International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp 223 231 2014 http://campusmstedu/ijde Oscillatory Behavior of Third-order Difference Equations with Asynchronous Nonlinearities Xi-Lan Liu Qinghai Nationalities University Department of Mathematics and Statistics Xining, 810007, China doclanliu@163com and doclanliu2002@aliyuncom Yang-Yang Dong Qinghai Nationalities University Department of Mathematics and Statistics Xining, 810007, China Abstract This paper studied the oscillatory and asymptotic properties of the third order difference equation with asynchronous nonlinearities Some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero were established, and one example was presented to illustrate the main results AMS Subject Classifications: 39A21 Keywords: Oscillation, third-order, asynchronous 1 Introduction In this paper, we are concerned with the following third-order difference equation of the form a n 2 x n + b n x n τ1 + c n x n+τ2 + q n x α n+1 σ 1 + p n x β n+1+σ 2 = 0 11 where {a n }, {b n }, {c n }, {p n } and {q n } are real positive sequences, α β are ratios of odd positive integers, τ 1, τ 2, σ 1, σ 2 are positive integers, and n N = {n 0, n 0 + 1, }, n 0 is a nonnegative integer Let θ = max{τ 1, σ 1 } By a solution of equation 11, we mean a real sequence {x n } defined for all n n 0 θ and satisfying equation 11 for all n N A nontrivial Received December 9, 2013; Accepted February 17, 2014 Communicated by Wan-Tong Li
224 X L Liu and Y Y Dong solution {x n } is said to be nonoscillatory if it is either eventually positive or eventually negative and it is oscillatory otherwise Since the neutral type difference equations have various applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problems, there was increasing interest in studying the oscillatory behavior of this kind of equations, the references therein Recently, many authors discussed the oscillation problems of neutral difference equations [1 10] Among of these, [8] considered the m-th order mixed linear neutral difference equation m x n + ax n k bx n+σ + qx n g + px n+h = 0 12 and obtained some oscillation theorems for the oscillation of all solutions of equation 12 EThandapani et al [10] considered the third-order, mixed neutral type difference equation a n 2 x n + b n x n τ1 + c n x n+τ2 + q n x β n+1 σ 1 + p n x β n+1+σ 2 = 0, 13 and established some sufficient conditions for the oscillation of all solutions of equation 13 It is easy to see that if α = β in 11, then it becomes 13, we call it is a synchronous case, if α β, then we call 11 is an equation with asynchronous nonlinearities To the best of our knowledge, there is no paper dealing with the asynchronous case, which inspires us to discuss the oscillatory and asymptotic behavior of solutions of equation 11 2 Oscillation Results In this section, we present some oscillation criteria for equation 11 Without loss the generality, we admit that a functional inequality holds for all sufficiently large n and the following conditions hold throughout of this paper H 1 {a n } is a positive nondecreasing sequence such that n=n 0 1 a n = ; H 2 {b n } and {c n } are real sequences such that 0 < b n < b and 0 < c n < c with b + c < 1 Lemma 21 See [4] Assume A 0 B 0, α 1 Then A + B α 2 α 1 A α + B α Similar to [4], we have the following lemmas Let y n = x n + b n x n τ1 + c n x n+τ2
Third-Order Difference Equations with Asynchronous Nonlinearities 225 Lemma 22 Let {x n } be a positive solution of equation 11 Then there are only two cases hold for n n 1 N sufficiently large: 1 y n > 0, y n > 0, 2 y n > 0, a n 2 y n 0; 2 y n > 0, y n < 0, 2 y n > 0, a n 2 y n 0 Lemma 23 Let y n > 0, y n > 0, 2 y n > 0, 3 y n 0 for all n N N Then for any ξ 0, 1, and some integer N 1, the following inequalities hold y n+1 n N y n 2 ξn 2 for n N 1 N 21 Lemma 24 Suppose that {x n } be a bounded positive solution of equation 11 with the upper bound M, y n satisfies 2 of Lemma 22 If 1 q r + M β α p r = 22 holds, then lim x n = 0 n=n 0 l=n a l r=l Proof Let {x n } be a positive solution of equation 11 satisfying x n M Since y n > 0 and y n < 0, then lim y n = µ 0 exists It can be proved that µ = 0 If not, then µ > 0, and for any ɛ > 0, we have µ + ɛ > y n eventually Choose Then we have 0 < ɛ < µ1 b c b + c x n = y n b n x n τ1 c n x n+τ2 > µ b+cy n τ1 > µ b+cµ+ɛ = dµ+ɛ > dy n, where d = µ b + cµ + ɛ µ + ɛ > 0 Further, a n y n q n d α y α n+1 σ 1 p n d β y β n+1 σ 2 d α q n + M β α p n y α n+1 τ 1 Summing the above inequality from n to, and using the relation y n µ, we obtain 2 y n dµ α 1 q r + M β α p r 23 a n r=n Summing the two sides of 23 from n to, we have y n dµ α 1 a s=n s r=s q r + M β α p r
226 X L Liu and Y Y Dong Summing from n 1 to leads to y n1 dµ α n=n 1 s=n 1 a s q r + M β α p r, r=s which is a contradiction to 22 Then µ = 0, which together with the inequality 0 < x n < y n implies that lim x n = 0 The proof is complete Let Q n = min{q n, q n τ1, q n+τ2 }, P n = min{p n, p n τ1, p n+τ2 } Theorem 25 Suppose that {x n } be a bounded positive solution of equation 11 with the upper bound M, and that condition 22 holds, σ 1 τ 1 and α, β 1 If there exists a positive real sequence {γ n } and an integer N 1 N such that for some ξ 0, 1 and δ > 0 lim sup γ s δ 4 α 1 ξn σ 1 α 2 α W s 1 + bα + 4 2 β 1 a s σ1 γ s 2 = 24 holds, then every such solution {x n } of equation 11 oscillates or lim x n = 0, where W s = Q s + M β α P s Proof Let {x n } be a nonoscillatory solution of equation 11, and x n M Without loss of the generality, assume that there exists an integer N n 0 such that x n, x n σ1, x n+σ2, x n τ1, x n+τ2 0, M] for all n > N, we have a n 2 y n + q n x α n+1 σ 1 + p n x β n+1+σ 2 + b α a n τ1 2 y n τ1 +b α q n τ1 x n+1 τ1 σ 1 + b α p n τ1 x β n+1 τ 1 +σ 2 + 2 β 1 a n+τ 2 2 y n+τ2 + 2 β 1 q n+τ 2 x α n+1+τ 2 σ 1 + 2 β 1 p n+τ 2 x β n+1+τ 2 σ 2 = 0 25 By Lemma 21 and β α in 25, we have a n 2 y n + b α a n τ1 2 y n τ1 + γ s 2 β 1 a n+τ 2 2 y n+τ2 + Q n 4 α 1 yα n+1 σ 1 + M β α P n 4 α 1 y α n+1+σ 2 0 26 By Lemma 22, there are two cases for y n to be considered Assume that case 1 holds for all n N 1 N It follows from y n > 0 that y n+σ2 > y n σ1 26 tells us a n 2 y n + b α a n τ1 2 y n τ1 + 2 β 1 a n+τ 2 2 y n+τ2 + R n 4 α 1 yα n+1 σ 1 0 27
Third-Order Difference Equations with Asynchronous Nonlinearities 227 Define ν 1 n = γ n a n 2 y n, n N 1 28 Then ν 1 n > 0 for n N 1 From 28, we see that ν 1 n = γ n a n 2 y n 2 y n σ1 ν 1 n + 1 + γ n ν 1 n + 1 γ n+1 By 11, we know that a n 2 z n = q n x α n+1 σ 1 p n x β n+1+σ 2 < 0, 29 and a n σ1 2 y n σ1 a n+1 2 y n+1 Thus from 28, we have Define ν 1 n γ n a n 2 y n ν 2 1 n + 1 ν 1 n + 1 + γ n n 210 γ n+1 γn+1a 2 n σ1 ν 2 n = γ n a n τ1 2 y n τ1, n N 1 211 Then ν 2 n > 0 for n > N 1, which together with 211 yields ν 2 n = γ n a n τ1 2 y n τ1 2 y n σ1 ν 2 n + 1 + γ n ν 2 n + 1 γ n+1 Note that σ 1 > τ 1 By 29, we find a n σ1 2 y n σ1 211, we obtain a n+1 τ1 2 y n+1 τ1 Hence by ν 2 n γ n γ n+1 ν 2 n + 1 + γ n a n τ1 2 y n τ1 n w 2 2n + 1 γ 2 n+1a n σ1 212 Similarly, define Then we have ν 3 n = γ n a n+τ2 2 y n+τ2, n N 1 213 ν 3 n γ n γ n+1 ν 3 n + 1 + γ n a n+τ2 2 y n+τ2 n ν 2 3n + 1 γ 2 n+1a n σ1 214
228 X L Liu and Y Y Dong Therefore 210, 212 and 214 show that ν 1 n + b α ν 2 n + 2 β 1 ν 3n R n yn+1 σ α γ 1 n + γ n ν 1n 2 + 1 4 α 1 n γ n+1 γn+1a 2 n σ1 +b α γ n ν 2 n + 1 ν 2n 2 + 1 n γ n+1 γn+1a 2 n σ1 + γ n ν 3 n + 1 ν 3n 2 + 1 2 β 1 n 215 γ n+1 γn+1a 2 n σ1 On the other hand, since {a n } is nondecreasing and 2 y n > 0 for n N 1 Then we have 3 y n < 0 for n N 1 By Lemma 23 for any ξ 0, 1, and n sufficiently large y n+1 σ1 ξn σ 1 216 2 Note that y n > 0, y n > 0 and 2 y n > 0 for n N 1 Thus y n = y N1 + y s n N 1 y N1 δn 2 217 for some δ > 0 and n sufficiently large From 216, 217 and β 1 we have yn+1 σ α 1 λα 1 ξn σ 1 α Combining the last inequality with 215, and completing 2 α the square on the right-hand side of the resulting inequality, we obtain ν 1 n + b α ν 2 n + 2 ν 3n β 1 γ n δ ξn σ 1 α 4 α 1 s=n 2 2 α [Q s + M β α P s ] Summing the last inequality from N 2 N 1 to n 1 implies that γ s δ ξn σ 1 α 4 α 1 + 1 + bα + a 2 β 1 n σ1 γ n 2 4γ n [Q 2 α s + M β α P s ] + 1 + bα + 2 β 1 a s σ1 γ s 2 4γ s νn 2 + b α ν 2 N 2 + 2 ν 3N β 1 2 218 Taking lim sup in the last inequality yields a contradiction to 24 The case 2 of Lemma 22 can be proved similarly, here it is omitted The proof is complete
Third-Order Difference Equations with Asynchronous Nonlinearities 229 Let γ n = n and α = β = 1 Then the following corollary is easily obtained Corollary 26 Suppose that {x n } be a bounded positive solution of equation 11 with the upper bound M, and α = 1 Assume that condition 24 holds and σ 1 > τ 1 If there is an integer N 1 N such that for some ξ 0, 1 and δ > 0 lim sup δ α 1 ξn σ 1 α s W 4 2 α s 1 + b + c a s σ1 = 4s holds, then every such solution {x n } of equation 11 oscillates or lim x n = 0, where W s = Q s + M β α P s Theorem 27 Suppose that {x n } be a bounded positive solution of equation 11 with the upper bound M Assume that condition 24 holds, σ 1 τ 1, and α β 1 If there exists a positive real sequence γ n, and an integer N 1 N such that for some ξ 0, 1 and δ > 0 lim sup γ s δ 4 α 1 ξn σ 1 α 2 α W s 1 + bα + 2 β 1 a s τ1 γ s 2 4γ s = 219 holds, then every such solution {x n } of equation 11 oscillates or lim x n = 0, where W s = Q s + M β α P s Proof Similar to Theorem 25, 26 is still true By Lemma 22, there are two cases for y n Assume that case 1 holds for all n N 1 N Then 27 holds In like manner, define ν 1 n = γ n a n 2 y n y n τ1, n N 1, and We have ν 2 n = γ n a n τ1 2 y n τ1 y n τ1, n N 1, ν 3 n = γ n a n+τ2 2 y n+τ2 y n τ1, n N 1 ν 1 n + b α ν 2 n + 2 β 1 ν 3n R n zn+1 τ α γ 1 n + γ n w 1n 2 + 1 4 α 1 n y n τ1 γ n+1 γn+1a 2 n τ1 +b α γ n ν 2 n + 1 w 2n 2 + 1 n γ n+1 γn+1a 2 n τ1 + γ n ν 3 n + 1 w 3n 2 + 1 2 β 1 n 220 γ n+1 γn+1a 2 n τ1
230 X L Liu and Y Y Dong On the other hand, we have for any ξ 0, 1 y n+1 σ1 = y n+1 σ 1 y n σ 1 ξn σ 1, y n τ1 y n τ1 2 for all n N 2 As in the proof of Theorem 25, we obtain ν 1 n + b α ν 2 n + 2 ν 3n γ β 1 n ω n + 1 + bα + a 2 β 1 n τ1 γ n 2 4γ n Summing the last inequality from N 2 to n 1, we obtain γ s δ ξn σ 1 α 4 α 1 [Q 2 α s + M β α P s ] + 1 + bα + a 2 β 1 s τ1 γ s 2 4γ s s=n 2 ν 1 N 2 + b α ν 2 N 2 + 2 ν 3N β 1 2 221 Taking lim sup on both sides of the last inequality yields a contradiction to 219 The case 2 can be proved similarly The proof is completed Let γ n = n and α = β = 1 Then, we can obtain the following result Corollary 28 Suppose that {x n } be a bounded positive solution of equation 11 with the upper bound M Assume that condition 22 holds and τ 1 σ 1 If lim sup s δ4 ξn σ 1 α α 1 W 2 α s 1 + b + c a s τ1 = 4s holds for all sufficiently large N, then every such solution {x n } of equation 11 oscillates or lim x n = 0, where W s = Q s + M β α P s 3 Example Next we present an example to illustrate the main results Example 31 Consider the third order difference equation 3 x n + 1 3 x n 1 + 1 3 x n+1 + 16n 10240 x5 n 2 + 31 30 4n x 3 n+1 = 0 31 Let a n = 1, b n = c n = 1 3, q n = 16n 10240, p n = 31 30 4n, α = β = 1, τ 1 = 1, τ 2 = 1, σ 1 = 3, σ 2 = 0 Take γ = 1 Then condition 22 holds On the other hand, condition 24 also holds Therefore by Theorem 25, every solution {x n } with x n 1 of the equation 31 oscillates or lim x n = 0 It is easy to see that {x n } = {2 n } is a solution of the equation 31
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