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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 209309, 13 pages doi:10.1155/2010/209309 Research Article Oscillatory Criteria for the Two-Dimensional Difference Systems Jin-Fa Cheng1 and Yu-Ming Chu2 1 2 Department of Mathematics, Xiamen University, Xiamen 361005, China Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China Correspondence should be addressed to Yu-Ming Chu, [email protected] Received 5 April 2010; Revised 3 June 2010; Accepted 17 June 2010 Academic Editor: Alberto Cabada Copyright q 2010 J.-F. Cheng and Y.-M. Chu. 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. We establish some necessary and suﬃcient conditions for oscillation of the solutions of the following two-dimensional diﬀerence system: Δxn fn, yn , Δyn −gn, xn , where fn, u and gn, u are strongly superlinear or sublinear functions. 1. Introduction We consider the following two-dimensional nonlinear diﬀerence system as follows: Δxn f n, yn , Δyn −gn, xn , 1.1 where Δxn xn1 − xn , Δyn yn1 − yn , fn, u and gn, u are strongly superlinear or sublinear functions. Now we pose some conditions on functions f and g: H1: ufn, u > 0 and ugn, u > 0 for u / 0; H2: fn, u and gn, u are continuous real-valued functions, and nondecreasing with respect to u; 2 Journal of Inequalities and Applications H3: it is ∞ fn, ±c ±∞ 1.2 nn0 for each c > 0. Definition 1.1. Suppose that f, g : N × R → R are real-valued functions. α and β are the quotients of positive odd numbers. 1 f and g are said to be strongly superlinear if there exist constants α > 0 and β > 0 with αβ > 1, such that fn, u/|u|α sgn u and gn, u/|u|β sgn u are nondecreasing with respect to |u| for each fixed n ∈ N. 2 f and g are said to be strongly sublinear if there exist constants α > 0 and β > 0 with αβ < 1, such that fn, u/|u|α sgn u and gn, u/|u|β sgn u are nonincreasing with respect to |u| for each fixed n ∈ N. The solutions of 1.1 are said to be nonoscillatory if {xn } or {yn } is eventually positive or negative. Otherwise the solutions are called oscillatory. β Some oscillation results for the diﬀerence system 1.1 in the case of gn, xn an xn with an > 0 have been established by many authors. In particular, if fn, yn bn yn and bn > 0, then the diﬀerence system 1.1 is reduced to the well-known second-order nonlinear diﬀerence equation: 1 β Δ Δxn an xn 0. bn 1.3 Also, if bn 1, then 1.3 becomes β 1.4 Δ2 xn an xn 0. Furthermore, if fn, yn bn ynα and α is a ratio of odd positive integers, then 1.1 reduces to the well-known quasilinear diﬀerence equation: Δ 1, 2 . 1 bn1/α Δxn 1/α β an xn 0. 1.5 For 1.4, the following well-known Theorem A was established by Hooker and Patula Journal of Inequalities and Applications 3 Theorem A. For 1.4, the following statements are true. (1) If 0 < β < 1, then every solution of 1.4 oscillates if and only if ∞ nβ an ∞. 1.6 n1 (2) If β > 1, then every solution of 1.4 oscillates if and only if ∞ nan ∞. 1.7 n1 For 1.3, if one denotes Bn n−1 bs 1.8 s0 and assumes that lim Bn n→∞ ∞ bs ∞, 1.9 s0 then the following theorem is proved in [3]. Theorem B. If 1.9 holds, then the following statements are true. (1) If 0 < β < 1, then every solution of 1.3 oscillates if and only if ∞ β Bn an ∞. 1.10 n1 (2) If β > 1, then every solution of 1.3 oscillates if and only if ∞ Bn an ∞. 1.11 n1 The problem of oscillation of second-order nonlinear diﬀerence equations has attracted the attention of many mathematicians because of its physical applications 2, 4 . For some results regarding the growth of solutions of some equations related to the above mentioned see book 5 , as well as the following papers 6–8 . It is an interesting problem to extend oscillation criteria for second-order nonlinear diﬀerence equations to the case of nonlinear two-dimensional diﬀerence systems since such systems include, in particular, the secondorder nonlinear, half-linear, and quasilinear diﬀerence equations that are the special cases of the nonlinear two-dimensional diﬀerence systems 5, 9, 10 . The main purpose of this paper is to establish some necessary and suﬃcient conditions for oscillation of the nonlinear two-dimensional diﬀerence systems. 4 Journal of Inequalities and Applications 2. Main Results In order to establish our main results, we need the following lemma. Lemma 2.1. Suppose that conditions H1–(H3 are satisfied. If {xn } and {yn } are nonoscillatory solutions of 1.1 for n > n0 , then sgn xn sgn yn . 2.1 Proof. Without loss of generality, we assume that xn is eventually positive; that is, xn > 0 for n > n0 > 0. From 1.1, we clearly see that Δyn < 0, then we know that either yn > 0 or yn < 0 eventually holds. If yn < yN1 < 0 for n > N1 > n0 , then we have Δxn f n, yn ≤ f n, yN < 0, 2.2 summing up from N1 to n, and by 1.2 of H3, we get xn − xN1 ≤ n−1 f s, yN −→ −∞ n −→ ∞. 2.3 sN1 This contradiction completes the proof of the lemma. Theorem 2.2. If f and g are strongly sublinear (i.e., 0 < αβ < 1), then a necessary and suﬃcient condition for 1.1 to oscillate is that ∞ nn1 g n, n−1 fs, c ∞ 2.4 sn0 for every c > 0, where n1 > n0 . Proof. Suﬃciency. If 1.1 has a nonoscillatory solution xn , then without loss of generality, we assume that xn is eventually positive. Then, by Lemma 2.1, for n0 suﬃciently large, yn > 0, Δxn > 0, Δyn < 0, for n ≥ n0 . 2.5 Since {yn } is decreasing, hence there exists c > 0 such that yn ≤ yN ≤ c, for N ≥ n0 . 2.6 Summing up Δxn f n, yn 2.7 Journal of Inequalities and Applications 5 from s n0 to n − 1, we obtain xn − xn0 n−1 n−1 f s, y n−1 fs, c s α f s, ys ≥ yn ≥ ynα , α y cα s sn0 sn0 sn0 n−1 c−α ynα xn ≥ 2.8 fs, c, sn0 and so gn, xn ≥ g n, c−α ynα n−1 2.9 fs, c . sn0 Therefore −Δyn gn, xn ≥ g n, c−α ynα n−1 fs, c . 2.10 sn0 Since n−1 c−α ynα fs, c ≤ sn0 n−1 2.11 fs, c, sn0 we have −Δyn ≥ g n, c−α ynα n−1 fs, c sn0 β n−1 g n, c−α ynα n−1 sn0 fs, c −α α c yn fs, c β sn0 c−α ynα n−1 fs, c sn0 β n−1 g n, n−1 sn0 fs, c −α α c yn ≥ fs, c β n−1 sn0 fs, c sn0 2.12 n−1 αβ yn c−αβ g n, fs, c , sn0 n−1 Δyn −αβ − αβ ≥ c g n, fs, c , yn sn0 and let n1 > n0 , we have n−1 in1 Δyi − αβ yi ≥c −αβ n−1 in1 g i, i−1 sn0 fs, c . 2.13 6 Journal of Inequalities and Applications From yn yn1 Δyn 1 1 du αβ yn − yn1 ≥ − αβ , αβ u ξ yn yn1 < ξ < yn , 2.14 ∞ i−1 ∞ Δyi 1 −αβ du ≥ − αβ ≥ c g i, fs, c , αβ yi1 u yi sn0 in1 in1 yn ∞ i−1 1 1 αβ g i, fs, c ≤ c du < ∞, uαβ c sn0 in1 2.15 we get ∞ yi in1 which leads to a contradiction. Necessity. If ∞ g n, nn1 n−1 < ∞ fs, c 2.16 sn0 for some c > 0, then there exist M > n0 > 0 and c/2 > d > 0, such that ∞ g n, n−1 fs, c 2.17 < d. sn0 nM Let X be the Banach space of all the real-valued sequences {xn } with the norm |xn | , x sup n−1 n≥M sn0 fs, c 2.18 let Ψ be the subset of X defined by Ψ {xn } ∈ X : n−1 fs, d ≤ xn ≤ sn0 n−1 fs, 2d 2.19 sn0 and let F : Ψ → X be the operator defined by Fxn n−1 sn0 ∞ s, d gi, xi . f is 2.20 Journal of Inequalities and Applications 7 Then the mapping F satisfies the assumptions of Knaster’s fixed-point theorem see 11, page 8 : F maps Ψ into itself and F is increasing. The latter statement is easy to see, and the former statement follows from Fxn ≥ n−1 fs, d, nn0 Fxn n−1 ∞ s, d gi, xi f sn0 ≤ n−1 ≤ n−1 ∞ i−1 s, d g i, fs, 2d f sn0 is is 2.21 sn0 ∞ i−1 n−1 ≤ s, d g i, fs, c fs, 2d f sn0 sn0 is sn0 for any {xn } ∈ Ψ. From Knaster’s fixed-point theorem, we know that there exists {xn } ∈ Ψ such that xn Fxn . Let yn d ∞ gs, xs , 2.22 sn then limn → ∞ yn d and Δyn −gn, xn . On the other hand, we have xn Fxn n−1 f i, yi . 2.23 in0 Then by 1.2 and the continuity of function f, we have that limn → ∞ xn ∞ and Δxn fn, xn , which leads to a contradiction and the proof of Theorem 2.2 is completed. Example 2.3. Considering the diﬀerence system, Δxn 2n 11/3 yn1/3 , Δyn − 2n 3 x5/3 , n 1n 2 n 2.24 n ≥ n0 . Here α 1/3, β 5/3, and f and g are strongly sublinear. It is easy to verify that the conditions of Theorem 2.2 are satisfied and hence all solutions are oscillatory. In fact, we clearly see that the sequence {xn , yn } {−1n , −1n1 /n 1} is such a solution for the diﬀerence system. Example 2.4. Considering the diﬀerence system, Δxn n n1 1/3 yn1/3 , 1 Δyn − 5/3 x5/3 , nn 1 n n 1 2.25 n ≥ n0 . 8 Journal of Inequalities and Applications Here α 1/3, β 5/3, and f and g are strong sublinear. We clearly see that the conditions of Theorem 2.2 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence {xn , yn } {n, n 1/n} is such a solution. Theorem 2.5. If f and g are strongly superlinear (i.e., αβ > 1), then a necessary and suﬃcient condition for 1.1 to oscillate is that ∞ f n, ∞ gs, c ∞ 2.26 sn n1 for every c > 0. Proof. Suﬃciency. If 1.1 has a nonoscillatory solution xn , then without loss of generality, we may assume that xn is eventually positive. Then by Lemma 2.1, we have for N1 suﬃciently large, Δxn > 0, xn > 0, Δyn < 0, for n > N1 . 2.27 Since {xn } is increasing, hence there exists c > 0 such that xn ≥ yN ≥ c, 2.28 for N > n0 . Summing up Δyn −gn, xn 2.29 from s n to ∞, we have −yn ≤ y∞ − yn − ∞ gs, xs , sn yn ≥ ∞ sn gs, xs ≥ ∞ gs, xs sn1 β xs β xs ≥ β xn1 ∞ ∞ gs, c −β β c x gs, c. n1 cβ sn1 sn1 2.30 Therefore Δxn f n, yn ≥ f β n, c−β xn1 ∞ gs, c . sn1 2.31 Journal of Inequalities and Applications β From xn ≥ c and c−β xn1 ∞ sn1 gs, c ≥ 9 ∞ sn1 gs, c, we get β α ∞ gs, c f n, c−β xn1 ∞ sn1 −β β gs, c Δxn ≥ c xn1 α β sn1 c−β xn1 ∞ gs, c sn1 α ∞ f n, ∞ sn1 gs, c −β β c xn1 ≥ ∞ gs, c α sn1 sn1 gs, c ∞ αβ −αβ xn1 c f n, gs, c , Δxn αβ xn1 ∞ Δxi in αβ xi1 sn1 ≥c −αβ f ∞ n, 2.32 gs, c , sn1 ≥c −αβ ∞ f ∞ i, in gs, c . si1 But xn1 1 1 Δxn du αβ xn1 − xn ≥ , xn < ξ < xn1 , αβ u ξ xn1 αβ xn ∞ xs1 ∞ ∞ ∞ 1 Δxs −αβ du ≥ ≥c f s, gt, c . αβ αβ sn xs u sn xs1 sn ts1 2.33 Therefore ∞ f sN1 s, ∞ ≤c gt, c ∞ αβ xN1 ts1 1 du < ∞, uαβ 2.34 which leads to a contradiction. Necessity. If ∞ f n, ∞ < ∞ gs, c 2.35 sn n1 for some c > 0, then there exists M > 0 large enough, such that ∞ nM f n, ∞ sn gs, c < c . 2 2.36 10 Journal of Inequalities and Applications Let X be the set of all bounded and real-valued sequences {xn } with the norm x sup|xn | 2.37 n≥M and Ψ be the subset of X defined by c Ψ {xn } ∈ X : ≤ xn ≤ c , 2 2.38 then Ψ is a bounded, convex, and closed subset of X. Let F : Ψ → X be the operator defined by ∞ Fxn c − f i, in ∞ gs, xs . 2.39 si Then F maps Ψ into Ψ. In fact, if {xn } ∈ Ψ, then c ≥ Fxn ≥ c − ≥c− ∞ iM ∞ i, gs, c f in si ∞ c i, gs, c ≥ . 2 si f ∞ 2.40 j Next, we show that F is continuous. Let {xn } be a convergent sequence in Ψ such that j limi → ∞ xn − xn 0, then from that Ψ is closed {xn } ∈ Ψ and the definition of F, we have ∞ ∞ ∞ ∞ j j − f i, gs, xs − Fxn f i, g s, xs Fx in n si in si ∞ ∞ ∞ j ≤ − f i, gs, xs . f i, g s, xs in si si 2.41 ∞ ∞ ∞ Since ∞ in fi, si gs, xs < in fi, si gs, c < ∞, now from the continuity of f and g together with the well-known Lebesgue’s dominated convergence theorem see 11, page 263 , we know that limj → ∞ Fxj n − Fxn 0 for xj − x → 0. Journal of Inequalities and Applications 11 Finally, we show that FΨ is precompact. Let {xm } ∈ Ψ, {xn } ∈ Ψ, then for large enough m > n we have ∞ ∞ ∞ ∞ |Fxm − Fxn | f i, gs, xs − f i, gs, xs im si in si m ∞ f i, gs, xs in si ≤ m 2.42 ∞ i, gs, c < ε f in si for any ε > 0. From Schauder’s fixed-point theorem see 11 , we know that there exists {xn } ∈ Ψ such that xn Fxn . Let yn ∞ gs, xs , 2.43 sn then limn → ∞ yn 0 and Δyn −gn, xn . On the other hand, we have xn Fxn c − ∞ f s, ys . 2.44 sn Therefore, limn → ∞ xn c and Δxn fn, yn , which leads to a contradiction. The proof of Theorem 2.5 is completed. Example 2.6. Considering the diﬀerence system, Δxn 23n1 yn3 , Δyn − 3 3 x , 2n n n ≥ n0 . 2.45 Here α 3, β 3, and f and g are strongly suplinear. We clearly see that the conditions of Theorem 2.5 are satisfied and hence all solutions are oscillatory. In fact, the sequence {xn , yn } {−1n , −1n1 /2n } is such a solution. Example 2.7. Considering the diﬀerence system, Δxn n n1 2/3 yn2/3 , 1 Δyn − 5/3 x5/3 , nn 1 n n 1 2.46 n ≥ n0 . 12 Journal of Inequalities and Applications Here α 2/3, β 5/3, and f and g are strong sublinear. However, it is easy to see that the conditions of Theorem 2.5 are not satisfied and hence there exists a nonoscillatory solution. In fact, the sequence {xn , yn } {n, n 1/n} is such a solution. β If we set fn, yn bn ynα , gn, xn an xn , then the diﬀerence system 1.1 is reduced to 1.5. From Theorems 2.2 and 2.5, we get the following results for 1.5. Corollary 2.8. If 0 < αβ < 1, then every solution of 1.5 oscillation if and only if ∞ nn1 an n−1 β bs ∞, 2.47 sn0 where n1 > n0 . Corollary 2.9. If αβ > 1, then every solution of 1.5 oscillation if and only if α ∞ ∞ bn as ∞. n1 2.48 sn Remark 2.10. It is easy to see that Theorems A and B are the special cases of our Corollaries 2.8 and 2.9, respectively. Acknowledgment The authors wish to thank the anonymous referees for their very careful reading of the paper and fruitful comments and suggestions. References 1 J. W. Hooker and W. T. Patula, “A second-order nonlinear diﬀerence equation: oscillation and asymptotic behavior,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 9–29, 1983. 2 V. L. Kocić and G. 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