OscillationofNonlinearFirstOrderNeutral Di erenceequations

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1 AppliedMathematics E-Notes, 1(2001), 5-10 c Availablefreeatmirrorsites ofhttp://math2.math.nthu.edu.tw/»amen/ OscillationofNonlinearFirstOrderNeutral Di erenceequations YingGaoandGuangZhang yz Received1June2000 Abstract In this note, oscillation criteria are obtained for a class of nonlinear neutral di erence equations. Oscillatory behaviors of neutral di erence equations of the form (x n p n x ) + q n x ¾ = 0; n = 0; 1; ::: (1) have been explored to some extent in a number of studies [1-3]. However, relatively few results are known for their nonlinear extensions of the form (x n p n x ) + q n max x s = 0; n = 0; 1; ::: : (2) Nonlinear functional equations involving the maximum function are important since they appear naturally in automatic control theory, see e.g. Popov [4]. Some of the qualitative theory of these equations has been developed recently, see e.g. [5-8]. In this note, we will consider their oscillatory behaviors. We will assume that and ¾ are positive integers, that fp n g and fq n g are nonnegative real sequences, and fq n g has a positive subsequence. Let ¹ = max f ; ¾g. By a solution of (1) or (2), we mean a real sequence fx n g 1 n= ¹ which satis es (1) or (2) respectively forn 0. Such asolution issaid tobe oscillatory if it is neithereventually positive nor eventually negative. It is easily seen that fx n g is an eventually positive solution of equation (1) if, and only if, f x n g is its eventually negative solution. However, such a property is not valid for equation (2). Instead, fx n g is an eventually positive solution of (2) if, and only if, f x n g is an eventually negative solution of the equation (x n p n x ) + q n min x s = 0; n = 0; 1;::: : (3) Thus, all solutions of (2) are oscillatory if, and only if, both (2) and (3) do not have any eventually positive solutions. MathematicsSubjectClassi cations: 39A10 y DepartmentofMathematics,YanbeiNormalCollege,Datong,Shanxi037000,P.R.China z SupportedbytheScienceFoundationofShanxiProvinceofChina 5

2 6 Di erence Equations LEMMA 1. Assume that there exists a nonnegative integer N 0 such that p N+j 1 for j = 0;1; 2; ::: : Let fx n g be an eventually positive solution of (2) or (3). Set y n = x n p n x (4) for all large n: Then y n > 0 eventually. The proof is similar to the proof of Lemma 1 in [2]. THEOREM 1. Assume that there exists a nonnegative integer N 0 such that p N+j 1 for j = 0; 1; 2;::: : Suppose further that either p n > 0 or q n does not vanish identically over any set of consecutive integers of the form fa; a + 1; ::; a + ¾g. Then equation (2) has an eventually positive solution if, and only if, the following functional inequality (x n p n x ) + q n max x s 0; n = 0; 1;::: (5) hasan eventuallypositivesolution; and equation (3) hasan eventually positive solution if, and only if, the functional inequality has an eventually positive solution. (x n p n x ) + q n min x s 0; n = 0; 1;::: (6) The proof of Theorem 1 is similar to that of Theorem 1 in [2], and is thus omitted. THEOREM 2. Assume that there exists a nonnegative integer N 0 such that p N+j 1 for j = 0; 1; 2; ::: : Suppose further that there exists some positive integer T such that the functional inequality y n + q n min p s i y 0 does not have any eventually positive solutions. Then all solutions of (2) oscillate. PROOF. If fx n g is an eventually positive solution of (2), then y n 0 and y n = x n p n x > 0 eventually. Thus, Hence, max x s x n = y n + p n x = y n + p n y + p n p x 2 = Y1 ::: y n + p n y + ::: + p i y (i+1) T X 1 max p i y : p s i y s i=0 max p s i y :

3 Y. Gao and G. Zhang 7 Substituting the last inequality into (2), we have y n + q n max p s i y 0; which is a contradiction. If fz n g is an eventually negative solution of (2), then x n = z n is an eventually positive solution of equation (3). Similarly, we have y n + q n min p s i y y n + q n This is also a contradiction. The proof is complete. For the equation we have the following result. max p s i y 0; (x n x ) + q n max x s = 0; n = 0; 1;:::; (7) THEOREM 3. Equation (7) has nonoscillatory solutions if, and only if, also has nonoscillatory solutions. 2 z q nz n = 0 (8) PROOF. Assume that fx n g is an eventually positive solution of (7). In view of Lemma 1, we see that there is an integer N 1 such that x > 0; y n = x n x > 0; and y n 0 for n N 1. Set m = minfx N1 ;x N1 +1; :::; x N1 1g. For n N 1 + ; there exists a positive integer k such that and N 1 + k n < N 1 + (k + 1) k k x n = x k + y j m + y j : j=0 Furthermore, since y n 0 for n N 1, and since n k + N = N 2, k 1 j=0 X y j = y (k 1) + y (k 2) + ::: + y n j=0 (y (k 1) + y (k 1) +1 + ::: + y (k 2) 1 ) +(y (k 2) + y (k 2) +1 + ::: + y (k 3) 1 +::: + (y n + y n+1 + ::: + y n+ 1 ) nx y i ;

4 8 Di erence Equations we have Let then z n > 0 and x n m + 1 nx y i : z n = m + 1 nx y i ; 2 z 1 + q n z n = y n + q n à m + 1 à = y n + q n max nx y i! m + 1 y n + q n max x s = 0: sx y i In view of Lemma 1, we know that equation (8) has an eventually positive solution. If fx n g is an eventually positive solution of the equation we can also prove that (x n x ) + q n min x s = 0; n = 0; 1; :::; (9) 2 z 1 + q n z ¾ 0 has an eventually positive solution. In view of Theorem 2 in [3], we know that equation (8) has an eventually positive solution. We now show that the converse holds. Let fz n g be an eventually positive solution of (8), then it is positive and concave for all large n, so that f z n g is eventually positive and nonincreasing. Thus it is easy to see that there exists a su± ciently large integer N such that 0 < z 1 z for n N. Let 8 < z 1 n N H n = (n N + ) z N 1 N n < N ; : 0 n < N and let x n = z N z N 1 + 1X H i ; n 0: In view of the de nition of H n, it is clear that 0 < x n < 1 for all n 0, that and that max fx N ; x N +1 ; :::; x N 1 g = z N z N 1 + ( 1) z N 1 z N ; i=0 x n x = H n = z 1 for n N. For any n which satis es N n N + 1, we see that x n = z 1 + x i= z i + x i=n! z i + z N = z n :

5 Y. Gao and G. Zhang 9 By induction, it is easy to prove that for any n which satis es N+k n < N+(k+1) where k = 0; 1;2; :::; x n z n is still valid. Thus (x n x ) + q n max x s 2 z 1 + q n z n = 0 as desired. Similarly, we have also Thus, we have x n The proof is complete. i=n z i + z N +¾ i=n z i + z N = z n+¾ : (x n x ) + q n min x s 2 z 1 + q n = 2 z 1 + q n z n = 0: As an important corollary, the equation (x n x ) + min z s+¾ (n + 1) 2 min x s = 0; n = 0; 1;2; :::: in view of Theorem 3, is oscillatory if, and only if, = > 1=4: We make several additional remarks. Let us say that a solution of (1) or (2) is strongly oscillatory if for any given nonnegative integer N; there is a corresponding integer m N such that x m x m+1 < 0: Assume that q n > 0 for n 0: In this case, let fx n g be a nonnegative solution of (1) or (2), and let y n be de ned by (4). We have already seen that y n > 0 and y n 0 eventually. Note that x n y n for all large n. THEOREM 4. Assume that there exists a nonnegative integer N 0 such that p N+j 1 for j = 0; 1; 2; ::: : Suppose further that q n > 0 for n 0: Then every nontrivial solution of equation (1) is strongly oscillatory if, and only if, the inequality (x n p n x ) + q n x ¾ 0; : n = 0; 1; ::: does not have any eventually nonnegative solutions, and equation (2) has anonnegative solution if, and only if, (5) has a nonnegative solution. THEOREM 5. Assume that there exists a nonnegative integer N 0 such that p N+j 1 for j = 0; 1; 2;::: : Assume further that q n > 0 for n 0; If there exists some integer T such that the functional inequality y n + q n max p s i y 0 does not have an eventually positive solution, then equation (2) does not have an eventually nonnegative solution. If there exists some integert such that the functional inequality y n + q n p i y 0

6 10 Di erence Equations does not have an eventually positive solution, then every solution of equation (1) is strongly oscillatory. THEOREM 6. Assume that q n > 0 for n 0: Then all solutions of the equation (x n x ) + q n x ¾ = 0; are strongly oscillatory if, and only if, equation (8) is oscillatory; and equation (7) has an eventually nonnegative solution if, and only if, (8) has an eventually positive solution. The proofs of Theorem 4, Theorem 5 and Theorem 6 are similar to the proofs of Theorem 1, Theorem 2 and Theorem 3 respectively. They will be omitted. Results analogous to Theorem 4, Theorem 5 and Theorem 6 are not valid for equation (3). For example, the sequence fsin(n¼=2) 1g is a nonpositive solution of the equation where fq n g is any real sequence. (x n x 4 ) + q n max s2[ 6;n] x s = 0; References [1] M. P. Chen and B. G. Zhang, The existence of bounded positive solutions of delay di erence equations, PanAmerican Math. J., 3(1)(1993), [2] G. Zhang and S. S. Cheng, Positive solutions of a nonlinear neutral di erence equation, Nonlinear Anal., 28(4)(1997), [3] G. Zhang and S. S. Cheng, Note on a discrete Emden-Fowler equation, PanAmerican Math. J., 9(3)(1999), [4] E. P. Povov, Automatic Regulation and Control, Nauka, Moscow, [5] G. Zhang and S. S. Cheng, Asymptotic dichotomy for nonoscillatory solutions of a nonlinear di erence equation, Applications Math., 25(4)(1999), [6] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di erence Equations of Higher Order with Applications, Mathematics and Applications, Vol. 256, Kluwer Academic Publishers, [7] G. Ladas, Open problems and conjectures, J. Di. Eqs. Appl., 1(1995), [8] G. Ladas, Open problems and conjectures, J. Di. Eqs. Appl., 2(1996),