ON NONCONTINUABLE SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH DELAY

Sufficient conditions for the existence of noncontinuable
solutions of n-th order differential equation with delay are
given.


Introduction
In the paper we study the problem of continuability of solutions of the n-th order differential equation with delays of the form y (n) = f t, y(τ 0 ), . . ., y (n−1) (τ n−1 ) where n ≥ 2, f is a continuous function defined on τ i (t) ≤ t for t ∈ R + and i = 0, 1, . . ., n − 1 .
We will suppose for the simplicity that inf t∈R + τ i (t) > −∞ for i = 0, 1, . . ., n − 1. Note, that C s (I), s ∈ {0, 1, . . .}, I ⊂ R + is the set of continuous functions on I that have continuous derivatives up to the order s.
We will study solutions on their maximal interval of existence to the right.Definition 1.2.Let y be defined on [T 1 , T ) ⊂ [0, ∞).Then y is called noncontinuable if T < ∞ and sup T 1 ≤t<T |y (n−1) (t)| = ∞ (i.e. if y cannot be defined at t = T ).A solution y is defined to be continuable if T = ∞.Definition 1.3.Let y be a noncontinuable solution of (1) on [T 1 , T ).y is called oscillatory if there exists a sequence of its zeros tending to T .Otherwise, y is called nonoscillatory.
In the two last decades the existence and properties of noncontinuable solutions are investigated mainly for equation (2).It is important to study the existence/nonexistence of such solutions.Properties and different types of continuable solutions of (2) are studied intensively now; it is necessary to know if solutions are continuable or not.Furthermore, noncontinuable solutions appear e.g. in water flow problems in one space dimension, see e.g.[14] (flood waves, a flow in sewerage systems).
The authors of the papers [8,9,10,11,27] stated sets of Cauchy initial conditions for which solutions are noncontinuable; the results are used mainly for solving the nonlinear limit-circle/limit-point problem.
On the other hand, in [10,12,26] a set of initial conditions is described for which solutions are continuable even in the superlinear case.
In the last decade, the problem of the existence of noncontinuable solutions with prescribed asymptotics on the right-hand side point T of the definition interval is studied.More precisely, let T ∈ (0, ∞).In [1,2,4,5,6], necessary and sufficient conditions for the existence of a solution y satisfying boundary value problem (for l = −1, C i and the first equality is missing); the solution y is defined in a left neighbourhood of T .It has to be stressed that the first authors who studied the problem (5) were Jaroš and Kusano [22] (for n = 2).
As concerns equation (1), see the monography [20].In [12] the continuability of solutions of a functional-differential system is investigated; note that (1) cannot be transformed into this system.
The goal of the paper is to generalize some results given above to equation (1).The following example shows that noncontinuable solutions exist.
Example 1.2.Consider the equation is the noncontinuable solution of the above given equation.

Nonexistence Results
In this section nonexistence results for noncontinuable solutions will be given.The first theorem is very simple but important.
Proof.Let, contrarily, y be a noncontinuable solution of (1) defined on [T 1 , T ), T > T 1 .If we define τ = max 0≤i<n τ i (T ), the assumptions of the theorem yield τ < T .Let 0 < ε < T − T 1 and τ = min . ., n − 1 and t ∈ J, and due to τ < T a constant N exists such that Note, that y is a solution of the equation z From this and from ( 6), y can be defined at t = T ; that contradicts the noncontinuability of y.
The following lemma is very useful.
Lemma 2.1 ([26] Lemma 2.1).Let λ > 1, K > 0, Q be a continuous nonnegative function on R + , and u be continuous and nonnegative on R + satisfying The next theorem is devoted to the sublinearity of f .
Then every solution y of (1) is continuable.
The following theorem describes a set of Cauchy initial conditions whose corresponding solutions are defined on R + , even in the superlinear case.
Then there exists ε > 0 such that a solution y of (1) defined in a right neighbourhood of t = 0 with Cauchy initial conditions is continuable.
Remark 2.1.The method of the proof of Theorem 2.3 was used in [12] for a different type of the differential equation.
Next, we look for sufficient conditions under which such solutions do not exist for Equation (1).So we study the nonexistence of noncontinuable solutions of (1) defined in a left neighbourhood of T ∈ (0, ∞) and satisfying either lim or l ∈ {0, 1, . . ., n − 2}, C i > 0 for i = 0, 1, . . ., l, Needed results for equations (2) and and a comparison theorem will be given in the following two lemmas.Proof.It follows from Theorem 2 in [4] (Theorems 3.1 and 3.2 in [6] or [3]) in case (i) (case (ii)).
Proof.For the sake of contradiction, let y be a solution of ( 1) with ( 33) and (24).Let us consider Equation ( 25) with (30).We prove similarly as in the proof of Theorem 2.4 that z (i) (t) ≥ y (i) (t), i = 0, 1, . . ., n−1, on the intersection of the definition intervals of y and z.Note, that z (i) and y (i) are nonnegative and nondecreasing for i = 0, 1, . . ., n−1.
Let k > l.We apply Lemma 2.2 (ii) with T = T 1 and l = l 0 .Then according to the definition of λ we have either l 0 < k and 1 Hence, the assumptions of Lemma 2.2(ii) hold and there exists no solution satisfying (36) and (37).Note, that k < n−1 2 in (32) is necessary for the second assumptions in (32) to be nonempty.
Example 1.2 shows that a solution satisfying (23) exists.Similarly, according to the following example, solutions satisfying (24) exist.

Existence Theorems
In this section existence results will be derived.We need the following main lemma that investigates properties of the equation on (−∞, T ) where s = n−kλ λ−1 , and Proof.Note, that τ (T ) = T and τ is the delay in (−∞, T ].The fact that (39) is a solution of (38) can be obtained by the direct computation.This solution is defined if and only if M ∈ R; i.e., if and only if s Here σ i and σi are numbers from Definition 1.1, applied on (38) and (1), respectively.
Proof.It can be proved by the same way as well known theorems on differential inequalities without delay, see e.g.[21].
The next theorem addresses a boundary value problem and provides sufficient conditions for the existence of noncontinuable solution of (1) in a left neighbourhood of a given number T > 0. Let z be a solution of (38), defined on [T 0 , T ) ⊂ [T 0 , T ) satisfying the initial conditions φi and t = T 1 for i = k and τ (T 0 ) ≤ t ≤ T 0 in case i = k.From this, (40), (46), from Lemma 3.2 (applied on (38) and on (38) with λ = λ + ε) and Theorem 2.1, it follows that z Let y be a solution of (1) defined on [T 1 , T ) satisfying (43); note that due to (40) and Theorem 2.1, y is defined on [T 1 , T ).Then Lemma 3.2, (42), ( 44) and (47) imply y (j) (t) ≥ z (j) (t), t ∈ [T 0 , T ) and y is noncontinuable as z has this property.
The next theorem solves the problem of existence of noncontinuable solutions for the initial problem.and (42) holds for T 1 ≤ t ≤ T , x i ≥ M, i = 0, 1, . . ., n − 1.Then there exists δ > 0 such that every solution y on (1) with the initial conditions is defined only on a finite interval [0, t 0 ); i.e., y is noncontinuable on [0, t 0 ) (t 0 depends on y).
Proof.If y is noncontinuable on [0, t 0 ) ⊂ [0, T ) then the statement is valid.Let y be defined on [0, T ], then we can prove by the same way as in the proof of Theorem 3.1 that y is noncontinuable; it is a contradiction.Note, that, as y (i) is nondecreasing, (48) implies y (i) > δ in a left neighbourhood of T 1 given by the proof of Theorem 3.1.Moreover, assumption (40) is not supposed as we only need to show that t 0 exists.