On Singular Solutions of a Second Order Differential Equation

Sufficient conditions are given under which all nontrivial
solutions of (g(a(t)y'))' + r(t)f(y) = 0 are proper where a >
0,r > 0, f(x)x>0 , g(x)x>0 for x is different from zero and g
is increasing.A sufficient conditions for the existence of a
noncontinuable solution is given.

It is clear that (1) is equivalent to the system y 1 = y, y 2 = g(a(t)y ), where g −1 is the inverse function to g.Hence, as the right-hand sides of (4) are continuous, the Cauchy problem for (1) has a solution.
Definition.Let y be a continuous function defined on [0, τ ) ⊂ R + .Then y is called oscillatory if there exists a sequence {t k } ∞ k=1 , t k ∈ [0, τ ), k = 1, 2, . . . of zeros of y such that lim k→∞ t k = τ and y is nontrivial in any left neighbourhood of τ .
Definition.A solution y of (1) is called proper if it is defined on R + and sup τ ≤t<∞ |y(t)| > 0 for every τ ∈ (0, ∞).It is called singular of the first kind if it is defined on R + , there exists τ ∈ (0, ∞) such that y ≡ 0 on [τ, ∞) and sup T ≤t<τ |y(t)| > 0 for every T ∈ [0, τ ).It is called singular of the second kind if it is defined on [0, τ ), τ < ∞, and cannot be defined at t = τ .A singular solution y is called oscillatory if it is an oscillatory function on [0, τ ).
In the sequel we will investigate only solutions that are defined either on R + or on [0, τ ), τ < ∞ and cannot be defined at = τ .Consider the equation with p-Laplacian where p > 0, A ∈ C 0 (R + ) and A > 0 on R + .This is a special case of (1) with g(z) = |z| p−1 z and a = A 1 p .It is widely studied now; see e.g.[3], [4], [8] and the references therein.
Recall the following sufficient conditions for the nonexistence of singular solutions of (5).
then there exists no singular solution of the first kind of (5).
then there exists no singular solution of the second kind of (5).(iii) Let the function A 1 p r be locally absolutely continuous on R + .Then every solution of (5) is proper.
Theorem A (iii) shows that if A and r are smooth enough, singular solutions do not exist.But the following theorem shows that singular solutions may exist.Theorem B ([3] Theorem 4).Let 0 < λ < p (0 < p < λ).Then there exists a positive continuous function r defined on R + such that the equation has a singular solution of the first (of the second) kind.
Note that the proof of Theorem B uses ideas from [5] and [6] for the case p = 1.
The goal of this paper is to generalize results of Theorems A and B to Eq. (1).EJQTDE, 2006 No. 8, p. 3 We begin our investigations with simple properties of singular solutions.
Lemma 1.Let y be a singular solution of (1) and τ be the number in its definition.Then y is oscillatory if and only if y is an oscillatory function on [0, τ ).
Proof.It follows directly from system (4) since, due to (2), y is an oscillatory function on [0, τ ) if and only if y 2 = g(a(t)y ) is oscillatory on the same interval.
Theorem 1. (i) Every singular solution of the first kind of (1) is oscillatory.(ii) If (3) holds, then every singular solution of the second kind of (1) is oscillatory.
Proof.(i) Let y be a singular solution of the first kind of (1) and τ < ∞ be the number from its definition.Suppose, contrarily, that y > 0 in a left neighborhood of τ (the case y < 0 can be studied similarly).Then (1) and (2) yield g(ay ) is decreasing and hence, ay is decreasing on I. From this and from Remark 1 (iii), we have y (τ ) = 0 and hence y > 0 on I; this contradicts the fact that y > 0 on I and y(τ ) = 0. (ii) Let y be a singular solution of the second kind of (1) defined on [0, τ ), τ < ∞.Suppose, contrarily, that y > 0 in a left neighbourhood I = [τ 1 , τ ) of τ (the case y < 0 can be studied similarly).Then (1) and ( 2) yield ay is decreasing on I and according to Remark 1 (ii) and Lemma 1 lim t→τ − y (t) = −∞.Hence y is positive and decreasing in a left neighbourhood of τ and rf (y) is bounded on I. From this, we have This contradiction proves the statement.
The following example shows that singular solutions of the second kind may be nonoscillatory if (3) does not hold.The first result for the nonexistence of singular solutions follows from more common results of Mirzov [8] that are specified for (1).
has the trivial solution on [t * , ∞) only, then (1) has no singular solution of the first kind.
(ii) If for every c 1 ≥ 0 and c 2 ≥ 0 the Cauchy problem has the upper solution defined on R + , then (1) has no singular solution of the second kind.
Proof.This follows from [8, Theorems 1.1 and 1.2 and Remark 1.1] setting , and let f be nondecreasing on R + .(i) If there exists a continuous function R(t) and a right neighbourhood for t ∈ R + and for z ∈ I, then (1) has no singular solution of the first kind.
(ii) For any c > 0 let there exist a continuous function R 1 (c, t) and a neighbourhood Then there exists no singular solution of the second kind of (1).
Proof.In our case, EJQTDE, 2006 No. 8, p. 5 (i) It is clear that (7) can be studied only for |z| ∈ I. Then t ∈ R + and z ∈ I. From this and from (9), Eq. ( 7) is sublinear in I, the trivial solution z ≡ 0 is unique, and the statement follows from Th. 2 (i).
(ii) We have 0 . From this and from (9), Eq. ( 8) is sublinear for large values of z, (8) has the upper solution defined on R + , and the statement follows from Theorem 2 (ii).r(s)ds The remainder of the proof is similar to that of Cor. 1 (i).
(ii) Similarly, M and d 2 (z) ≤ M 1 |z| p for |z| ≥ z 0 , and so From this, equation ( 8) is sublinear for large |z|, the problem (8) has the upper solution defined on R + , and the statement follows from Theorem 2 (ii).
where r 0 and r 1 are nonnegative, nondecreasing and continuous functions.
Moreover, y is not singular of the first kind, and if (3) holds, then y is proper.
The contradiction proves that y is not singular of the second kind and, according to Remark 1 (i), it is proper.
Theorem 4. Let the assumptions of Theorem 3 be valid and let Then for 0 ≤ s < t < b we have Proof.The proof is similar to that of Theorem 3 since Remark 3. Inequalities ( 12) and ( 15) are proved in [7] for Equation ( 5) with p = 1 and a ≡ 1, in [3] for g(z) = |z| p−1 z with p > 0, and in [8] for Equation (6).EJQTDE, 2006 No. 8, p. 8 Corollary 3. Let ar be locally absolute continuous on R + .Let ρ and ρ 1 be given by (11) and (14), respectively.(i) If ar is nondecreasing on R + , then for an arbitrary solution y of (1), ρ is nondecreasing and ρ 1 is nonicreasing on R + .(ii) If ar is nonincreasing on R + , then for an arbitrary solution y of (1), ρ is nonincreasing and ρ 1 is nondecreasing on R + .
In [1] there is an example of Eq. ( 1) with a ≡ 1, g(z) ≡ z, f (z) = |z| λ sgn z and 0 < λ < 1 for which there exists a proper solution y with infinitely many accumulation points of zeros.The following corollary gives a sufficient condition under which every solution of (1) has no accumulation point of zeros in its interval of definition.
Corollary 4. If ar is locally absolute continuous on R + , then every nontrivial solution y of (1) has no accumulation point of its zeros and has no double zero in its interval of definition.
Proof.Let τ be an accumulation point of zeros or a double zero of a solution y of (1) lying in its definition interval.Hence, y(τ ) = 0 and y (τ ) = 0.Then, ȳ(t) = y(t) on [0, τ ] and ȳ(t) = 0 for t > τ is a singular solution of the first kind of (1) that contradits Theorem 3. Proof.Let ar be nondecreasing on R + .As all assumptions of Corollary 3 are fulfilled, ρ 1 is nonincreasing and the statement follows from ρ 1 (t k ) = y(t k ) 0 f (σ)dσ and (2).If ar is nonincreasing, the proof is similar.
The following corollary generalizes Theorem B and it shows that singular solutions may exist if ar is not locally absolutely continuous on R + .M if y(t) = 0. From this and from (2), the function r is positive and continuous on R + , and hence y is also a solution of ( 5).If 0 < p < λ, then the proof is similar.
Example 1 shows that the statement of Theorem 3 does not hold if (3) is not valid; singular solutions of the second kind may exist.The following theorem gives sufficient conditions for the existence of such solutions.

EJQTDE, 2006
No. 8, p. 2 Remark 1. (i) According to (2) every nontrivial solution of (1) is either proper, singular of the first kind, or singular of the second kind.(ii) A solution is singular of the second kind if and only if lim t→τ − sup |y (t)| = ∞.(iii) If y is a singular solution of the first kind then y(τ ) = y (τ ) = 0.

EJQTDE, 2006 No. 8 , p. 6 Remark 2 . 1 pTheorem 3 .
Theorem A (i), (ii) is special case of Corollary 2 with g(z) = |z| p−1 z, a = A , and M = 1.The following theorem generalizes Theorem A (iii); sufficient conditions for the nonexistence of singular solutions are posed on the functions a and r only.Let the function ar be locally absolute continuous on R + , y be a nontrivial solution of (1) defined on [0, b), b ≤ ∞, ar = r 0 − r 1 on R + , and

Corollary 5 .
Let ar be locally absolute continuous and nondecreasing (nonincreasing) on R + .Let y be a solution of (1) defined on [0, b), b ≤ ∞, and {t k } N k=1 , N ≤ ∞, be a (finite or infinite) increasing sequence of zeros of y lying in [0, b).Then the sequence of local extrema {|y(t k )|} N k=1 is nonincreasing (nondecreasing).