Effect of Nonlinear Perturbations on Second Order Linear Nonoscillatory Differential Equations

The aim of this paper is to show that any second order nonoscil-latory linear differential equation can be converted into an oscillating system by applying a " sufficiently large " nonlinear perturbation. This can be achieved through a detailed analysis of possible nonoscillatory solutions of the perturbed differential equation which may exist when the perturbation is " sufficiently small ". As a consequence the class of oscillation-generating perturbations is determined precisely with respect to the original nonoscillatory linear equation .


Introduction
Consider the perturbed second order linear differential equation of the form (A) (p(t)x ′ ) ′ + q(t)x + Q(t)|x| γ sgn x = 0, where γ is a positive constant with γ = 1, and p(t), q(t) and Q(t) are continuous functions on [a, ∞), a ≧ 0, such that p(t) > 0 and Q(t) ≧ 0 for t ≧ a. Equation (A) is called superlinear or sublinear according as γ > 1 or γ < 1.
Assume that the second order linear differential equation (B) (p(t)x ′ ) ′ + q(t)x = 0 is nonoscillatory, that is, all of its nontrivial solutions are nonoscillatory.It is natural to expect that (A) inherits the nonoscillatory character from (B) as long as the perturbation Q(t)|x| γ sgn x remains "small", and that application of a "sufficiently large"perturbation might turn (B) into an oscillating system (A), which means the oscillation of all of its solutions.The objective of this paper is to verify the truth of this expectation by presenting the results on nonoscillation and oscillation of (A) implying, respectivley, preservation of nonoscillation and generation of oscillation of (B), which are determined by the convergence or divergence of the integrals where {u(t), v(t)} is a fundamental system of solutions of (B) consisting of a (uniquely determined) principal solution u(t) and a non-principal solution v(t).
We recall that (Hartman [4]) u(t) and v(t) satisfy and it holds that (1.3) v(t) ∼ cu(t) t t 0 ds p(s)u(s) 2 for some constant c > 0, where the symbol ∼ denotes the asymptotic equivalence: In Section 2, in order to gain useful information about the structure of nonoscillatory solutions of (A), we classify the set of nonoscillatory solutions of (A) into three types according to their asymptotic behavior at infinity.Next, in Section 3 we give sharp conditions for the existence of solutions belonging to these types.Some examples illustrating the nonoscllation theorems obtained are also presented.Finally, in Section 4 we consider the case where the function Q(t) in equation (A) is of alternating sign and show that the results of Section 3 can be extended to this case.Moreover, we establish sharp oscillation criteria for (A) on the basis of oscillation and nonoscillation results due to Belohorec [2] and Kiguradze [6].

Classification of positive solutions
In this section we classify the set of all nonoscillatory solutions of (A) according to their asymptotic behavior as t → ∞.It suffices to restrict our attention to the totality of eventually positive solutions of (A) since if x(t) is a solution of (A), then so is −x(t).
The subsequent development is based on the fact that the linear differential operator in equation (B) can be represented as in terms of the principal solution u(t) of (B) which may be assumed to be positive on [t 0 , ∞), t 0 ≧ a, so that the analysis of the equation (A) is reduced to that of the equation In dealing with (2.2) a crucial role is played by the function Let x(t) be a positive solution of (A) on [t 0 , ∞), and put x(t) = u(t)y(t).Then, y(t) is a positive solution of (2.2) on [t 0 , ∞).It follows from (2.2) that (p(t)u(t) 2 y ′ (t)) ′ ≦ 0, t ≧ t 0 , which shows that p(t)u(t) 2 y ′ (t) is nonincreasing for t ≧ t 0 .We claim that y ′ (t) > 0, t ≧ t 0 .In fact, if y ′ (t 1 ) < 0 for some t 1 > t 0 , then we have Dividing the above inequality by p(t)u(t) 2 and integrating it from t 1 to t, we obtain from which, letting t → ∞, we find that y(t) → −∞ as t → ∞, a contradiction to the assumed positivity of y(t).Therefore, we must have y ′ (t) > 0 for t ≧ t 0 , so that y(t) is increasing on [t 0 , ∞).Integrating the inequality we get y(t) − y(t 0 ) ≦ c t t 0 ds p(s)u(s) 2 = c[P (t) − P (t 0 )], t ≧ t 0 , from which it follows that there exist positive constants c 1 and c 2 such that (2.4) c 1 ≦ y(t) ≦ c 2 P (t) for all sufficiently large t.
Multiplying (2.4) by u(t) and taking (1.3) into account, we obtain the following result.
Lemma 2.1.Let x(t) be a positive solution of (A) on [t 0 , ∞).Then there exist positive constants c 1 and c 2 such that for all sufficiently large t.
On the basis of this lemma and in view of (1.1) it is convenient to classify the set of all positive solutions x(t) of (A) into the following three types according to their asymptotic behavior as t → ∞: x(t) ∼ cu(t) for some c > 0; (II) where the symbol ≺ is used to mean EJQTDE, 2010 No. 34, p. 3

Existence of positive solutions
A natural question arises: Is it possible to obtain conditions under which equation (A) possesses a positive solution of the types (I), (II) or (III) defined at the end of Section 2? Our purpose here is to show that an almost complete answer to this question can be provided.
Theorem 3.1.(i) Equation (A) possesses a positive solution x(t) such that x(t) ∼ cu(t) for some c > 0 if and only if (ii) Equation (A) possesses a positive solution x(t) such that x(t) ∼ cv(t) for some c > 0 if and only if Proof: Let x(t) be a positive solution of (A) on [t 0 , ∞) and put y(t) = x(t)/u(t).Then, y(t) > 0 is a solution of (2.2) for t ≧ t 0 , and in view of (2.4) y(t) falls into one of the following three types: for some c > 0, which naturally correspond, respectively, to the types (I), (II) and (III) for the solutions x(t) of (A).Integrating (2.2) on [t, ∞), we have Put c = lim t→∞ y(t).Then, there are two possibilities: either c = ∞ or c ∈ (0, ∞).If c = ∞, then (3.5) will be used as an integral equation for y(t) of the type (II').If c ∈ (0, ∞), then integration of (3.3) from t to ∞ yields which will be used as an integral equation for y(t) of the type (I').
Proof of Statement (i).Let x(t) be a positive solution of (A) on [t 0 , ∞) such that x(t) ∼ cu(t) for some c > 0.Then, y(t) = x(t)/u(t) is a solution of (2.2) such that y(t) ∼ c, so that y(t) satisfies (3.6) for t ≧ t 0 .This means in particular that implying the validity of (3.1).Therefore, (3.1) is a necessary condition for (A) to possess a solution of the type (I).
Assume now that (3.1) is satisfied.Let c > 0 be any fixed constant and choose T ≧ t 0 sufficiently large so that Define the set Y by which is a closed convex subset of the locally convex space C[T, ∞) of continuous functions on [T, ∞) equipped with the topology of uniform convergence on compact subintervals of [T, ∞).Consider the integral operator F defined by This shows that y ∈ Y implies F y ∈ Y , and hence Then, it can be shown with the help of the Lebesgue dominated convergence theorem that F y n (t) → F y 0 (t) on any compact subinterval of [T, ∞), which means that F y n → F y 0 in C[T, ∞).This implies that F is a continuous mapping.It is clear that the set F (Y ) is uniformly bounded on [T, ∞).Differentiating (3.9), we see that Proof of Statement (ii).Let x(t) be a positive solution of (A) such that x(t) ∼ cv(t) for some c > 0. Put y(t) = x(t)/u(t).Then, it is a solution of (2.2) such that y(t) ∼ cP (t), and so y(t) satisfies (3.4), which means in particular that Using y(t) ∼ cv(t) in the above inequality, we obtain Assume that (3.2) is satisfied.Let c > 0 be any fixed constant and choose T ≧ t 0 so that Define the integral operator G and the set Z by (3.12) Then it can be verified without difficulty that G is a self-map on Z and sends Z into a relatively compact subset of C[T, ∞).Therefore, by the Schauder-Tychonoff fixed point theorem there exists a fixed point z ∈ Z of G, which satisfies from which it readily follows that z(t) is a solution of the differential equation (2.2) such that z(t) ∼ c ′ P (t) for some c ′ ∈ [c, 2c].The function x(t) = u(t)z(t) then gives a solution of equation (A) such that x(t) ∼ c ′′ v(t) for some c ′′ > 0. This completes the proof of Theorem

It remains to study the existence or nonexistence of type (II)-solutions of equation (A).
In the case where (A) is sublinear a necessary and sufficient condition for the existence of such solutions can be given as the following theorem shows.Theorem 3.2.Let 0 < γ < 1. Equation (A) possesses a positive solution x(t) such that u(t) ≺ x(t) ≺ v(t) if and only if (3.2) holds and Proof: (The "only if"part) Suppose that (A) has a solution x(t) such that u(t) ≺ x(t) ≺ v(t) as t → ∞.Put y(t) = x(t)/u(t).Then, y(t) is a solution of (2.2) such that 1 ≺ y(t) ≺ P (t), and it satisfies (3.5) which can be expressed as (3.15) Note that the following two inequalities follow from (3.15): Integrating the above from t 0 to t, we have which clearly implies (3.2).Thus, (3.2) is a necessary condition for the sublinear equation (A) to possess a solution of the type (II).To prove the necessity of (3.14) we proceed as follows.Suppose to the contrary that In view of (3.3) (with c ′ = 0) and the decreasing nature of p(t)u(t) 2 y ′ (t), we obtain This is a contradiction, and so (3.14) must be satisfied.Hence (3.14) is also necessary for (A) to have a positive solution of the type (II) (We note that the above argument is an extended adaptation of the one given in [8]).
(The "if"part) Assume that (3.2) and (3.14) hold.Let c > 0 be any fixed constant and choose T ≧ t 0 such that and define the integral operator H by (3.23) It can be verified routinely that H maps W continuously into a relatively compact subset of W .By the Schauder-Tychonoff theorem H has a fixed point w in W . From the integral equation w = Hw we see that w(t) is a positive solution of (2.2) on [T, ∞) and satisfies EJQTDE, 2010 No. 34, p. 8 lim t→∞ p(t)u(t) 2 w ′ (t) = 0, which implies that lim t→∞ w(t)/P (t) = 0. On the other hand, using (3.16), we have which, combined with (3.14), implies that lim t→∞ w(t) = ∞.Accordingly, the solution w(t) of equation (2.2) satisfies 1 ≺ w(t) ≺ P (t), and so the function x(t) = u(t)w(t) is a solution of (A) satisfying u(t) ≺ x(t) ≺ v(t).This completes the proof of Theorem 3.2.
Remark 3.2.The problem of characterizing the existence of a solution of type (II) for the superlinear case (γ > 1) of (A) seems to be difficult.In this case we conjecture that (A) has such a solution if and only if (3.1) holds and but we have so far been able to prove the "only if"part of the conjecture.
Proof: Let x(t) be a solution of (A) such that u(t) ≺ x(t) ≺ v(t).Then, y(t) = x(t)/u(t) is a solution of (2.2) such that 1 ≺ y(t) ≺ P (t), and it satisfies (3.16).Let Ψ(t) denote the right hand side of (3.16): Then, it satisfies This implies that ).The truth of (3.24) is proved by reductio ad absurdum.Assume that (3.24) fails to holds, that is, Then, from (3.3) with c ′ = 0 combined with (3.26) and (3.27) we see that Accordingly, we have which is a contradiction.This shows that (3.24) must be satisfied.
We present examples illustrating Theorems 3.1 and 3.It is known (Hille [5], Swanson [10]) that equation (3.37) has a linearly independent positive solutions From Theorem 3.1 applied to the nonlinear perturbation of (3.37) and that (3.41) has a solution x(t) such that x(t) ∼ cL n−1 (t) 1 2 log n t for some c > 0 if and only if Applying Theorem 3.2, we see that the sublinear equation (3.41) with 0 < γ < 1 possesses a solution x(t) such that L n−1 (t) For the equation (3.45) wihch is a special case of (3.41) with n = 1 and Q(t) = 1/t Therefore, (3.45) possesses a solution x(t) such that t Remark 3.3.We notice that x(t) = t 1 2 log 2 t is also a solution of (3.45) in the case γ > 1.This example suggests that the conjecture stated in Remark 3.2 would be true because in this case EJQTDE, 2010 No. 34, p. 12

Oscillation criteria and remarks
We have assumed so far that the function Q(t) in equation (A) is nonnegative.A question naturally arises: To what extent does a perturbation Q(t)|x| γ sgn x with Q(t) of alternating sign affect the oscillatory behavior of equation (B)?The aim of this section is to give an affirmative answer to this question by showing that necessary and sufficient conditions for all nontrivial solutions of (A) to be oscillatory can be established to the case of sign-changing Q(t) provided the positive part Q + (t) = 1 2 (|Q(t)| + Q(t)) of Q(t) is much larger than its negative part Q − (t) = The above results are due to Kiguradze [6] and Belohorec [2] (see also [7], Theorems 18.1 and 18.2) and give rise to the following proposition.Then, for all nontrivial solutions of (C) to be oscillatory, it is necessary and sufficient that the condition (4.1) (respectively (4.2)) be fulfilled.
Tychonoff fixed point theorem are fulfilled for F , and we conclude that there exists y ∈ Y such that y = F y, which is the integral equation (3.6) with t 0 replaced by T .Differentiating this integral equation, we see that y = y(t) is a solution of equation (B) of the type (I') such that y(t) ∼ c as t → ∞.The function x(t) = u(t)y(t) provides a solution of (A) on [T, ∞) such that x(t) ∼ cu(t) as t → ∞.It follows that (3.1) is a sufficient condition for the existence of a type (I)-solution of equation (A).This completes the proof of the first statement of Theorem 3.1.