(t) ≥ (δ 1 + 1)P(t) (3.13) and ∞ s q(r)P 2 (r)drds (3.14) for t ≥ t 3 .From (3.9) it follows that Therefore, (3.10) Then it is easy to check that x * is a solution of (1.1).From (3.11) and (3.12) it follows that   Proof.Assume, to the contrary, that there exists a solution x(t) of (1.1) such that x(t) > 0 on [t 1 , ∞) for some t 1 ≥ t 0 .Integrating (1.1) on [t 0 , t], we have where  In particular, in both cases, (1.1) has no positive non-monotone solution.
Proof.Since 1/P(t) is decreasing, we have 1/P(t) is bounded from above and according to (1.3), we obtain: About the method of lower and upper solutions method in the second-order differential equations we refer reader to [8].The next principle gives the relation between the well-ordered lower and upper solutions with the reverse-ordered first derivatives.As a consequence we easily derive the following criterion for non-monotonicity of solutions.
Proof of Corollary 5.3.From assumption that α(t) and β(t) are two non-monotone functions, there exist two sequences s n and t n , s n → ∞ and t n → ∞ as t → ∞, and n 0 ∈ N such that α (s n ) < 0 and β (t n ) > 0, n ≥ n 0 .Now, taking into account the conclusion (5.5), from previous we derive that x (s n ) ≤ α (s n ) < 0 and x (t n ) ≥ β (t n ) > 0, n ≥ n 0 .
It verifies that x (t) is a sign-changing function, that is, x(t) is a non-monotone function on [t 0 , ∞).
It is easy to check that, for all a, b ∈ R such that a > 1 and a + b > 2, the coefficients p(t) = t a and q(t) = t −b , t ≥ t 0 > 0, satisfy both conditions (2.2) and (2.3).
which implies that F is well defined on Y and maps Y into itself.Here and hereafter, C[t 2 , ∞) is regarded as the Fréchet space of all continuous functions on [t 2 , ∞) with the topology of uniform convergence on every compact subinterval of [t 2 , ∞).Lebesgue's dominated convergence theorem shows that F is continuous on Y. Now we claim that F (Y) is relatively compact.We note that F (Y) is uniformly bounded on every compact subinterval of [t 2 , ∞), because of F (Y) ⊂ Y.By the Ascoli-Arzelà theorem, it suffices to verify that F (Y) is equicontinuous on every compact subinterval of [t 2 , ∞). ≥ t 2 .Let I be an arbitrary compact subinterval of [t 2 , ∞).Then we see that {(F y) (t) : y ∈ Y} is uniformly bounded on I, because of (3.1) and Remark 3.1.The mean value theorem implies that F (Y) is equicontinuous on I.Now we are ready to apply the Schauder-Tychonoff fixed point theorem to the mapping F .Then there exists a y * ∈ Y such that y * = F y * .Therefore, lim t→∞ y * 1.1) on [t 2 , ∞) and (3.6) implies 1≤ x * (t) ≤ δ 1 + δ 2 + 2, t ≥ t 2 provided lim inf t→∞ θ(t) = lim sup t→∞ θ(t).The proof is complete.Proof of Theorem 3.4.There exist t 1 > t 0 , δ 1 > 0 and δ 2 > 0 such that t→∞ x * (t) < lim sup t→∞ x * (t) < ∞, C[t 3 , ∞) : (δ 1 + 1)P(t) ≤ y(t) ≤ (δ 1 + 3)P 2 (t) for t ≥ t 3 }.
Therefore, F is well defined on Y and maps Y into itself.By the same argument as in the proof of Theorem 3.2, we can conclude that F is continuous on Y and F (Y) is relatively compact.By applying the Schauder-Tychonoff fixed point theorem to the mapping F , there exists a y * ∈ Y such that y * = F y * .By L'Hospital's rule, we observe that and (3.14)imply that (F y)(t) ≤ (δ 1 + 1)P(t) + P 2 (t)≤ (δ 1 + 2)P 2 (t) + P 2 (t) = (δ 1 + 3)P 2 (t), t ≥ t 3 .
), t ≥ t 3 (t) is a sign-changing function, and thus, x * (t) is a non-monotone positive solution of (1.1).The proof is complete. *