ON SINGULAR SOLUTIONS FOR SECOND ORDER DELAYED DIFFERENTIAL EQUATIONS

Asymptotic properties of singular solutions of the second order
differential equations with delay are investigated.

The relationship between a solution y of (1) and a solution (y 1 , y 2 ) of the system (2) is (3) y 1 (t) = y(t) and y 2 (t) = a(t) y ′ (t) p−1 y ′ (t) , and when discussing a solution y of (1), we will often use (3) without mention.
Definition 2. Let y be a solution of (1) defined on [0, T ), T ≤ ∞. It is called singular of the 1st kind if T = ∞, τ ∈ (0, ∞) exists such that y ≡ 0 on [τ, ∞) and y is nontrivial in any left neighbourhood of τ . Solution y is called singular of the 2nd kind if T < ∞ and put τ = T . It is called proper if T = ∞ and it is nontrivial in any neighbourhood of ∞. Singular solutions of either 1st or 2nd kind are called singular.
Note, that a solution of (1) is either proper, or singular or trivial on (ϕ 0 , ∞). Singular solutions of the second kind are sometimes called noncontinuable. When discussing singular solutions, τ will be the number in Definition 2 in all the paper without mention.

From this and from
Definition 3. Let y be a singular solution of (1) of the 1st kind (of the 2nd kind).
Then it is called oscillatory if there exists a sequence of its zeros tending to τ and it is called nonoscillatory otherwise.
In the monography of Kiguradze and Chanturia [13] it is a good overview of results for p = 1 and a = 1.
Eq. (5) may have singular solutions. Heidel [11] (Coffman, Ulrych [9]) proved the existence of an equation of type (5), a ≡ 1, p = 1 with singular solutions of the 1st kind (of the 2nd kind) in case λ < p (λ > p); in this case r is continuous but not of locally bounded variation. If a and r are smooth enough, then singular solutions of (5) do not exist (see Theorem A below). As concerns to Eq. (1), the existence of singular solutions of the second kind are investigated in [4] in case r ≤ 0. The existence and properties of singular solutions of either the first kind or of the second kind in case r ≥ 0 seem not to be studied at all.
The following theorem sums up results concerning to Eq. (5).
Theorem A. Let r ∈ C 0 (R + ) and r(t) > 0 on R + . (i) If λ ≥ p, then there exists no singular solution of (5) of the 1st kind.
(ii) If λ ≤ p, then there exists no singular solution of (5) of the 2nd kind.
(iii) If a solutions are obtained for differential equations of the third and fourth orders, see also [3]. About uniform estimates of solutions of quasi-linear ordinary differential equations see [2]. In [16] estimates of singular solutions of the second kind of a system of second order differential equations (of the form (5)) are derived.
Theorem B ( [16], Theorem 2). Let r ∈ C 0 (R + ) and r(t) > 0 on R + . Let λ > p, y be a singular solution of (5) of the second kind, T ∈ [0, τ ), τ − T ≤ 1, r 0 = max T ≤s≤τ r(s), C 0 = 2 λ+2 in case p > 1 and C 0 = 2 2λ+1 in case p ≤ 1. Then a positive constant C = C(p, λ, τ, r 0 ) exists such that It is important to study the existence of proper/singular solutions. When studying solutions of (1) and (5), some authors sometimes investigate properties of solutions that are defined on R + only without proving the existence of them. Moreover, sometimes, proper solutions have crucial role in a definition of some problems, see e.g. the limit-point/limit-circle problem in [6], [8]. Furthermore, noncontinuable solutions appear e.g. in water flow problems (flood waves, a flow in sewerage systems), see e.g. [4].
Our goal is to study properties of singular solutions and to extend Theorems A and B to (1).
For convenience, we define the constants and the function If y is a solution of (1), then we set on its interval of existence Notice that F (t) ≥ 0 for every solution of (1) and

Singular solutions of the 2nd kind
The following theorem shows that such solutions do not exist in case λ ≤ p.
Theorem 1. If λ ≤ p, then all solutions of (1) are defined on R + Proof. It is proved in Lemma 7 in [6] for r < 0, for arbitrary r the proof is the same, it is necessary to replace r by |r|.
The following theorem gives us basic properties.
Theorem 2. Let y be a singular solution of (1) of the second kind. Then it is oscillatory and ϕ(τ ) = τ . If, moreover, R ∈ C 1 (R + ), then ϕ(t) ≡ t in any left neighbourhood of τ .
Let y be a singular solution of (1) and ϕ(t) ≡ t on a left neighbourhood J on τ . Then y is a singular solution of (5) on J. A contradiction with Theorem A(iii) proves that ϕ(t) ≡ t in any left neighbourhood of τ .
Remark 2. According to Theorem 1 there exists no singular solution of (1) of the second kind in case ϕ(t) < t on R + ; all solutions are defined on R + . This fact was used by many authors for special types of (1), see e.g. [10], [4] (r < 0).
The following two lemmas serve us for estimate of solutions. Lemma 1. Let ω > 1, t 0 ∈ R + , K > 0, Q be a continuous nonnegative function on [t 0 , ∞) and u be continuous and nonnegative on [t 0 , ∞) satisfying Proof. It is proved in Lemma 2.1 in [14] for m = ω and p = 1.
|y(s)| and Then T = ∞ and y is defined on R + .
Proof. Suppose, contrarily, that y is singular of the 2nd kind. Then T = τ < ∞ and denote by It follows from (2) that Hence, for t 0 ≤ s ≤ t < T we have From this Then (16) and (17) imply (13) and (14), and according to Lemma 1, (15) is valid. As T < ∞, y 2 is bounded on J. A contradiction with (4) proves the statement.
Remark 3. Note that Lemma 2 is valid even if we suppose r ≥ 0 instead of r > 0 on R + .
The next theorem derives an estimate from below of a singular solution of the second kind. Then Especially, a left neighbourhood I of τ exists such that Proof. Let y be a singular solution of (1) of the 2nd kind defined on [0, τ ). Let t ∈ [T, τ ) be fixed. Definē note thatr andā are continuous on R + and are linear on [τ, 2τ −t]. Furthermore, we have Consider an auxilliary equation (22) ā(t)|z ′ | p−1 z ′ +r(t)|z(ϕ)| λ sgn z(ϕ) = 0 .
Then z = y is the singular solution of (22) of the second kind defined on [0, τ ). Suppose that (18) is not valid for t =t, i.e. (23) holds. We apply Lemma 2 and Remark 3 with T = τ and t 0 =t. Then it follows from (20), (21) and (23) that all assumptions of Lemma 2 are valid. Hence, z is defined on R + and the contradiction with z to be singular proves that (18) is valid. Furthermore, a left neighbourhood I of t = τ exists such that and (20) follows from this and from (18).
Remark 5. The used method of the proof of Theorem 2 is due to Pekárková [16] (for ϕ(t) ≡ t).
Corollary 1. Every singular solution of (1) of the second kind is unbounded.
Remark 6. In case ϕ(t) ≡ t, Theorem 3 gives us similar estimate than Theorem B but it can be used also for τ − t > 1.
Proof. Let y be a singular solution of the 2nd kind. Then according to Lemma 2 and Corollary 2 it is oscillatory and unbounded. Hence, an increasing sequence {t k } ∞ k=1 exists such that lim t→∞ t k = τ , y has the local extreme at t k and Then y ′ (t k ) = 0, max ϕ(t k )≤s≤t k |y(s)| = |y(t k )|, and the statement follows from (19).

Singular solution of the 1st kind
This paragraph begins with some basic properties Theorem 4. Let y be a singular solution of (1) of the first kind. Then it is oscillatory and ϕ(τ ) = τ . Moreover, (i) if R ∈ C 1 (R + ), then ϕ(t) ≡ t in any left neighbourhood of τ ; (ii) if R ∈ C 1 (R + ), λ ≥ p and ϕ is nondecreasing in a left neighbourhood J of τ , then a left neighbourhood J 1 of τ exists such that ϕ(t) < t on J 1 .
Proof. Let y be a singular solution of (1) of the first kind. Then y(t) = 0 for t ≥ τ (24) and y(t) ≡ 0 in any left neighbourhood of τ .
As, according to (2), y 2 is decreasing on I 1 and (24) implies y 2 (τ ) = 0 we have y 2 > 0 on I 1 ; hence, y ′ > 0 on I 1 . The contradiction with (27) and (24) proves that y is oscillatory. Case (i). The proof follows from Theorem A(iii) by the same way as in the proof of Theorem 1.
The following result is a consequence of Theorem 2 and Theorem 4.
Lemma 3. Let y be a singular solution of the 1st kind, let T ∈ [0, τ ) be such that EJQTDE, 2012 No. 3, p. 8 Proof. Let y be a singular solution of the 1st kind. Then (9) implies F (s) for t ∈ I. From this and from (7), (8) and (30) for t ∈ I and t ≤ s ≤ τ where C 1 = δKC. Hence, The following theorem gives us an estimate from above of singular solutions of the 1st kind.
Theorem 6. Let y be a singular solution of (1) of the 1st kind and M > 0 be such that ϕ ′ (t) ≤ M in a left neighbourhood S of τ .
(i) Let λ ≥ p and m > 0. Then a positive constant K and a left neighbourhood J of τ exist such that on J.