Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients

and Applied Analysis 3 is conditionally oscillatory with the oscillation constant K 1/4. It is known see 22 that the equation [ r t y′ t ]′ γs t t2 y t 0, 1.5 where r, s are positive periodic continuous functions, is conditionally oscillatory as well. We also refer to 23 and 24–29 which generalize 23 for the discrete case, see 30 . Since the Euler difference equation Δyk γ k 1 k yk 1 0 1.6 is conditionally oscillatory with the oscillation constant K 1/4 see 31 , it is natural to analyse the conditional oscillation of ∗∗ . Note that the announced result is more general than the results known in the continuous case, because ∗∗ has almost periodic coefficients. The conditional oscillation of discrete equations with constant coefficients can be generalized in other ways. Point out 32 , where an oscillation constant is characterized. The constant from 32 coincides with our oscillation constant if the considered coefficients are asymptotically constant. Solutions of the second-order Sturm-Liouville difference equations with periodic coefficients are studied in 33 see also 34, 35 . In 36 , the half-linear differential equations of the second order with the Besicovitch almost periodic coefficients are considered and an oscillation theorem for these equations is obtained. In the last years, many results dealing with the conditional oscillation of second-order equations and two-term equations of even order appeared. The two-term difference equation of even order −1 n Δn ( Γ k 1 Γ k − α 1 Δ yk ) qkyk n 0, 1.7 where Γ denotes the gamma function, is studied in 9, 10 . Results concerning the half-linear difference equation Δ [ rkΦ ( Δyk )] qkΦ ( yk 1 ) 0, 1.8 where rk > 0, Φ ( y ) ∣y ∣α−1 sgny, α > 1, 1.9 can be found in 37 for rk 1, qk γ k 1 −α and also in 38, 39 for dynamic half-linear equations on time scales, see 40–42 . The paper is organized as follows. In Section 2, we mention only necessary preliminaries and an auxiliary result. Our main result is proved in Section 3, where the particular case concerning the equation with periodic coefficients is formulated as well. The paper is finished by concluding remarks and simple examples. 4 Abstract and Applied Analysis 2. Preliminaries We begin this section recalling some elements of the oscillation theory of the Sturm-Liouville difference equation Δ ( rkΔyk ) qkyk 1 0, rk > 0, k ∈ N. 2.1 For more details, we can refer to books 43, 44 and references cited therein. We recall that an interval a, a 1 , a ∈ N, contains the generalized zero of a solution {yk} of 2.1 if ya / 0 and yaya 1 ≤ 0. Equation 2.1 is said to be conjugate on {a, . . . , a n}, n ∈ N, if there exists a solution which has at least two generalized zeros on a, . . . , a n 1 or if the solution {ỹk} satisfying ỹa 0 has at least one generalized zero on a, . . . , a n 1 . Otherwise, 2.1 is said to be disconjugate on {a, . . . , a n}. Since Sturmian theory is valid for difference equations, all solutions of 2.1 have either a finite or an infinite number of generalized zeros on N. Hence, we can categorize these equations as oscillatory and nonoscillatory. Definition 2.1. Equation 2.1 is called non-oscillatory provided a solution of 2.1 is disconjugate at infinity, that is, there exists N ∈ N such that 2.1 is disconjugate on any set N,N m ∩ N, m ∈ N. Otherwise, we say that 2.1 is oscillatory. Since we study a special case of 2.1 , when the coefficients are almost periodic, we also mention the basics of the theory of almost periodic sequences. Here, we refer to each one of books 45, 46 . Definition 2.2. A real sequence {fk}k∈Z is called almost periodic if, for any ε > 0, there exists a positive integer p ε such that any set consisting of p ε consecutive integers contains at least one integer l with the property that ∣fk l − fk ∣ < ε, k ∈ Z. 2.2 We say that a sequence {gk}k 1 is almost periodic if there exists an almost periodic sequence {fk}k∈Z for which fk gk, k ∈ N. The above definition of almost periodicity is based on the Bohr concept. Now we formulate a necessary and sufficient condition for a sequence to be almost periodic. The following theorem is often used as an equivalent definition the Bochner one of almost periodicity for k ∈ Z. Theorem 2.3. A sequence {fk}k∈Z ⊂ R is almost periodic if and only if any sequence of the form {fk h n }, where h n ∈ Z, n ∈ N, has a uniformly convergent subsequence with respect to k. Proof. See 45, Theorem 1.26 . Corollary 2.4. Let {fk} be almost periodic. The sequence {1/fk} is almost periodic if and only if inf ∣fk ∣; k ∈ N > 0. 2.3 Abstract and Applied Analysis 5 Proof. The corollary follows from 45, Theorem 1.27 and 47, Theorem 1.9 or directly from Theorem 2.3 . It suffices to use that 2.3 implies inf{|fk|} > 0 for any almost periodic sequence {fk} if k ∈ Z. Note that there exist nonzero almost periodic sequences {fk} for which 2.3 is not satisfied see, e.g., 48, Theorem 3 . Theorem 2.5. If {fk} is an almost periodic sequence, then the limitand Applied Analysis 5 Proof. The corollary follows from 45, Theorem 1.27 and 47, Theorem 1.9 or directly from Theorem 2.3 . It suffices to use that 2.3 implies inf{|fk|} > 0 for any almost periodic sequence {fk} if k ∈ Z. Note that there exist nonzero almost periodic sequences {fk} for which 2.3 is not satisfied see, e.g., 48, Theorem 3 . Theorem 2.5. If {fk} is an almost periodic sequence, then the limit M ({ fk }) : lim n→∞ fk fk 1 · · · fk n n 1 2.4 exists uniformly with respect to k. Proof. See 45, Theorem 1.28 . Definition 2.6. Let {fk} be almost periodic. The number M {fk} introduced in 2.4 is called the mean value of {fk}. Remark 2.7. For any positive almost periodic sequence {fk}, we have M {fk} > 0. Indeed, if we put ε f1/2 and find a corresponding p ε in Definition 2.2, then we obtain M ({ fk }) ≥ f1 2p ε > 0. 2.5 In the proof of our main result, we use an adapted Riccati technique. The classical Riccati technique deals with the so-called Riccati difference equation, which we obtain from 2.1 using the substitution wk rk Δyk/yk , that is, we obtain the equation Δwk qk w2 k wk rk 0, k ∈ N. 2.6 Putting ζk −kwk, we adapt 2.6 to our purposes. A direct calculation leads to the equation Δζk 1 k [ k k 1 qk ζk k 1 ζ2 k krk − ζk ] , k ∈ N. 2.7 We also mention two lemmas which we use to prove the main result. Lemma 2.8. Let the equation Δ ( rkΔyk ) qkyk 1 0, k ∈ N, 2.8 where sup{rk; k ∈ N} < ∞ and rk, qk > 0, k ∈ N, be non-oscillatory. For any solution {ζk} of the associated equation 2.7 , there exists k0 ∈ N such that, if ζk0 m < 0 for some m ∈ N, then ζk m < 0, k ≥ k0. 6 Abstract and Applied Analysis Proof. Let {yk} be a solution of the non-oscillatory equation 2.8 for which ykyk 1 > 0, k ≥ k0. From 43, Lemma 6.6.1 it follows that the sequence {wk}k k0 , where wk rk Δyk/yk , is decreasing. Further, 43, Theorem 6.6.2 implies that limk→∞wk 0. Thus, the sequence {wk}k k0 is positive, that is, ζk −kwk < 0, k ≥ k0. Lemma 2.9. If there exists a solution {ζk} of the associated equation 2.7 satisfying ζk < 0 for all k ≥ k0, then 2.8 is non-oscillatory. Proof. The statement of the lemma follows from 44, Theorem 6.16 . 3. Oscillation Constant This section is devoted to the main result of our paper. After its proof, within the concluding remarks, we formulate as a corollary the result which deals with periodic equations. This corollary is the discrete counterpart of the main result of 49 . Theorem 3.1. Let the equation Δ rkΔxk γsk k 1 k xk 1 0, k ∈ N, 3.1 where γ ∈ R and {rk} and {sk} are positive almost periodic sequences satisfying inf{rk; k ∈ N} > 0, 3.2 be arbitrarily given. Let K : [ 4M ({ r−1 k }) M {sk} ]−1 . 3.3 Then, 3.1 is oscillatory for γ > K and non-oscillatory for γ < K. Proof. At first, let us prepare several estimates which we will use to prove the theorem. Henceforth, for given γ / K, we will consider α ∈ N and θ > 0 such that