Existence of Oscillatory Solutions of Singular Nonlinear Differential Equations

and Applied Analysis 3 Consequently, the condition u′ 0 0 1.11 is necessary for each solution of 1.7 . Denote usup sup{u t : t ∈ 0,∞ }. 1.12 Definition 1.2. Let u be a solution of 1.7 . If usup < L, then u is called a damped solution. If a solution u of 1.7 satisfies usup L or usup > L, then we call u a bounding homoclinic solution or an escape solution. These three types of solutions have been investigated in 15–18 . Here, we continue the investigation of the existence and asymptotic properties of damped solutions. Due to 1.11 and Definition 1.2, it is reasonable to study solutions of 1.7 satisfying the initial conditions u 0 u0 ∈ L0, L , u′ 0 0. 1.13 Note that if u0 > L, then a solution u of the problem 1.7 , 1.13 satisfies usup > L, and consequently u is not a damped solution. Assume that L0 > −∞, then f L0 0, and if we put u0 L0, a solution u of 1.7 , 1.13 is a constant function equal to L0 on 0,∞ . Since we impose no sign assumption on f x for x < L0, we do not consider the case u0 < L0. In fact, the choice of u0 between two zeros L0 and 0 of f has been motivated by some hydrodynamical model in 11 . A lot of papers are devoted to oscillatory solutions of nonlinear differential equations. Wong 19 published an account on a nonlinear oscillation problem originated from earlier works of Atkinson and Nehari. Wong’s paper is concerned with the study of oscillatory behaviour of second-order Emden-Fowler equations y′′ x a x ∣∣y x ∣∣γ−1y x 0, γ > 0, 1.14 where a is nonnegative and absolutely continuous on 0,∞ . Both superlinear case γ > 1 and sublinear case γ ∈ 0, 1 are discussed, and conditions for the function a giving oscillatory or nonoscillatory solutions of 1.14 are presented; see also 20 . Further extensions of these results have been proved for more general differential equations. For example, Wong and Agarwal 21 or Li 22 worked with the equation ( a t ( y′ t )σ)′ q t f(y t ) 0, 1.15 where σ > 0 is a positive quotient of odd integers, a ∈ C1 is positive, q ∈ C , f ∈ C1 , xf x > 0, f ′ x ≥ 0 for all x/ 0. Kulenović and Ljubović 23 investigated an equation ( r t g ( y′ t ))′ p t f(y t ) 0, 1.16 4 Abstract and Applied Analysis where g u /u ≤ m, f u /u ≥ k > 0, or f ′ u ≥ k for all u/ 0. The investigation of oscillatory and nonoscillatory solutions has been also realized in the class of quasilinear equations. We refer to the paper 24 by Ho, dealing with the equation ( tn−1Φp ( u′ ))′ tn−1 N ∑ i 1 αit Φqi u 0, 1.17 where 1 < p < n, αi > 0, βi ≥ −p, qi > p − 1, i 1, . . . ,N, Φp y |y|p−2y. Oscillation results for the equation ( a t Φp ( x′ ))′ b t Φq x 0, 1.18 where a, b ∈ C 0,∞ are positive, can be found in 25 . We can see that the nonlinearity f y |y|γ−1y in 1.14 is an increasing function on having a unique zero at y 0. Nonlinearities in all the other 1.15 – 1.18 have similar globally monotonous behaviour. We want to emphasize that, in contrast to the above papers, the nonlinearity f in our 1.7 needs not be globally monotonous. Moreover, we deal with solutions of 1.7 starting at a singular point t 0, and we provide an interval for starting values u0 giving oscillatory solutions see Theorems 2.3, 2.10, and 2.16 . We specify a behaviour of oscillatory solutions in more details decreasing amplitudes—see Theorems 2.10 and 2.16 , and we show conditions which guarantee that oscillatory solutions converge to 0 Theorem 3.1 . The paper is organized in this manner: Section 2 contains results about existence, uniqueness, and other basic properties of solutions of the problem 1.7 , 1.13 . These results which mainly concern damped solutions are taken from 18 and extended or modified a little. We also provide here new conditions for the existence of oscillatory solutions in Theorem 2.16. Section 3 is devoted to asymptotic properties of oscillatory solutions, and the main result is contained in Theorem 3.1. 2. Solutions of the Initial Problem 1.7 , 1.13 Let us give an account of this section in more details. The main objective of this paper is to characterize asymptotic properties of oscillatory solutions of the problem 1.7 , 1.13 . In order to present more complete results about the solutions, we start this section with the unique solvability of the problem 1.7 , 1.13 on 0,∞ Theorem 2.1 . Having such global solutions, we have proved see papers 15–18 that oscillatory solutions of the problem 1.7 , 1.13 can be found just in the class of damped solutions of this problem. Therefore, we give here one result about the existence of damped solutions Theorem 2.3 . Example 2.5 shows that there are damped solutions which are not oscillatory. Consequently, we bring results about the existence of oscillatory solutions in the class of damped solutions. This can be found in Theorem 2.10, which is an extension of Theorem 3.4 of 18 and in Theorem 2.16, which are new. Theorems 2.10 and 2.16 cover different classes of equations which is illustrated by examples. Abstract and Applied Analysis 5 Theorem 2.1 existence and uniqueness . Assume that 1.2 – 1.5 , 1.8 , 1.9 hold and that there exists CL ∈ 0,∞ such that 0 ≤ f x ≤ CL for x ≥ L 2.1and Applied Analysis 5 Theorem 2.1 existence and uniqueness . Assume that 1.2 – 1.5 , 1.8 , 1.9 hold and that there exists CL ∈ 0,∞ such that 0 ≤ f x ≤ CL for x ≥ L 2.1 then the initial problem 1.7 , 1.13 has a unique solution u. The solution u satisfies u t ≥ u0 if u0 < 0, u t > B if u0 ≥ 0, for t ∈ 0,∞ . 2.2 Proof. Let u0 < 0, then the assertion is contained in Theorem 2.1 of 18 . Now, assume that u0 ∈ 0, L , then the proof of Theorem 2.1 in 18 can be slightly modified. For close existence results, see also Chapters 13 and 14 of 26 , where this kind of equations is studied. Remark 2.2. Clearly, for u0 0 and u0 L, the problem 1.7 , 1.13 has a unique solution u ≡ 0 and u ≡ L, respectively. Since f ∈ Liploc , no solution of the problem 1.7 , 1.13 with u0 < 0 or u0 ∈ 0, L can touch the constant solutions u ≡ 0 and u ≡ L. In particular, assume that C ∈ {0, L}, a > 0, u is a solution of the problem 1.7 , 1.13 with u0 < L, u0 / 0, and 1.2 , 1.8 , and 1.9 hold. If u a C, then u′ a / 0, and if u′ a 0, then u a / C. The next theorem provides an extension of Theorem 2.4 in 18 . Theorem 2.3 existence of damped solutions . Assume that 1.2 – 1.5 , 1.8 , and 1.9 hold, then for each u0 ∈ B, L , the problem 1.7 , 1.13 has a unique solution. This solution is damped. Proof. First, assume that there exists CL > 0 such that f satisfies 2.1 , then, by Theorem 2.1, the problem 1.7 , 1.13 has a unique solution u satisfying 2.2 . Assume that u is not damped, that is, sup{u t : t ∈ 0,∞ } ≥ L. 2.3 By 1.3 – 1.5 , the inequality F u0 ≤ F L holds. Since u fulfils 1.7 , we have u′′ t p′ t p t u′ t f u t for t ∈ 0,∞ . 2.4 Multiplying 2.4 by u′ and integrating between 0 and t > 0, we get