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1532 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 Then every solution of (4) is oscillatory on [ t , ∞) 0 T . We next study a Philos-type oscillation criteria for (4). First, Let us introduce the class of functions R which will be extensively used in the sequel. 2 Let D= {( ts , ) ∈T : t≥ s≥t} .The function H ∈ Crd ( D, ) is said to belong to the class R by H ∈R, if H (,) tt 0, t t 0 = ≥ ; H (, ts) 0, t s t 0 0 > > ≥ , (23) Δs and H has a continuous Δ− partial derivative H (, ts) with respect to the second variable. Theorem 2 Assume that (H1) - (H5) hold. Let g (t) be as defined in Theorem 1, and Hh , ∈C rd ( D, ) such that H ∈R. Furthermore, suppose that there exists a positive rd-continuous function ϕ() t satisfies Hts (, ) ≤ ϕ() s , (24) Htt (, ) 0 Δ Δ g () s h(,) t s s γ ( γ+ 1) −H (, t s) − H(, t s) = ( H(, t s)) , (25) σ σ g () s g () s and for all sufficiently largeT ∗ , we have 1 t ∗ limsup ∫ [ β (, s T ) g() s Q() s H(,) t s t→∞ t0 Htt (, ) 0 α()( s h (,)) t s − ] Δ s =∞. γ + 1 − γ+ 1 γ ( γ + 1) g ( s) (26) Then every solution of (4) is oscillatory on [ t , ∞) 0 T . Proof Suppose (4) has a nonoscillatory solution x (t), without loss of generality, say xt () > 0, x( τ ( t)) > 0, x( δ ( t)) > 0, for all t ≥ t , for some t 1 1 ≥ t 0 . By (H2) - (H5), proceed as in the proof of Theorem 1, we get that (11) holds for all t ≥ t . Again we define wt () as in the proof of 1 Theorem 1, then there exists t 2 ≥ t 1 , sufficiently large such that for all t ∗ ≥ t and for t t ∗ Δ ≥ , (22) holds and let g () t 2 + Δ be replaced by g () t in (22), thus Δ Δ g () t σ β (, tt) gtQt () () ≤− w() t + w() t 2 σ g () t γ gt () − α ()( t g ()) t 1 γ σ λ σ λ ( w ( t)) . (27) Multiplying both the sides of (27), with t replaced by s, by H (t, s) and integrating with respect to s from t ∗ to t, we obtain t ∫ ∗ Hts (,) β (, st) gsQs () () Δs≤ 2 t Δ t Δ t g () s σ −∫ ∗H (, tsw ) ( s) Δ s+ ∗Hts (, ) w( s) s t ∫ Δ t σ g () s t γ gs () σ λ −∫ ∗ Hts (, ) ( w( s)) Δs. t 1 γ σ λ α ()( s g ()) s Integrating by parts formula and using (23) and (25), we get t ∗ ∗ ∫ ∗ Hts ( , ) β( st , ) gsQs ( ) ( ) Δs≤ Htt ( , ) wt ( ) + t 2 1 λ t hts (, )( Hts (, )) σ γ Htsgs (, ) ( ) (28) − σ λ ∫ ∗[ w ( s) − ( w ( s)) ] Δs. t σ 1 γ σ λ g () s α ()( s g ()) s And applying Lemma 1, we obtain 1 λ (, )( (, )) − σ γ (, ) ( ) σ λ w () s − ( w ()) s σ 1 γ σ λ h t s H t s Htsgs g () s α ()( s g ()) s α ≤ ( γ + 1) g ( s) γ + 1 ( h ( t, s)) ( s) − γ+ 1 γ . From the last inequality and (24), (28), we have 1 ( h ( t, s)) α( s) [ (, s t ) g() s Q() s H(,) t s ] Δs Htt g s γ + 1 t − ∫ β − t0 2 γ+ 1 γ (, ) ( γ + 1) ( ) 0 ∗ ∗ ∗ t ≤ ϕ( t ) w( t ) + ∫ ϕ( s) β( s, t ) g( s) Q( s) Δ s T , we have © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1533 t ∗ limsup ∫ [ β ( s, T ) g( s)( q( s) − p( s)) t→∞ T Δ γ + 1 α()(( s g ())) s + − ] Δ s =∞. γ+ 1 γ ( γ + 1) g ( s) (29) Then every solution of (5) is either oscillatory on [ t , ∞) 0 T or tends to zero. proof Suppose that x is an eventually positive solution of (5), say xt () > 0, x( τ ( t)) > 0, x( δ ( t)) > 0for all t ≥ t for 1 some t ≥ t . We consider only this case, because the 1 0 proof for the case that x is eventually negative is similar. In the view of (5), by (H2) - (H5), and there exists t ≥ t such that for all t ≥ t , we have 2 1 2 Δ ( ( z ) γ γ α ) Δ ≤−( q( t) − p( t)) x ( δ( t)) < 0 , (30) γ then α( z Δ ) is strictly decreasing on [ t 2 , ∞ ). Hence z (t) Δ and z () t are of constant sign eventually. We claim that x() t is bounded. If not, there exists { t k } ⊆ [ t 2 , ∞ ), such that lim t =∞ ,lim x( t ) =∞, and k→∞ k k→∞ k x( t ) = max{ x( s): t ≤ s≤ t }. k Since lim τ ( t ) =∞, we can choose a large k such that k→∞ 0 k τ ( t ) > t , and by (H2), we obtain that k x( τ( t )) = max{ x( s) : t ≤ s≤τ( t )} Therefore, for all large k, k 0 0 ≤ max{ x( s) : t ≤ s ≤ t } = x( t ). z( τ ( t )) ≥ x( t ) −c x( τ ( t )) ≥ (1 − c ) x( t ) k k 0 k 0 k , and lim zt ( ) =∞. From (H1) and (30), as in the proof of k→∞ k Theorem 1 [4], there exists t 3 ≥ t 2 such that for all t ≥ t , 3 we have In view of (5), (30) and (31), we get 0 k Δ zt () > 0, z () t > 0. (31) Δ ( ( z ) γ γ α ) Δ ≤−( q( t) − p( t)) z ( δ( t)) < 0. (32) Now by using the same proof of Theorem 1, we get a contradiction with (29). Thus x (t) is bounded and hence z (t) is bounded. Also, by using (H1) and the same proof of Theorem 1 Δ in [4], there exist t 4 ≥ t 3 such that z () t > 0on [t 4 , ∞). There are two cases. Δ Case 1 zt () > 0 and z () t > 0. As in the proof of Theorem 1, we get a contradiction with (29). Δ Case 2 zt () < 0and z ( t) > 0 . We claim lim xt ( ) = 0 . k k k t→∞ Assume not, then there exists { t } ⊆[ t , ∞) such that k 5 lim t =∞, lim xt ( ) = : b> 0 and x( t ) = max{ x( s): t ≤ s k k k 0 k→∞ t k k→∞ ≤ }. But, by x( τ ( t )) ≤ x( t ), we get k k 0 > zt ( ) ≥ xt ( )(1 −c) →b(1 − c) > 0, as k→∞. k k 0 0 Which is a contradiction. This completes the proof. Corollary 4 Assume that (H1) - (H5) hold, Furthermore, suppose that for all sufficiently large T ∗ , and for δ ( T ) > T ∗ , we have t ∗ α() s limsup ∫ ( sβ ( s, T )( q( s) − p( s)) − ) Δ s =∞. t→∞ T γ+ 1 γ ( γ + 1) s Then every solution of (5) is either oscillatory on [ t , ∞) 0 T or tends to zero. Corollary 5 Assume that (H1) - (H5) hold, Furthermore, suppose that for all sufficiently large T ∗ , and for δ ( T ) > T ∗ , we have t ∗ limsup ∫ β (, sT)(() qs − ps ()) Δ s=∞. t→∞ T Then every solution of (5) is either oscillatory on [ t , ∞) 0 T or tends to zero. We next study a Philos-type oscillation criteria for (5). Theorem 4 Assume that (H1) - (H5) hold. Let g (t) be as defined in Theorem 1, and H, h∈C rd ( D, ) such that H ∈R . Furthermore, suppose that there exists a positive rd-continuous function ϕ () t such that (24), (25) hold, and for all sufficiently largeT ∗ , we have 1 t ∗ limsup ∫ { β ( s, T ) g( s)( q( s) − p( s)) H( t, s) t→∞ t0 Htt (, ) 0 α()( s h (,)) t s − } Δ s =∞. γ + 1 − γ+ 1 γ ( γ + 1) g ( s) (33) Then every solution of (5) is either oscillatory on [ t , ∞) 0 T or tends to zero. Proof Suppose that (5) has a nonoscillatory solution x (t), without loss of generality, say xt () > 0, x( τ ( t)) > 0, x( δ ( t)) > 0, for all t ≥ t , for some t 1 1 ≥ t 0 . By (H2) - (H5), we obtain that (30) holds for all t ≥ t , and zt () and 1 Δ z () t are of constant sign eventually. Similar to the proof of Theorem 3, we claim that x() t is bounded. If not, there exists{ t } ⊆[ t , ∞ ), for all large k, there exists t k 1 2 ≥ t 1 , such that (31) and (32) hold for t ≥ t . Again we define wt () as 2 in the proof of Theorem 1, then there exists t 3 ≥ t 2 , sufficiently large such that for t ∗ ≥ t and for t ≥ t ∗ , we 3 find β (, tt) gt ()( qt () − pt ()) 3 Δ g () t γ g() t ≤− w () t + w () t − ( w ()). t g t t g t Δ σ σ λ σ 1 γ σ λ () α ()( ()) And similar to the proof of the theorem 3, we obtain 1 t ∫ [ β ( s, t ) g( s)( q( s) − p( s)) H( t, s) t Htt (, ) 0 3 0 γ + 1 ( h ( t, s)) α( s) − γ+ 1 γ ] s ( ∗ ϕ t ) w ( ∗ t ) − Δ ≤ + ( γ + 1) g ( s) (34) © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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