The Ratio of Eigenvalues of the Dirichlet Eigenvalue Problem for Equations with One-Dimensional p-Laplacian

and Applied Analysis 3 and using 1.9 we have ( tanpt )′ 1 − sinpt ( cospt )′ cospt 1 ∣ ∣tanpt ∣ ∣. 1.10 Like for p 2, tanpt > t for t ∈ 0, πp/2 and tanpt < t for t ∈ −πp/2, 0 , which is equivalent to ∣ ∣sinp ∣ ∣p > tΦ ( sinpt ) cospt 1.11 for t ∈ −πp/2, πp/2 , t / 0. A similar formula to 1.10 for cotp is related to the Riccati equation associated with 1.1 . Namely, if x t / 0 is a solution of 1.1 in some interval I ⊂ R, then the function w Φ x′/x solves the Riccati equation w′ − c t λ p − 1|w|q 0, q : p p − 1 . 1.12 In particular, from 1.6 [ Φ ( cotpt )]′ −p − 11 ∣cotpt ∣p] − p − 1 ∣sinpt ∣p < 0, t / kπp. 1.13 Let x be a nontrivial solution of 1.1 and consider the half-linear Prüfer transformation see 2, 10, Section 1.1.3 x t r t sinpφ t , x′ t r t cospφ t . 1.14 Then using the same procedure as in case of the classical linear Prüfer transformation one can verify that φ and r are solutions of φ′ ∣cospφ ∣∣p − c t − λ p − 1 ∣sinpφ ∣p, 1.15 r ′ Φ ( sinpφ ) cospφ [ 1 − c t − λ p − 1 ] r. 1.16 From 1.15 , φ′ > 0 at the points where x t 0, that is, where φ t nπp, n ∈ N. Also, solutions of 1.15 behave similarly as in the linear case which means that the eigenvalues of 1.1 , 1.2 are simple, form an increasing sequence λn → ∞ and the corresponding eigenfunction xn has exactly n − 1 zeros in πp, 0 . Moreover, if c t ≡ 0, then λn p − 1 n with the associated eigenfunction xn t sinpnt. 4 Abstract and Applied Analysis 2. Preliminary Computations To prove our main result, we will use the half-linear Prüfer transformation in a modified form. Therefore, we rewrite 1.1 into the form −Φx′′ c t Φ x p − 1zpΦ x 2.1 with z > 0. Note that due to the fact that c t ≥ 0, all eigenvalues of 1.1 , 1.2 are positive. Let x x t, z be a nontrivial solution of 2.1 for which x 0 0. For this solution we introduce the Prüfer angle φ and radius r by x t r t z sinpφ t, z , x′ t r t cospφ t, z . 2.2 Differentiating the first equation and comparing it with the second one we obtain r ′ z sinpφ r z φ′ cospφ r cospφ. 2.3 Equation 2.1 can be written as −x′′ c t p − 1 ∣x′ ∣2−pΦ x z ∣x′ ∣2−pΦ x 2.4 and similarly one can rewrite 1.6 as sinpt ′′ −|cospt|Φ sinpt . Differentiating the second equation in 2.2 and substituting into 2.4 we have r ′cospφ − r ∣cospφ ∣2−pΦ ( sinpφ ) φ′ c t − p − 1zp−1 ( p − 1zp r ∣cospφ ∣2−pΦ ( sinpφ ) . 2.5 Multiplying 2.3 by z cospt, 2.5 by −sinpφ, adding the resulting equations and dividing them by cospφwe get φ′ z − c t ( p − 1zp−1 ∣sinpφ ∣p. 2.6 By a similar computation, we get the equation for the radius r r ′ r c t ( p − 1zp−1 ( sinpφ ) cospφ. 2.7 Concerning the dependence of φ φ t, z on the eigenvalue parameter z, we have from 2.6 d dz φ′ t, z : φ̇′ 1 c t zp ∣sinpφ ∣p − pc t ( p − 1zp−1 φ̇ Φ ( sinpφ ) cospφ. 2.8 Abstract and Applied Analysis 5 Sometimes, we will skip the argument z of r and φ when its value is not important or it is clear what value we mean. The last equation can be regarded as a first-order linear nonhomogeneous differential equation for φ̇. Multiplying this equation by the integration factorand Applied Analysis 5 Sometimes, we will skip the argument z of r and φ when its value is not important or it is clear what value we mean. The last equation can be regarded as a first-order linear nonhomogeneous differential equation for φ̇. Multiplying this equation by the integration factor