Báo cáo hóa học: "Research Article Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations" docx

Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 438130, 18 pages doi:10.1155/2008/438130 Research Article Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations Ond ˇ rej Do ˇ sl ´ y 1 and Simona Fi ˇ snarov ´ a 2 1 Department of Mathematics and Statistics, Masaryk University, Jan ´ a ˇ ckovo n ´ am. 2a, 66295 Brno, Czech Republic 2 Department of Mathematics, Mendel University of Agriculture and Forestry in Brno, Zem ˇ ed ˇ elsk ´ a1, 61300 Brno, Czech Republic Correspondence should be addressed to Ond ˇ rej Do ˇ sl ´ y, dosly@math.muni.cz Received 23 August 2007; Accepted 26 November 2007 Recommended by John R. Graef We consider the half-linear second-order difference equation Δr k ΦΔx k   c k Φx k1 0, Φx : |x| p−2 x, p>1, where r, c are real-valued sequences. We associate with the above-mentioned equa- tion a linear second-order difference equation and we show that oscillatory properties of the above- mentioned one can be investigated using properties of this associated linear equation. The main tool we use is a linearization technique applied to a certain Riccati-type difference equation correspond- ing to the above-mentioned one. Copyright q 2008 O. Do ˇ sl ´ y and S. Fi ˇ snarov ´ a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we deal with oscillatory properties of solutions of the half-linear second-order difference equation Δ  r k Φ  Δx k   c k Φx k1 0, Φx : |x| p−2 x, p > 1, 1.1 where r, c are real-valued sequences and r k > 0. This equation can be regarded as a discrete counterpart of the half-linear differential equation  rtΦ  x     ctΦx0 1.2 which attracted considerable attention in the recent years. We refer to the books in 1, 2 and the references given therein. The basic qualitative theory of 1.1 has been established in the series of papers in 3–7 and it is summarized in the books 8, Chapter 3 and 2, Chapter 8. 2 Advances in Difference Equations It is known that oscillatory properties of 1.1 are very similar to those of the second-order Sturm-Liouville difference equation which is a special case of p  2in1.1: Δ  r k Δx k   c k x k1  0. 1.3 In particular, the discrete linear Sturmian theory extends verbatim to 1.1, and hence this equa- tion can be classified as oscillatory or nonoscillatory. We will recall elements of the oscillation theory of 1.1 in more detail in the next section. The basic idea of the discrete linearization technique which we establish in this paper is motivated by the paper of Elbert and Schneider 9, where the second-order half-linear differ- ential equation  Φ  x     γ p t p Φx2  p − 1 p  p−1 δt t p Φx0,γ p :  p − 1 p  p , 1.4 is viewed as a perturbation of the Euler-type half-linear differential equation  Φ  x     γ p t p Φx0, 1.5 and oscillatory properties of 1.4 are studied via the linear equation  ty     δt t y  0 1.6 under the assumption that  ∞ t δs/s ds ≥ 0forlarget. In particular, the following statements are presented in 9. i Let p ≥ 2 and let linear equation 1.6 be nonoscillatory. Then 1.4 is also nonoscillatory. ii Let p ∈ 1, 2 and let half-linear equation 1.4 be nonoscillatory. Then linear equation 1.6 is also nonoscillatory. The linearization technique for 1.2 has been further developed in 10–12; see also references given therein. In our paper, we introduce a similar linearization technique for the investigation of os- cillatory properties of 1.1. This equation is regarded as a perturbation of the nonoscillatory equation of the same form: Δ  r k Φ  Δx k   c k Φ  x k1   0, 1.7 and oscillatory properties of solutions of 1.1 are related to those of the linear second-order difference equation Δ  R k Δy k   C k y k1  0, 1.8 where R k  2 q r k h k h k1   Δh k   p−2 ,C k   c k − c k  h p k1 , 1.9 O. Do ˇ sl ´ yandS.Fi ˇ snarov ´ a3 with q  p/p − 1 being the conjugate number of p, and with a certain distinguished solution h of 1.7. This enables to apply the deeply developed linear oscillation theory when investi- gating oscillations of half-linear equation 1.1. As we will see in the next sections, compared to the continuous case, the linearization technique is technically more difficult in the discrete case since a nonlinear function which appears in the so-called modified Riccati equation is considerably more complicated in the discrete case. The paper is organized as follows. In the next section, we recall basic oscillatory proper- ties of 1.1, including a quadratization formula for a certain nonlinear function which plays an important role in subsequent sections of the paper. In Section 3, we present a discrete version of the above-mentioned result of Elbert and Schneider 9.InSection 4, we show that under certain additional restriction on properties of solutions of 1.7 we do not need to distinguish between the cases p ≥ 2andp ∈ 1, 2. The last section of the paper is devoted to an application of the results of the previous sections of the paper. 2. Preliminaries Oscillatory properties of 1.1 are defined using the concept of the generalized zero which is defined in the same way as for 1.3see, e.g., 8, Chapter 3 or 2, Chapter 7.Asolutionx of 1.1 has a generalized zero in an interval m, m 1 if x m /  0andx m x m1 r m ≤ 0. Since we suppose that r k > 0 oscillation theory of 1.1 generally requires only r k /  0, a generalized zero of x in m, m  1 is either a “real” zero at k  m  1 or the sign change between m and m  1. Equation 1.1 is said to be disconjugate in a discrete interval m, n if the solution x of 1.1 given by the initial condition x m  0, x m1 /  0, has no generalized zero in m, n  1. Equation 1.1 is said to be nonoscillatory if there exists m ∈ N such that it is disconjugate on m, n for every n>m,and it is said to be oscillatory in the opposite case. If x is a solution of 1.1 such that x k /  0insomediscreteintervalm, ∞,thenw k  r k ΦΔx k /x k  is a solution of the associated Riccati-type equation R  w k  :Δw k  c k  w k  1 − r k Φ  Φ −1  r k  Φ −1  w k    0. 2.1 Moreover, if x has no generalized zero in m, ∞,thenΦ −1 r k Φ −1 w k  > 0, k ∈ m, ∞.If we suppose that 1.1 is nonoscillatory, among all solutions of 2.1 there exists the so-called distinguished solution w which has the property that there exists an interval m, ∞ such that any other solution w of 2.1 for which Φ −1 r k Φ −1 w k  > 0, k ∈ m, ∞,satisfiesw k > w k , k ∈ m, ∞. Therefore, the distinguished solution of 2.1 is, in a certain sense, minimal solution of this equation near ∞.If w is the distinguished solution of 2.1, then the associated solution of 1.1 given by the formula x k  k−1  jm  1 Φ −1  w j r j  2.2 is said to be the recessive solution of 1.1see 13. Note that in the linear case p  2asolution x of 1.3 is recessive if and only if ∞  1 r k x k x k1  ∞. 2.3 4 Advances in Difference Equations Our first statement presents a comparison theorem for distinguished solutions of 2.1 and 2.4 given below. Lemma 2.1 see 13. Let 1.1 be nonoscillatory and let c k ≥ c k for large k. Further, let w k , v k be distinguished solutions of the corresponding generalized Riccati equations 2.1 and  R  v k  :Δv k  c k  v k  1 − r k Φ  Φ −1  r k  Φ −1  v k    0, 2.4 respectively. Then there exists m ∈ Z such that w k ≥ v k for k ∈ m, ∞. In particular, if c k ≥ 0 and  ∞ r 1−q k  ∞,then w k ≥ 0 for large k. The next statement relates nonoscillation of 1.1 to the existence of a certain solution of the Riccati inequality associated with 2.1. Lemma 2.2 see 2, Theorem 8.2.7. Equation 1.1 is nonoscillatory if and only if there exists a sequence w k satisfying r k  w k > 0 and R  w k  ≤ 0 2.5 for large k. The next statement is the discrete version of the generalized Leighton-Wintner oscillation criterion. In this criterion, 1.1 is viewed as a perturbation of 1.7. Lemma 2.3 see 13. Let h be the positive recessive solution of nonoscillatory equation 1.7.If ∞   c k − c k  h p k1  ∞, 2.6 then 1.1 is oscillatory. The last auxiliary oscillation results of this section are Hille-Nehari non-oscillation cri- teria for linear difference equation 1.3. Lemma 2.4 see 14. Suppose that c k ≥ 0, r k > 0,  ∞ r −1 k  ∞,and  ∞ c k < ∞.If lim inf k→∞  k−1  1 r j  ∞  jk c j  > 1 4 , 2.7 then 1.3 is oscillatory. If lim sup k→∞  k−1  1 r j  ∞  jk c j  < 1 4 , 2.8 then 1.3 is nonoscillatory. O. Do ˇ sl ´ yandS.Fi ˇ snarov ´ a5 For the remaining part of this section, we suppose that 1.7 is nonoscillatory and we let h be its solution such that h k > 0forlargek. Further, put G k : r k h k Φ  Δh k  2.9 and define the function Hk,v : v  r k h k1 Φ  Δh k  − r k  v  G k  h p k1 Φ  h q k Φ −1  r k  Φ −1  v  G k  . 2.10 Lemma 2.5. Put v k : h p k  w k − w k  , 2.11 where w k  r k ΦΔh k /h k  is a solution of 2.4 and w k is any sequence satisfying r k  w k /  0.Then Δv k   c k − c k  h p k1  H  k, v k   h p k1 R  w k  . 2.12 In particular, if w k is a solution of 2.1,then Δv k   c k − c k  h p k1  H  k, v k   0. 2.13 Moreover, Hk, v ≥ 0 for v>−r k h k ΦΔh k h p−1 k  with the equality if and only if v  0. Proof. By a direct computation and using the fact that w k is a solution of 2.4,weobtain Δv k  h p k1  w k1 − w k1  − v k  h p k1  w k1  c k − r k w k Φ  Φ −1  r k  Φ −1  w k   − v k  h p k1  w k1  c k − r k Φ  Δh k h k1  − v k  h p k1  w k1  c k  − r k h k1 Φ  Δh k  − v k . 2.14 Next, since v k  h p k w k − G k , we have r k  v k  G k  Φ  h q k Φ −1  r k  Φ −1  v k  G k   r k h p k w k Φ  h q k Φ −1  r k  Φ −1  h p k w k   r k w k Φ  Φ −1  r k  Φ −1  w k  , 2.15 and hence Δv k   c k − c k  h p k1  H  k, v k   h p k1  w k1  c k − r k  v k  G k  Φ  h q k Φ −1  r k  Φ −1  v k  G k    h p k1 R  w k  . 2.16 6 Advances in Difference Equations If w k is a solution of 2.1,thenv k satisfies 2.13. We prove the nonnegativity of the function Hk,v for v>−r k h k ΦΔh k h p−1 k  as follows. By a direct computation, we have H v  k, v   1 − r q k h q k h p k1  h q k Φ −1  r k  Φ −1  v k  G k  p , H vv k, v qr q k h q k h p k1   v k  G k   q−2  h q k Φ −1  r k  Φ −1  v k  G k  p1 . 2.17 Hence H v k, v0 if and only if v  0 and the function Hk, v is convex with respect to v for v satisfying h −q k Φ −1 r k Φ −1 v  G k  > 0 which is equivalent to v>−r k h k ΦΔh k h p−1 k .This proves the last statement of Lemma 2.5. Lemma 2.6. Let R, G be defined by 1.9 and 2.9, respectively, and suppose that G k > 0 for k ∈ N. Then we have the following inequalities for v ≥ 0 and k ∈ N :  R k  v  Hk,v ≥ v 2 ,p∈ 1, 2,  R k  v  Hk,v ≤ v 2 ,p≥ 2. 2.18 Proof. In this proof, we write explicitly an index by a sequence only if this index is different from k; that is, no index means the index k. In addition to 2.17,wehave H vvv k, 0 q r 2 h 2 h 2 k1 Δh 2p−3  q − 2h k1 − 2q − 1Δh  . 2.19 Denote Fk,v :R k  vHk,v − v 2 .ThenwehaveF v k, 00  F vv k, 0 and F vvv k, 0RH vvv k, 03H vv k, 0  2 rhh k1 Δh p−1  q − 2h k1 − 2q − 1Δh   3q rhh k1 Δh p−2  1 rhh k1 Δh p−1  2q − 2h Δh2 − qΔh   q − 2 rhh k1 Δh p−1 h  h Δh q − 2 rhh k1 Δh p−1  h  h k1  . 2.20 Consequently, sgn Fk, vsgn q − 2 2.21 in some right neighborhood of v  0. Further, we have F vv k, v2H v k, vR  vH vv k, v − 2  − 2r q h q h p k1  h q Φ −1 rΦ −1 v  G  p  qr q h q h p k1 v  G q−2 R  v  h q Φ −1 rΦ −1 v  G  p1  r q h q h p k1  h q Φ −1 rΦ −1 v  G  p1  − 2r q−1 h q − 2Φ −1 v  Gqv  G q−2 R  v  . 2.22 O. Do ˇ sl ´ yandS.Fi ˇ snarov ´ a7 Denote by Av the expression in brackets in the last expression. By a direct computation, we have Avq − 2Φ −1 v  GqR − Gv  G q−2 − 2r q−1 h q , 2.23 hence sgn Avsgn F vv k, vsgn q−2 for large v, and from the computation of F vvv k, 0, we also have q − 2Av > 0 in some right neighborhood of v  0. Since A  vq − 2v  G q−3  q − 1v  GqR − G  2.24 has no positive root observe that q − 1v  GqR − G0 if and only if v  −1/q − 1rhΔh p−2 h k1  h < 0, this means that q − 2Av and hence also q − 2F vv k, v have a constant sign for v ∈ 0, ∞. Therefore, the function Fk, v is convex for q ≥ 2 and concave for q ≤ 2, and this together with 2.21 implies the required inequalities. 3. (Non-)oscillation criteria: p ≥ 2 versus p ∈ 1, 2 In this section, we suppose that 1.7 is nonoscillatory and possesses a positive increasing so- lution h. We associate w ith 1.1 the linear Sturm-Liouville second-order difference equation Δ  R k Δy k   C k y k1  0, 3.1 where R and C are given by 1.9,thatis, R k  2 q r k h k h k1  Δh k  p−2 ,C k   c k − c k  h p k1 . 3.2 The results of this section can be regarded as a discrete version of the results given in 9. Theorem 3.1. Let p ≥ 2, c k ≥ c k for large k, ∞  1 R k  ∞, 3.3 and suppose that linear equation 3.1 with R, C given by 1.9 is nonoscillatory. Then half-linear equation 1.1 is also nonoscillatory. Proof. The proof is based on Lemma 2.2. Nonoscillation of 3.1 implies the existence of a solu- tion v of the associated Riccati equation Δv k  C k  v 2 k R k  v k  0 3.4 such that R k  v k > 0forlargek. Moreover, since 3.3 holds and C k ≥ 0forlargek,by Lemma 2.1 v k ≥ 0forlargek.ByLemma 2.6,wehaveR k  vHk,v ≤ v 2 ; hence v is also a solution of the inequality Δv k  C k  H  k, v k  ≤ 0. 3.5 Now, substituting for v  h p w − w,where w  rΦΔh/h, we see from Lemma 2.5 that w is a solution of Riccati inequality 2.5. Moreover, r k  w k  r k  h −p k v k  w k > 0 since v k ≥ 0andh is a nonoscillatory solution of 1.7; that is, the corresponding solution of the associated Riccati equation w satisfies r k  w k > 0. Therefore, 1.1 is nonoscillatory. 8 Advances in Difference Equations Theorem 3.2. Let p ∈ 1, 2, c k ≥ c k for large k, and let h be the recessive solution of 1.7. If half- linear equation 1.1 is nonoscillatory, then linear equation 3.1 is also nonoscillatory. Proof. We proceed similarly as in the previous proof. Nonoscillation of 1.1 implies the ex- istence of the distinguished solution w of the associated Riccati equation 2.1 such that w k  r k > 0forlargek. Put again v  h p w − w,where w is the distinguished solution of 2.4.Thenv solves the equation Δv k  C k  H  k, v k   0, 3.6 and by Lemma 2.1,wehavew k ≥ w k for large k, hence v k ≥ 0, and therefore R k  v k > 0for large k.ByLemma 2.6, Δv k  C k  v 2 k R k  v k ≤ 0. 3.7 This means that 3.1 is nonoscillatory by Lemma 2.2. 4. Criteria without restriction on p Throughout this section, we suppose that R k , C k ,andG k are given by 1.9 and 2.9, respec- tively, and that 1.7 is nonoscillatory. Theorem 4.1. Let c k ≥ c k for large k and let h k > 0 be the recessive solution of 1.7 such that ∞   c k − c k  h p k1 < ∞. 4.1 Further, suppose that condition 3.3 holds and lim k→∞ r k h k Φ  Δh k   ∞. 4.2 If there exists ε>0 such that the equation Δ  R k Δy k  1 − εC k y k1  0 4.3 is oscillatory, then 1.1 is also oscillatory. Proof. Let ε>0 be such that 4.3 is oscillatory i.e., ε<1. Suppose, by contradiction, that 1.1 is nonoscillatory, and let x k be its recessive solution. Denote by w k  r k ΦΔx k / x k  and w k  r k ΦΔh k /h k  the distinguished solutions of the Riccati equations 2.1 and 2.4, respectively, and put v k : h p k w k − w k . Since c k ≥ c k for large k, it follows from Lemma 2.1 that w k ≥ w k , and hence also v k ≥ 0forlargek. According to Lemma 2.5,wehave Δv k  −C k − H  k, v k  . 4.4 Hence v k is nonnegative and n onincreasing for large k, and this means that there exists a limit of v k such that 0 ≤ lim k→∞ v k < ∞. 4.5 O. Do ˇ sl ´ yandS.Fi ˇ snarov ´ a9 Next, let N ∈ N be sufficiently large, k>N. Summing 4.4 from N to k,weobtain v N − v k1  k  jN C j  k  jN H  j, v j  , 4.6 and hence v N ≥ k  jN C j  k  jN H  j, v j  . 4.7 Letting k →∞and using condition 4.1,wehave ∞  H  k, v k  < ∞. 4.8 Substituting z k  v k /G k into Hk,v k ,weobtain H  k, G k z k   G k z k  r k h k1 Φ  Δh k  − r k  z k  1  h p k1 Φ  h k /Δh k Φ −1  z k  1  :  H  k, z k  . 4.9 Now, it follows from conditions 4.2 and 4.5 that z k → 0ask →∞. Hence we may approx- imate the function  Hk,z by the second-degree Taylor polynomial at the center z  0 k is regarded as a p arameter . By a direct computation, we have  Hk,00,  H z k, 00,  H zz k, 0 qr k h k  Δh k  p h k1 , 4.10 and hence  Hk,z qr k h k  Δh k  p 2h k1 z 2  o  z 2  as z −→ 0. 4.11 The term oz 2  is of the form  H zzz k, ξz 3 for some ξ ∈ 0,z. By a direct computation, we have  H zzz k, 0qr k h k  Δh k  p h 2 k1  q − 2h k1 − 2q − 1Δh k  , 4.12 that is,    H zzz k, 0   ≤ qr k h k  Δh k  p h k1  |q − 2| 2q − 1  . 4.13 Since  H zzz k, z is continuous with respect to z near z  0, there exists a constant M>0 such that    H zzz k, ξ   ≤ Mr k h k  Δh k  p h k1 , 4.14 10 Advances in Difference Equations and hence 4.11 can be written in the form  H k, zqr k h k  Δh k  p 2h k1 z 2  1  o1  as z −→ 0 4.15 and the convergence o1 → 0asz → 0 is uniform with respect to k. This means that there exists N 1 such that q − εr k h k  Δh k  p 2h k1 z 2 k <  H  k, z k  < q  εr k h k  Δh k  p 2h k1 z 2 k for k ≥ N 1 , 4.16 and consequently ∞ > ∞  H  k, v k   ∞   H  k, z k  > q − ε 2 ∞  r k h k  Δh k  p h k1 z 2 k  q − ε 2 ∞  r k h k  Δh k  p v 2 k h k1 G 2 k  q − ε 2 ∞  v 2 k r k h k h k1  Δh k  p−2 . 4.17 Taking into account condition 3.3, it follows that v k → 0ask →∞. Thus we can apply Taylor’s formula to the function  Fk,v :R k  vHk,v at the center v  0. By a direct computation see also the proof of Lemma 2.6,wehavek is regarded again as a parameter  F k, 00,  F v k, 00,  F vv k, 02, 4.18 and hence  F k, vv 2  o  v 2   v 2  1  o1  as v −→ 0. 4.19 Similarly as in the case of  Hk,z, the convergence o1 → 0asv → 0 is uniform with respect to k because of 4.2 and 2.20. Hence Hk,v v 2 R k  v  1  o1  as v −→ 0. 4.20 Consequently, there exists N 2 >N 1 such that  1 − ε 2  v 2 k R k  v k <H  k, v k  <  1  ε 2  v 2 k R k  v k for k ≥ N 2 . 4.21 Since R k  2 q G k h k1 Δh k  2 q G k  1  h k Δh k  > 2 q G k , 4.22 from conditions 4.2 and 4.22 we have v k R k −→ 0ask −→ ∞ . 4.23 [...]... “Generalized discrete Riccati equation and oscillation of half-linear difference equations,” Mathematical and Computer Modelling, vol 34, no 3-4, pp 257–269, 2001 ˇ a 6 P Reh´ k, “Oscillation criteria for second order half-linear difference equations,” Journal of Difference Equations and Applications, vol 7, no 4, pp 483–505, 2001 ˇ a 7 P Reh´ k, “Oscillation and nonoscillation criteria for second order linear... and P Reh´ k, Half-Linear Differential Equations, vol 202 of North-Holland Mathematics Studies, s ´ Elsevier, Amsterdam, The Netherlands, 2005 ˇ a 3 P Reh´ k, “Hartman-wintner type lemma, oscillation, and conjugacy criteria for half-linear difference equations,” Journal of Mathematical Analysis and Applications, vol 252, no 2, pp 813–827, 2000 ˇ a 4 P Reh´ k, “Oscillatory properties of second order half-linear. .. 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Methods, and Applications, vol 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000 21 M Cecchi, Z Doˇ l´ , and M Marini, Half-linear equations and characteristic properties of the princisa pal solution,” Journal of Differential Equations, vol 208, no 2, pp 494–507, 2005 22 M Cecchi, Z Doˇ l´ , and M Marini, “Corrigendum to Half-linear. .. particular, we were able to formulate inequalities for the function H in Lemma 2.6 only under more restrictive assumptions than in the continuous case This is also the reason why assumptions of non- oscillation criteria formulated in Section 3 are more restrictive than those of oscillation criteria for 1.2 given in 10, 16 Now we comment in more detail on assumptions of Theorems 4.1 and 4.2 of the previous... k Cj j N j N Rj / 1 ε vj 4.34 Letting k → ∞ and using 4.1 , we have ∞ 2 vk Rk / 1 ε vk < ∞ 4.35 This, together with conditions 3.3 and 4.5 , implies that vk → 0 as k → ∞ Hence by the Taylor formula for the function F k, v : Rk v H k, v at the center v 0 see the computations in the proof of Theorem 4.1 and observe that 4.29 is still sufficient for the uniform convergence o 1 → 0 as v → 0 in 4.19 , we... Half-linear equations and characteristic propsa erties of the principal solution”,” Journal of Differential Equations, vol 221, no 1, pp 272–274, 2006 23 M Cecchi, Z Doˇ l´ , and M Marini, “Limit and integral properties of principal solutions for half-linear sa differential equations,” Archivum Mathematicum, vol 43, no 1, pp 75–86, 2007 24 M Cecchi, Z Doˇ l´ , and M Marini, “Nonoscillatory half-linear difference... > 0 Since vk ≥ 0 for large k, we have rk wk rk hk vk wk > 0 and nonoscillation of 1.1 follows from Lemma 2.2 O Doˇ ly and S Fiˇ narov´ s ´ s a 13 5 Remarks and applications We start this section with a discussion of the continuous counterparts of the results presented in the previous sections In 15 and the subsequent papers 10, 16, 17 , 1.2 is viewed as a perturbation of another half-linear differential... differs ´ ˜ ential equations,” Journal of Inequalities and Applications, no 5, pp 535–545, 2005 16 O Doˇ ly, “Perturbations of the half-linear Euler-Weber type differential equation,” Journal of Mathes ´ matical Analysis and Applications, vol 323, no 1, pp 426–440, 2006 ˇ 17 O Doˇ ly and J Rezn kov´ , “Oscillation and nonoscillation of perturbed half-linear Euler differential s ´ a equation,” Publicationes . Article ID 438130, 18 pages doi:10.1155/2008/438130 Research Article Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations Ond ˇ rej Do ˇ sl ´ y 1 and. Therefore, the function Fk, v is convex for q ≥ 2 and concave for q ≤ 2, and this together with 2.21 implies the required inequalities. 3. (Non-)oscillation criteria: p ≥ 2 versus p ∈ 1, 2 In. “Oscillation and nonoscillation criteria for half-linear second order dif- ferential equations,” Hiroshima Mathematical Journal, vol. 36, no. 2, pp. 203–219, 2006. 11 O. Do ˇ sl ´ yandM. ¨ Unal, “Half-linear