ON EXISTENCE OF PROPER SOLUTIONS OF QUASILINEAR SECOND ORDER DIFFERENTIAL EQUATIONS

In the paper, the nonlinear differential equation (a(t)|y 0 | p 1 y 0 ) 0 + b(t)g(y 0 ) + r(t)f(y) = e(t) is studied. Sufficient conditions for the nonexistence of singular solutions of the first and second kind are given. Hence, sufficient conditions for all nontrivial solutions to be proper are derived. Sufficient conditions for the nonexistence of weakly oscillatory solutions are given.


Introduction
In this paper, we study the existence of proper solutions of a forced second order nonlinear differential equation of the form (a(t)|y | p−1 y ) + b(t)g(y ) + r(t)f (y) = e(t) where p > 0, a ) and a > 0 on R + .
A special case of Equation ( 1) is the unforced equation (a(t)|y | p−1 y ) + b(t)g(y ) + r(t)f (y) = 0. (2) We will often use of the following assumptions and g(x)x ≥ 0 on R + .
Note, that a singular solution y of the 2-nd kind is sometimes called noncontinuable.
Definition 2. A proper solution y of (1) is called oscillatory if there exists a sequence of its zeros tending to ∞.Otherwise, it is called nonoscillatory.A nonoscillatory solution y of (1) is called weakly oscillatory if there exists a sequence of zeros of y tending to ∞.
It is easy to see that (1) can be transformed into the system the relation between a solution y of (1) and a solution of ( 5) is y An important problem is the existence of solutions defined on R + or of proper solutions (for Equation (2)).Their asymptotic behaviour is studied by many authors (see e.g.monographs [7], [9] and [10], and the references therein).So, it is very important to know conditions under the validity of which all solutions of (1) are defined on R + or are proper.For a special type of the equation of ( 2), for the equation sufficient conditions for all nontrivial solutions to be proper are given e.g. in [1], [8], [9] and [10].It is known that for half-linear equations, i.e., if f (x) = |x| p sgn x, all nontrivial solutions of (4) are proper, see e.g.[6].For the forced equation ( 1) with (3) holding, a ∈ C 1 (R + ), ) and b ≡ 0, it is proved in [2] that all solutions are defined on R + , i.e., the set of all singular solutions of the second kind is empty.On the other hand, in [4] and [5] examples are given for which Equation ( 6) has singular solutions of the first and second kinds (see [1], as well).Moreover, Lemma 4 in [3] gives sufficient conditions for the equation (a(t)y ) + r(t)f (y) = 0 to have no proper solutions.
In the present paper, these problems are solved for (1).Sufficient conditions for the nonexistence of singular solutions of the first and second kinds are given, and so, sufficient conditions for all nontrivial solutions of (2) to be proper are given.In the last section, simple asymptotic properties of solutions of (2) are given.
Note that it is known that Equation ( 6) has no weakly oscillatory solutions (see e.g.[10]), but as we will see in Section 4, Equation (1) may have them.
It will be convenient to define the following constants: We define the function R : R + → R by For any solution y of (1), we let and if (3) and r > 0 on R + hold, let us define For any continuous function h : R + → R, we let h In this section, the nonexistence of singular solutions of the second kind will be studied.The following theorem is a generalization of the well-known Wintner's Theorem to (1).
Then there exist no singular solution y of the second kind of (1) and all solutions of (1) are defined on R + .
The following result shows that singular solutions of the second kind of (1) do not exist if r > 0 and R is smooth enough under weakened assumptions on f .Then Equation (1) has no singular solution of the second kind and all solutions of (1) are defined on R + .
Remark 2. It is clear from the proof of Theorem 2 (ii) that if b ≡ 0, then assumption (4) is not needed in case (ii).
Remark 3. Note that the condition |g(x)| ≤ |x| p in (i) can not be improved upon even for Equation (2).
Example 1.Let ε ∈ (0, 1).Then the function y = 1 is a singular solution of the second kind of the equation (i) The result of Theorem 2 is obtained in [2] in case b ≡ 0 using a the similar method.
Remark 5. Theorem 2 is not valid if r < 0 on an interval of positive measure, see e.g.Theorem 11.3 in [9] (for (6) and p = 1).The existence of singular solutions of the second kind for ( 1) is an open problem.
EJQTDE, 2007 No. 5, p. 7 3 Singular solutions of the first kind In this section, the nonexistence of singular solutions of the first kind mainly for (2) will be studied.The following lemma shows that e(t) has to be trivial in a neighbourhood of ∞ if Equation (1) has a singular solution of the first kind.
Lemma 1.Let y be a singular solution of the first kind of (1).Then e(t) ≡ 0 in a neighbourhood ∞.
In what follows, we will only consider Equation (2).( Then there exist no singular solution of the first kind of Equation (2).
Proof.Assume that y is a singular solution of the first kind and τ is the number from Definition 1.Using system (5), we have From the definition of τ , ( 22), ( 24) and (25), we have An integration of the first equality in ( 5) and (25) yield EJQTDE, 2007 No. 5, p. 8 and from (24) we obtain Similarly, an integration of the second equality in ( 5) and ( 21  Then Equation (2) has no singular solution of the first kind.