Generalized Variational Principles on Oscillation for Nonlinear Nonhomogeneous Differential Equations
نویسندگان
چکیده
and Applied Analysis 3 Lemma 2.1 see 10 . Suppose that X and Y are nonnegative. Then λXYλ−1 −X ≤ λ − 1 Y, λ > 1, 2.1 where the equality holds if and only if X Y . Now, we will give our main results. Theorem 2.2. Assume (S2) holds. Suppose further that for any T ≥ t0, there exist T ≤ s1 < t1 ≤ s2 < t2 such that e t ⎧ ⎨ ⎩ ≤ 0, t ∈ s1, t1 , ≥ 0, t ∈ s2, t2 . 2.2 Let u ∈ C1 si, ti , and nonnegative functions G1, G2 satisfying Gi u si Gi u ti 0, gi u Gi u are continuous and gi u t α 1 ≤ α 1 α Gi u t for t ∈ si, ti , i 1, 2. If there exists a positive function ρ ∈ C1 t0,∞ , such that Q ρ i u : ∫ ti si ρ t ⎡ ⎣q t Gi u t − ( α γ )α p t ∣∣u′ t ∣∣ G 1/ α 1 i u t ∣∣ρ′ t ∣∣ α 1 ρ t )α 1⎤ ⎦dt > 0, 2.3 for i 1, 2. Then 1.1 is oscillatory. Proof. Suppose to the contrary that there is a nonoscillatory solution y t of 1.1 . First, we consider the case when y t > 0 eventually. Assume that y t > 0 on T0,∞ for some T0 ≥ t0. Set w t ρ t p t Ψ ( y t )∣∣y′ t ∣∣α−1y′ t f ( y t ) , t ≥ T0. 2.4 Then differentiating 2.4 and making use of 1.1 , it follows that for all t ≥ T0, w′ t −ρ t q t ρ t e t f ( y t ) ρ ′ t ρ t w t − |w t | α 1 f ′ ( y t ) [ p t ρ t Ψ ( y t )∣∣f(y t )∣∣α−1 ]1/α . 2.5 By assumptions, we can choose s1, t1 ≥ T0 with s1 < t1 so that e t ≤ 0 on the interval I1 s1, t1 . For t ∈ I1 and in view of 1.5 and 2.5 ,w t satisfies the inequality ρ t q t ≤ −w′ t ρ ′ t ρ t w t − γ |w t | α 1 /α p1/α t ρ1/α t . 2.6 4 Abstract and Applied Analysis Multiplying G1 u t through 2.6 and integrating 2.6 from s1 to t1, using the fact that G1 u s1 G1 u t1 0, we obtain ∫ t1 s1 G1 u t ρ t q t dt ≤ ∫ t1 s1 G1 u t [ −w′ t ρ ′ t ρ t w t − γ |w t | α 1 /α p1/α t ρ1/α t ] dt −G1 u t w t |1 s1 ∫ t1 s1 Gi u t ′w t dt ∫ t1 s1 G1 u t ρ′ t ρ t w t dt − ∫ t1 s1 G1 u t γ |w t | α 1 /α p1/α t ρ1/α t dt
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Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities
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