OSCILLATION OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

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1 Journal of Applied Analysis Vol. 5, No. 2 29), pp OSCILLATION OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS Y. SHOUKAKU Received September, 28 and, in revised form, February 6, 29 Abstract. In this paper we obtain oscillation criteria for solutions of homogeneous and nonhomogeneous cases of second order neutral differential equations with positive and negative coefficients. Our results improve and extend the results of Manojlović, Shoukaku, Tanigawa and Yoshida [6].. Introduction In this paper we consider the oscillation of the second order neutral delay differential equations 2 Mathematics Subject Classification. Primary: 34K4, 34K99, 34C. Key words and phrases. Second order neutral differential equations, oscillation, positive and negative coefficients. ISSN c Heldermann Verlag.

2 282 Y. SHOUKAKU [ rt) + [ xt) ± i H h i t)xα i t))] ] E) p i t)xβ i t)) q i t)xγ i t)) = ft), t >. It is assumed throughout this paper that: H) H, P, Q are bounded starting segments of positive integers i.e., H = {, 2,..., H }, P = {, 2,..., P }, Q = {, 2,..., Q }), P Q ; H2) rt) C[, );, )), ft) C[, );R), h i t) i H), p i t) i P ), q i t) i Q) C[, ); [, )); H3) dt < ; rt) H4) α i t)i R) C[, );R), α i t) >, α it) t, lim t α i t) =, lim inf t α i t) >, β i t) i R) C[, );R), β i t) >, β it) t, lim t β i t) =, γ i t) i R) C[, );R), γ i t) >, γ it) t, lim t γ i t) = ; H5) h i t) h i i H), where h i are nonnegative constants; H6) ft) C[, );R) and there exists a function F t) C 2 [, );R) which satisfies [rt)f t)] = ft), rt)f t) C [, );R) and lim t F t) = F ) is finite; H7) there exists a mapping ψ : Q P and k i t)i P ) C[, ); [, )) such that p i t) ρ jt)q j ρ j t)), i ψq), ψj)=i k i t) = j Q p i t), i P \ ψq), lim inf k i t t) >, lim sup β i t) < t for some i P, where ρ i t) γi β ψi) t)) i Q); H8) γ i t) β ψi) t) i Q), then ρ i t) t. Definition. By a solution of E) we mean a continuous function xt) which is defined for t T, and satisfies sup{ x : t t } > for all t T, where T = min{α, β, γ} and α = inf t> { min i H α it) }, β = inf t> { } { } min β it), γ = inf min γ it). t>

3 OSCILLATION OF SOLUTIONS 283 Definition 2. A solution of E) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory. In recent years, oscillatory and nonoscillatory properties of solutions of neutral differential equations with positive and negative coefficints have been developed extensively see, for example, [3], [6], [7], [8], [9], []). Parhi and Chand [8], Manojlović, Shoukaku, Tanigawa and Yoshida [6], Karpuz, Manojlović, Öcalan and Shoukaku [3], and Weng and Sun [] obtained some oscillatory criteria for equation E) with rt) =. For the case /rs)ds = or rt) = ) and q i t) =, many authors see, for example, [], [2], [4], [5] and []) have investigated oscillatory behavior of solutions of second order neutral equations. In 27, sufficient conditions for oscillation of solutions of the equations E) with /rs)ds = were given by Manojlović, Shoukaku, Tanigawa and Yoshida [7]. However, it seems that no work has been done on oscillation of the equations E) with /rs)ds <. In this paper we derive sufficient conditions for oscillation of the equations E) under the hypothesis H3). In Section 2 we give the homogeneous case of equation E). Finally, Section 3 is concerned with the oscillation results for solutions of the nonhomogeneous equations E). 2. Oscillation of solutions of homogenous equations In this section we obtain the following oscillation criteria for the equations [ [ rt) xt) ± ] h i t)xα i t))] E ± ) i H + p i t)xβ i t)) q i t)xγ i t)) =, t >. Theorem. If s <, ) rs) ρ i s) then every solution of E + ) oscillates or satisfies lim t xt) =. Proof. Suppose that xt) is a nonoscillatory solution of E + ) which does not satisfies lim t xt) =. Without any loss of generality, we assume that xt) > for t t. Letting

4 284 Y. SHOUKAKU zt) = xt)+ i H h i t)xα i t)) for t t, then we observe that [ z t) = xt) + i H h i t)xα i t)) t t rs) ] s ρ i s) q i ξ)xγ i ξ))dξds 2) t q i s)xγ i s))ds. 3) rt) ρ i t) Multiplying 3) by rt) and differentiating both sides, we obtain rt)z t) ) [ [ = rt) xt) + ] ] h i t)xα i t)) i H [ ] t q i s)xγ i s))ds ρ i t) = p i t)xβ i t)) + q i t)xγ i t)) { } q i t)xγ i t)) ρ it)q i ρ i t))xγ i ρ i t))) = p i t)xβ i t)) + ρ it)q i ρ i t))xβ ψi) t)) = p i t)xβ i t)) i ψq) \ψq) = i ψq) \ψq) p i t)xβ i t)) + p it) ψj)=i j Q i ψq) ψj)=i j Q p i t)xβ i t)), t t. ρ jt)q j ρ j t)) xβ it)) ρ jt)q j ρ j t))xβ i t)) Using the conditon H7), we see that rt)z t) ) = k i t)xβ i t)), t t 4) for some t t, which means that rt)z t) is nonincreasing. Hence, we consider z t) or z t) < for t t. Since zt) is a monotone function,

5 OSCILLATION OF SOLUTIONS 285 there exists lim t zt), and so, we have zt) or zt) < on [t 2, ) for some t 2 t. Case. zt) < for t t 2. We claim that xt) is bounded from above. If this is not the case, then there exists a number t 3 t 2 such that Then we have zt 3 ) < and max t 2 t t 3 xt) = xt 3 ). 5) > zt 3 ) =xt 3 ) + i H h i t 3 )xα i t 3 )) t3 s t rs) h i t 3 ) i H h i t 3 ) i H ρ i s) t3 q i ξ)xγ i ξ))dξds t rs) s ρ i s) s rs) ρ i s) xt 3) xt 3), which is a contradiction. Therefore, xt) is bounded from above. There exists a positive constant L such that xt) L and L = lim sup xt). 6) t Thus we obtain, from 2), zt) xt) L xt) L t t rs) rs) Taking the superior limit as t yields lim zt) t s ρ i s) s s rs) ρ i s) ρ i s). L. 7) This is a contradiction. Case 2. zt) for t t 2. We discuss the following two subcases: Subcase. z t). Integrating 4) over [t 2, t] yields > rt 2 )z t 2 ) rt)z t) + rt 2 )z t 2 ) = t t 2 k i s)xβ i s))ds,

6 286 Y. SHOUKAKU which implies that k i x β i ) L [t 2, )). From Property [3] it follows that xt) L [t 2, )). On the other hand, from Property 2 [3], it is easy to see that { } Xt) xt) + i H h i t)xα i t)) L [t 2, )). 8) From 3) we can rewrite as X t) = z t) + t q i s)xγ i s))ds rt) ρ i t) for t t 2. Hence, Xt) is a nondecreasing function, which shows that Xt) Xt 2 ), t t 2. This implies that Xt) / L [t 2, ), which contradicts 8). Subcase 2. z t) <. Then we see that lim t zt) = µ [, ). I) µ >. Integrating 4) over [t 2, t], we obtain t t 2 k i s)xβ i s))ds = rt)z t) + rt 2 )z t 2 ) rt)z t). Next, dividing this inequality by rt) and integrating over [t 2, t] yields t s ) k i ξ)xβ i ξ))dξ ds zt) + zt 2 ). rs) t 2 Thus we have t 2 u t 2 k i s)xβ i s))ds ) t ) t 2 rs) ds zt 2 ) < for some u [t 2, t], and therefore xt) L [t 2, )). It is easy to see that Xt) L [t 2, )), that is, 8) holds. On the other hand, it follows from 2) that zt) Xt), t t 2. Since µ >, there exists a t 4 t 2 such that for < ε < µ. Then we observe that zt) µ ε Xt) zt) µ ε, which contradicts the fact that 8) holds.

7 OSCILLATION OF SOLUTIONS 287 II) µ =. If xt) is not bounded from above. There exists a sequence {t n } n= such that lim t n = and max xt) = xt n ). 9) n t 2 t t n Hence we obtain lim zt n) n s rs) ρ i s) lim xt n). ) n On the other hand, if xt) is bounded from above. There exists a positive constant L satisfying 6), and then 7) holds. It follows from 7) and ) that lim t xt) =. This completes the proof of theorem. Example. We consider the equation [ [e t xt) + ] ] 2 xt π) et + 5 e t 2π 2 et + 3 e t 5π/2 ) xt 2π) 3 e t 5π/2 x ) x t 52 ) π t π ) 2 5 e t 2π xt) =, t >. ) Here H = {}, P = Q = {, 2}, h t) = /2 and α t) = t π, p t) = 2 et + 3 e t 5π/2, β t) = t 5 2 π, p 2 t) = 2 et + 5 e t 2π, β 2 t) = t 2π, q t) = 3 e t 5π/2, γ t) = t π 2, q 2t) = 5 e t 2π, γ 2 t) = t. Let ψ : Q P satisfy ψ) =, ψ2) = 2, then ρ t) = t 2π, ρ 2 t) = β 2 t) and k t) = 2 et + ) 3 e t 5π/2 3 e t π/2 >, k 2 t) = 2 et + ) 5 e t 2π 5 e t >. It is clear that lim inf k t) =, t s e s s s 2π 3 e ξ 5π/2 dξds + e s = 3 3e 2π + 5e π/2 5e 5π/2) <. 3 s 2π 5 e ξ 2π dξds

8 288 Y. SHOUKAKU Therefore, Theorem implies that every solution xt) of the equation ) oscillates or tends to zero as t. In fact, xt) = sin t is an oscillatory solution of this equation. Next, we will present a criterion for oscillation of E ). Before starting theorem, we need the following lemma which is an improvement of the result in Kusano and Naito [5]. Lemma. If vt) is a positive function of for some t >, then it satisfies for all sufficiently large t t, where rt)v t)), t t 2) vt) πt)rt)v t) 3) πt) = t rs) ds. Proof. From 2) it follows that rt)v t) is nonincreasing [T, ) for some T t, i.e., rξ)v ξ) rs)v s), ξ > s T. Dividing this inequality by rξ) and integrating over [s, t], we obtain < vt) vs) + rs)v s) Then we consider two cases: I) v t). From 4) we find t < vt) vs) + rs)v s) s rξ) dξ, t > s > T. 4) s rξ) dξ, which means 3). II) v t) <. 4) and H3) imply that vt) is bounded from above. Letting t in 4), we see that 3) holds. Theorem 2. If and i H h i + s 5) rs) ρ i s) πs)k j s)πβ j s))ds = 6)

9 OSCILLATION OF SOLUTIONS 289 for some j P, then every solution of E ) oscillates or satisfies lim t xt) =. Proof. Suppose that xt) is a nonoscillatory solution of E ) whch does not satisfy lim t xt) =. Then we see that xt) > for t t. We denote by wt) =xt) i H h i t)xα i t)) t t rs) s ρ i s) q i ξ)xγ i ξ))dξds, 7) then as in the proof of Theorem, it follows from E ) that rt)w t) ) = k i t)xβ i t)), t t 8) for some t t. Therefore, rt)w t) is nonincreasing, and hence w t) < or w t) for t t. Since wt) is monotone function, there exists lim t wt), moreover, we find that wt) > or wt) on [t 2, ) for some t 2 t. Case. wt) for t t 2. If xt) is not bounded from above, then there exists a number t 3 t 2 such that 5) holds. Hence, we obtain t3 wt 3 ) = xt 3 ) h i t 3 )xα i t 3 )) q i ξ)xγ i ξ))dξds i H t rs) ρ i s) h i s i H rs) ρ i s) xt 3), which is a contradiction. Therefore, xt) is bounded from above. There exists a positive constant L such that 6) holds. We have wt) xt) L i H h i L so that, taking sperior limit as t yields lim wt) t i H which is a contradiction. h i rs) s rs) ρ i s) s s ρ i s), L, Case 2. wt) > for t t 2. We discuss the following two subcases:

10 29 Y. SHOUKAKU Subcase. w t). We see that wt) wt 4 ) for some t 4 t 2. Since lim t πt) =, we observe that wt 5 ) πt), t t 5 for some t 5 t 4. Consequently, we obtain wt) πt) for t t 5. Subcase 2. w t) <. In view of 8), we see that for some t 6 t 2, which means that rt)w t)), t t 6 rt)w t) rt 6 )w t 6 ) c <, t t 6. 9) Substituting 3) into 9), we have wt) c πt) for t t 6. For the above subcases, it follows that so that, from 7) wt) c πt), t t 6, xt) c πt), t t 6. Then, it is clear from 8) that rt)w t)) + c k i t)πβ i t)), t t 6. 2) Multiplying 2) by πt) and integrating over [t 6, t], we obtain πt)rt)w t) + wt) πt 6 )rt 6 )w t 6 ) wt 6 ) t + c πs)k i s)πβ i s))ds, t 6 which implies, in view of 3), c t for t t 6. complete. t 6 πs)k j s)πβ j s))ds πt 6 )rt 6 )w t 6 ) + wt 6 ) This contradicts the condition 6), and hence the proof is Example 2. We consider the equation [ [e t xt) ] ] 4 xt 2π) + e 2t + 34 et + 3 ) e 2t 4π xt 2π) et + ) 2 e 2t 5π x t 52 ) π + e 2t xt π) 3 e 2t 4π xt) 2 e 2t 5π x t π ) =, t >. 2) 2

11 OSCILLATION OF SOLUTIONS 29 Here H = {}, P = {, 2, 3}, Q = {, 2}, h t) = /4 and α t) = t 2π, p t) = e 2t et + 3 e 2t 4π, β t) = t 2π, p 2 t) = 3 4 et + 2 e 2t 5π, β 2 t) = t 5 2 π, p 3 t) = e 2t, β 3 = t π, q t) = 3 e 2t 4π, γ t) = t, q 2 t) = 2 e 2t 5π, γ 2 t) = t π 2. Let ψ) =, ψ2) = 2, then we can easy to check that ρ t) = ρ 2 t) = t 2π, k t) = e 2t + 34 et + 3 ) e 2t 4π 3 e 2t >, 3 k 2 t) = 4 et + ) 2 e 2t 5π 2 e 2t π >. So, Theorem 2 is applicable to 2) since and lim inf k t) =, t s 4 + e s s s 2π 3 e 2ξ 4π dξds + e s = e 4π + 3e π 3e 5π) < 36 s 2π 2 e 2ξ 5π dξds e s {e 2s + 34 es + 3 e 2s 4π 3 e 2s } e s+2π ds = for j =. Therefore all solutions oscillate or tend to zero as t. For example, xt) = cos t is such an oscillatory solution. 3. Oscillation of solutions of nonhomogenous equations In this section we consider equations E ± ) with forcing term [ [ rt) xt) + ] h i t)xα i t))] Ẽ±) i H + p i t)xβ i t)) q i t)xγ i t)) = ft), t >. Theorem 3. If ) holds, then every solution of Ẽ+) is oscillatory or satisfies lim t xt) =.

12 292 Y. SHOUKAKU Proof. Suppose that xt) is a nonoscillatory solution of Ẽ+) which does not satisfy lim t xt) =. We assume that xt) > for t t. There exists a constant ε > such that F t) F ) + ε. If we define Zt) = zt) F t) + F ) + ε, 22) where zt) is defined by 2), then we obtain from Ẽ+) rt)z t) ) = k i t)xβ i t)), t t 23) for some t t, and then rt)z t) or rt)z t) < eventually. Since rt) > and Zt) is a monotone function, we see that Zt) or Zt) < on [t 2, ) for some t 2 t. Case. Zt) < for t t 2. If xt) is not bounded from above, then there exists a number t 3 t 2 satisfying 5). Then we have Zt 3 ) =xt 3 ) + i H h i t 3 )xα i t 3 )) t3 s q i ξ)xγ i ξ))dξds t rs) ρ i s) F t 3 ) + F ) + ε h i t 3 ) s i H rs) ρ i s) xt 3) F t 3 ) + F ) + ε and leads to the following contradiction Zt 3 ) h i t 3 ) s i H rs) ρ i s) xt 3). Hence xt) is bounded from above, then there exists a positive constant L satisfying 6). We see that Zt) xt) L s F t) + F ) + ε. 24) rs) ρ i s) By taking the superior limit as t, we lead to the following contradiction lim Zt) t s L. 25) rs) ρ i s) Case 2. Zt) for t t 2. We discuss the following two subcases:

13 OSCILLATION OF SOLUTIONS 293 Subcase. Z t). Proceeding as the same proof of Theorem, we consider xt) L [t 2, )) and Xt) L [t 2, )). We see that so that [Xt) F t) + F )] = Z t) + rt) t ρ i t) q i s)xγ i s))ds, lim [Xt) F t) + F )] = lim Xt) = µ, ). t t There exists a t 3 t 2 such that Xt) > µ ε, for any ε with < ε < µ. However, this contradicts Xt) L [t 2, )). Subcase 2. Z t) <. From Zt) it follows that lim t Zt) = l [, ). I) If l >, then, as in the proof of Theorem, we observe that xt) L [t 2, )). Since zt) = Xt) t t rs) s ρ i s) q i ξ)xγ i ξ))dξds Xt), we see that zt) L [t 2, )). On the other hand, we can choose < ε < l and we have lim zt) = lim Zt) ε = l ε l, t t then there exists a t 4 t 2 such that zt) > l ε for any ε with < ε < l. This implies that zt) / L t 4, )), which is a contradiction. II) l =. If xt) is not bounded from above, there exists a sequence {t n } n= such that 9) is satisfied. Then, we see that lim Zt n) n s q i ξ)xγ i ξ))dξds rs) ρ i s) lim xt n). n Next if xt) is bounded from above. There exists a positive constant L such that 6) is satisfied. We obtain the inequality 24), which means that l s rs) ρ i s) L, by letting the superior limit as t. We observe from above inequalities that lim t xt) =. This is a contradiction. The proof is complete.

14 294 Y. SHOUKAKU Example 3. We consider the equation [ t + ) 2 = 6 [ xt) + t 2 ] ] 2 xt ) + t + ) 2 t + 7 ) 2 8 t + ) 2 x t ) xt) 2, t >. 26) t + ) 2 Here H = P = Q = {}, 2 t + 7 ) 2 t + )2 8 rt) =, α t) = t, p t) = 6 t + ) 2, β t) = t 8, q t) = 2 3, γ 2 t) = t, ft) = t + ) 2. Let ψ) =, it is clear that ρ t) = β t) and 2 t + 7 ) 2 8 k t) = t + ) >. Moreover, there exists a function F t) = 6/t + ) 2, and the condition ) is satisfied, since lim inf k t) = 4 t 3 >, 6 s 2 s + ) 2 3 dξds = 2 <. s /8 Therefore, Theorem 3 implies that every solution xt) of the equation 26) oscillates or tends to zero limit as t. In fact, xt) = /t + ) 2 is a solution of the equation 26) which tends to zero as t. For any continuous function θt) we use the notation [θt)] ± = max{±θt), }. Theorem 4. Assume that H9) there exists a oscillatory function F t) such that F ). If 5) and πs)k j s)[c πβ j s)) ± F β j s))] + ds = 27) for some j P, then every solution of Ẽ ) oscillates.

15 OSCILLATION OF SOLUTIONS 295 Proof. Suppose that xt) is a nonoscillatory solution of Ẽ ). Hence we see that xt) > for t t. We set the function as follows: W t) = wt) F t), where wt) is defined by 7). Then, we consider from Ẽ ) rt)w t) ) = k i t)xβ i t)), t t 28) for some t t. Therefore, we obtain rt)w t) is a nondecreasing function. From the monotonicity of W t) it follows that W t) > or W t) on [t 2, ) for some t 2 t. Case. W t) for t t 2. If xt) is not bounded from above, there exists a sequence {t n } n= such that 9) is satisfied. Then we obtain W t n ) =xt n ) i H h i t n )xα i t n )) 29) which implies that tn s t rs) h i i H ρ i s) q i ξ)xγ i ξ))dξds F t n ) s rs) ρ i s) W t n ) F t n ). xt n) F t n ), This is a contradiction. Hence, xt) is bounded from above. There exists a positive constant L satisfying 6). We see that W t) xt) L i H h i L rs) s ρ i s) F t). Taking superior limit as t, we obtain lim W t) t h i s i H rs) ρ i s) L lim inf F t) >, t which is a contradiction. Case 2. W t) > for t t 2. From the same proof of Theorem 2, we obtain W t) c πt), t t 3 for some t 3 t 2. Then, we conclude that xt) W t) + F t) c πt) + F t)

16 296 Y. SHOUKAKU for some t 4 t 3. It follows from xβ i t)) > that xβ i t)) [c πβ i t)) + F β i t))] +, t t 4. 3) Substituting 3) into 28) yields rt)w t)) + k i t)[c πβ i t)) + F β i t))] + 3) for t t 4. Multiplying 3) by πt) and integrating over [t 4, t], we have t t πs)k j s)[c πβ j s)) + F β j s))] + ds <. This contradicts the condition 27) and completes the proof of Theorem 4. Theorem 5. Assume that F ) = hold. If 5) and 27) hold for some j P, then every solution of Ẽ ) oscillates or satisfies lim t xt) =. Proof. The proof is similar to that of Theorem 4 and hence will be omitted. Example 4. Consider the equation [ [e t xt) ] ] 4 xt 2π) + e 3t x t 32 ) π et + ) 2 e 2t 3π xt 2π) 2 e 2t 3π xt) e 3t + 34 ) et x t π ) 2 = e 2π + e 6π) e t e8π + 34 ) e2π e 3t + e 5π e 3π) e 6t, 2 t >. 32) Here H = {}, P = {, 2, 3}, Q = {}, h t) = /4, α t) = t 2π p t) = e 3t, β t) = t 3 2 π, p 2t) = e 3t et, β 2 t) = t π 2, p 3 t) = 3 4 et + 2 e 2t 3π, β 3 t) = t 2π, q t) = 2 e 2t 3π, γ t) = t, ft) = e 2π + e 6π) e t e8π + 34 ) e2π e 3t + e 5π e 3π) e 6t. 2 Let ψ) =, then we see that conditions H7) are satisfied since k t) = e 3t 2 e 2t, k 2 t) = p 2 t), k 3 t) = p 3 t),

17 OSCILLATION OF SOLUTIONS 297 where ρ t) = β t). Moreover, there exists a function F t) = e 2π + e 6π) e 2t e8π + 34 ) e2π e 4t + e 5π e 3π) e 7t. 84 Then conditions 5) and 27) reduce to lim inf k t) =, t s 4 + e s s 3π/2 2 e 2ξ 3π dξds = 4 + e 3π ) <, 2 and { e s e 3s } [c 2 e 2s e s+3π/2 ± F s 32 )] π ds =, + hence, by Theorem 5 every solution xt) of the equation 32) is oscillatory or tends to zero as t. One such solution is xt) = sin t + e 4t. Remark. In Theorem 5, if we assume that F ), then also the theorem holds by putting W t) = wt) F t) + F ). References [] Dzurina, I., Stavroulakis, I. P., Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 4 23), [2] Grace, S. R., Lalli, B. S., Oscillations of nonlinear second order neutral delay differential equations, Rad. Mat ), [3] Karpuz, B., Manojlović, J., Öcalan, Ö., Shoukaku, Y., Oscillation criteria for a class of second order neutral delay differential equations, Appl. Math. Compt. 2 29), [4] Koplatadze, R., Kvinikadze, G., Stavroulakis, I. P., Oscillation of second order linear delay differential equations, Funct. Differ. Equ. 7 2), [5] Kusano, T., Naito, M., Nonlinear oscillation of second rder differential equations with retarded arguments, Ann. Mat. Pura Appl. 4) 6 975), [6] Manojlović, J., Shoukaku, Y., Tanigawa, T., Yoshida, N., Oscillation criteria for second order differential equations with positive and negative coefficients, Appl. Math. Comput. 8 26), [7] Manojlović, J., Shoukaku, Y., Tanigawa, T., Yoshida, N., Oscillatory behavior of second order neutral differential equations with positive and negative coefficients, Appl. Appl. Math. 3 28), 7. [8] Parhi, N., Chand, S., Oscillation of second order neutral delay differential equations with positive and negative coefficients, J. Indian Math. Soc. N.S.) ),

18 298 Y. SHOUKAKU [9] Shen, J. H., Stavroulakis, I. P., Tang, X. H., Hille type oscillation and nonoscillation criteria for neutral equations with positive and negative coefficients, Stud. Univ. Zilina Math. Ser. 4 2), [] Tanaka, S., Oscillation criteria for a class of second order forced neutral differential equations, Math. J. Toyama Univ ), 7 9. [] Weng, A., Sun, J., Oscillation of second order delay differential equations, Appl. Math. Comput ), Yutaka Shoukaku Faculty of Engineering Kanazawa University Kakuma, Kanazawa Ishikawa 92-92, Japan shoukaku@t.kanazawa-u.ac.jp