Oscillation of second-order nonlinear difference equations with sublinear neutral term

Transcription

1 Mathematica Moravica Vol. 23, No. (209), 0 Oscillation of second-order nonlinear difference equations with sublinear neutral term Martin Bohner, Hassan A. El-Morshedy, Said R. Grace and Ilgin Sağer Abstract. We establish some new criteria for the oscillation of secondorder nonlinear difference equations with a sublinear neutral term. This is accomplished by reducing the involved nonlinear equation to a linear inequality.. Introduction This paper deals with oscillatory behavior of all solutions of nonlinear second-order difference equations with a sublinear neutral term of the form () ( a n ( x n + p n x α n k)) + qn x β n+ m = 0. We assume that (H ) 0 < α < and β > 0 are ratios of positive odd integers, (H 2 ) {a n }, {p n }, {q n }, n n 0, are positive real sequences, lim p n = 0 and s=n 0 a s <, (H 3 ) k N and m N 0. Let ξ = max{k, m }. By a solution of (), we mean a real sequence {x n } defined for all n n 0 ξ that satisfies () for n n 0. A solution of () is said to be oscillatory if its terms are neither eventually positive nor eventually negative, and otherwise it is called nonoscillatory. Equation () is said to be oscillatory if all its solutions are oscillatory. In recent years, there has been a great interest in establishing criteria for the oscillation and asymptotic behavior of solutions of various classes of second-order difference equations, see, 2, 4, 9 2, 5, 8, 20, 2, 24] and the references cited therein. However, it seems that there are no known results regarding the oscillation of second-order difference equations with positive 200 Mathematics Subject Classification. Primary: 39A0, 39A2. Key words and phrases. Neutral term, nonlinear difference equation, oscillation. Full paper. Received 28 November 208, accepted 28 December 209, available online 7 January 209. c 208 Mathematica Moravica

2 2 Nonlinear difference equations with sublinear neutral term sublinear neutral term. More exactly, the existing literature does not provide any criteria which ensure oscillation of all solutions of (). In view of this motivation, our aim in this paper is to present sufficient conditions which ensure that all solutions of () are oscillatory. For related results concerning second-order differential equations with sublinear neutral term, we refer the reader to 3,6, 7,23]. Some related results concering second-order dynamic equations on time scales can be found in 6 8, 3, 4, 9, 22]. 2. Main Results For n n 0 for some n 0 n 0, we let A n =. a s s=n For convenience, for some 0 < ν and n n 0, we set and y n = x n + p n x α n k, P n = p n A α n k A +( α)ν n 0 Q n = q n A (+ν)(β ) n+ P β n+ m. In the following, we establish a new oscillation result for () when β. Theorem 2.. Let β. Assume (H ) (H 3 ). If n ] (2) lim sup Q s A s+ =, 4a s A s+ then () is oscillatory. s=n 0 Proof. Let x n be a nonoscillatory solution of (), say x n > 0, x n+ m > 0, x n k > 0, and y n > 0 for n n for some n n 0. It is easy to see that y n > 0, n n, and () becomes (3) (a n y n ) + q n x β n+ m = 0. Thus (a n y n ) 0 for n n, which implies that y n is bounded. Also, the decreasing nature of a n y n implies that (I) y n > 0 or (II) y n < 0 for n n n. Therefore, y n converges, and hence lim y n = lim (x n + p n x α n k ) = lim x n, since lim p n = 0. Now, we consider Case (I). Since y n is a positive increasing sequence, there exist n 2 n and d > 0 such that (4) x n > d for n n 2.

3 M. Bohner, H. El-Morshedy, S. Grace and I. Sağer 3 Substituting (4) into (3), we get (5) (a n y n ) + q n d β < 0 for n n 2. Summing (5) from n 2 to n, we obtain n (6) a n y n a n2 y n2 + d β But (2) implies that which together with (6) yields s=n 2 q s < 0 for n n 2. n=n 2 q n =, lim a n y n =, a contradiction due to the eventual positivity of a n y n. Next, we consider Case (II). Define the sequence {v n } by (7) v n = a n y n y n for n n. Then v n < 0 for n n. Also, the decreasing nature of a n y n implies that (8) y s a n a s y n for s n n. Summing (8) from n to r n, we obtain which, by letting r, leads to (9) i.e., ( r y r y n a n y n a s=n s ), a n y n y n A n for n n, (0) v n A n for n n. On the other hand, we find from (9) that ( ) yn = A n y n y n A n A n A n A n+ for n n, and thus () Now, = A n y n + yn an A n A n+ 0, y n A n y n k A n k for n n + k. x n = y n p n x α n k y n p n y α n k for n n + k,

4 4 Nonlinear difference equations with sublinear neutral term and using (), we obtain A α n k (2) x n y n p n A α yn α = n ( A α ) n k p n A α yn α y n. n Since y n /A n is positive and increasing, we get y n (3) y n =: γ > 0 for n n. A n A n Since {A n } is positive and converging to zero, there exists n 3 n + k such that (4) 0 < A ν n γ for all n n 3. Hence, by (3) and (4), (5) y n A +ν n for n n 3. Using (5) in (2), we get ( (6) x n By (6), from (3), we have (7) A α n k p n A α A (+ν)(α ) n n (a n y n ) = q n x β n+ m q n P β n+ m yβ n+ m ) y n = P n y n for n n 3. q n P β n+ m yβ n+ for n n 3, where we also used the decreasing nature of y n in the last estimate. Now (7), in view of (5), leads to (8) (a n y n ) q n A (+ν)(β ) n+ P β n+ m y n+ = Q n y n+ for n n 3. Taking the difference of both sides of (7) and using the decreasing nature of a n y n, we get (9) v n = y n (a n y n ) a n ( y n ) 2 y n y n+ = (a n y n ) y n vn 2 y n+ a n y n+ (a n y n ) y n+ v2 n a n for n n 3, where we have used again the decreasing nature of y n. Combining (9) and (8), we have (20) v n Q n v2 n a n for n n 3.

5 M. Bohner, H. El-Morshedy, S. Grace and I. Sağer 5 Using (20), we get (A n v n ) =v n A n + A n+ v n = v n a n + A n+ v n v n a n A n+ Q n A n+v 2 n A n+ Q n + a n 4a n A n+, and summing this resulting inequality from n 3 to n and using (0) yields n ] Q s A s+ A n3 v n3 A n+ v n+ 4a s A s+ s=n 3 contradicting (2). This completes the proof. + A n3 v n3 < for n n 3, When β =, we have the following immediate corollary from Theorem 2.. Corollary 2.. Let β =. Assume (H ) (H 3 ). If n ] (2) lim sup q s P s+ m A s+ =, 4a s A s+ then () is oscillatory. s=n 0 Next, we establish an oscillation result when 0 < β <. Theorem 2.2. Let 0 < β <. Assume (H ) (H 3 ). If n ] (22) lim sup Lq s P β s+ m A s+ = 4a s A s+ for some L > 0, s=n 0 then () is oscillatory. Proof. Let x n be a nonoscillatory solution of (), say x n > 0, x n+ m > 0, x n k > 0, and y n > 0 for n n for some n n 0. Proceeding as in the proof of Theorem 2., we obtain the two cases (I) y n > 0 or (II) y n < 0 for n n. Next, we consider only Case (II) as Case (I) can be treated similarly as in the proof of Theorem 2.. Recall that y n is positive and decreasing with lim y n = lim x n. Then we have either lim y n = d > 0 or lim y n = 0. The first case implies that lim x n = d. Thus, there exist d 2 > 0 and n N such that x n d 2 for all n n. Hence we can obtain a contradiction similarly as in Case (I). The other case implies that for K := L /(β ) > 0, there exists n 2 N such that (23) 0 < y n < K for all n n 2.

6 6 Nonlinear difference equations with sublinear neutral term Now proceeding as in the proof of Theorem 2., we obtain (7), which with (23) yields 0 (a n y n ) + q n P β n+ m yβ n+ = (a n y n ) + q np β n+ m y n+ y β n+ (a n y n ) + q np β n+ m y n+ K β = (a n y n ) + Lq n P β n+ m y n+ for n n 3, with some n 3 n 2. The remainder of the proof is similar to that of Theorem 2. and hence is omitted. 3. Examples and Remarks First, we give two examples for the case β >. Example 3.. Consider the second-order equation ( ( )) (24) n(n + ) x n + xα n k n 2 + (n + ) 6 x 5 3 n+ m = 0, n N. Here, 0 < α < is a ratio of positive odd integers, β = 5/3, the delays are k N and m N 0, and a n = n(n + ), p n = n 2, and q n = (n + ) 6. We let ν =. It is easy to see that (H 2 ) holds. Also, Moreover, A n = n and Q n A n+ = (n+) 3 4a n A n+ P n = n α (n k) α. ] 5 3 (n + m) α (n + m k) α 4n. Thus, ( ) lim Q n A n+ =. 4a n A n+ Therefore, (2) of Theorem 2. is satisfied, and hence (24) is oscillatory. Example 3.2. Consider the second-order equation ( ( )) (25) n(n + ) x n + xα n k n 2 + (n + ) 2 x 5 3 n+ m = 0, n N. Here, all data are the same as in Example 3. except q n = (n + ) 2,

7 M. Bohner, H. El-Morshedy, S. Grace and I. Sağer 7 and therefore Q n A n+ = (n + ) 3 for n N with some N N, and thus M ( ) M Q n A n+ = 4a n A n+ n=n ] 5 3 (n + m) α (n + m k) α n 2, n=n M n=n ( Q n A n+ ) 4n as M. 4n Hence, (2) of Theorem 2. is satisfied, and thus (25) is oscillatory. Next, we give an example in the case β =. Example 3.3. Consider the second-order equation (26) ( ( )) (n n(n + ) x n + 3 k)xn k 8n 4 + n + n x n+ m = 0, n N. Here, α = /3, β =, the delays are k N and m N 0, and n k a n = n(n + ), p n = 3 8n 4 and q n = n + n. We let ν = /2. It is easy to see that (H 2 ) holds. Also, Moreover, A n = n q n P n+ m A n+ and P n = 2. 4a n A n+ = q n 2(n + ) 4n = 4n. Therefore, (2) of Corollary 2. is satisfied, and hence (26) is oscillatory. Finally, we present an example in the case 0 < β <. Example 3.4. Consider the second-order equation (27) ( x n + 4 (α )(n ) (kα+)/2 x α ) n k 2 n + n8 n x β n+ m = 0, n N. Here, 0 < α < and 0 < β < are ratios of positive odd integers, the delays are k N and m N 0, and a n = 2 n, p n = 4 (α )(n ) (kα+)/2 and q n = n8 n. We let ν =. It is easy to see that (H 2 ) holds. Also, A n = 2 n and P n = 2.

8 8 Nonlinear difference equations with sublinear neutral term Moreover, Lq n P β n+ m A n+ 4a n A n+ = Ln2 2n β 2 2n 2, which tends to infinity for any constant L > 0. Therefore, (22) of Theorem 2.2 is satisfied, and hence (27) is oscillatory. Remark 3.. The results of this paper are presented in a form that makes it easy to study extensions to higher-order equations. It would also be of interest to use the approach here to study () with α >, i.e., () with superlinear neutral term. Remark 3.2. Another possibility for extension of the presented results would be to consider the time-scales 5, 8] analogue of (). References ] R. P. Agarwal, Difference equations and inequalities: Theory, methods, and applications, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2nd Ed., ] R. P. Agarwal, M. Bohner, S. R. Grace, D. O Regan, Discrete oscillation theory, Hindawi Publishing Corporation, New York, ] R. P. Agarwal, M. Bohner, T. Li, C. Zhang, Oscillation of second-order differential equations with a sublinear neutral term, Carpathian J. Math., 30 () (204), 6. 4] R. P. Agarwal, S. R. Grace, D. O Regan, Oscillation theory for difference and functional differential equations, Kluwer Academic Publishers, Dordrecht, ] M. Bohner, S. G. Georgiev, Multivariable dynamic calculus on time scales, Springer, Cham, ] M. Bohner, S. R. Grace, I. Jadlovská, Oscillation criteria for second-order neutral delay differential equations, Electron. J. Qual. Theory Differ. Equ., 207, No. 60, 2. 7] M. Bohner, T. Li, Oscillation of second-order p-laplace dynamic equations with a nonpositive neutral coefficient, Appl. Math. Lett., 37 (204), ] M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkhäuser Boston, Inc., Boston, MA, ] C. Dharuman, J. R. Graef, E. Thandapani, K. S. Vidhyaa, Oscillation of second order difference equation with a sub-linear neutral term, J. Math. Appl., 40 (207), ] H. A. El-Morshedy, Oscillation and nonoscillation criteria for half-linear second order difference equations, Dynam. Systems Appl., 5 (3-4) (2006),

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10 0 Nonlinear difference equations with sublinear neutral term Martin Bohner Department of Mathematics and Statistics Missouri S&T Rolla, MO USA address: Hassan A. El-Morshedy Department of Mathematics Faculty of Science Damietta University New Damietta 3457 Egypt address: Said R. Grace Department of Engineering Mathematics Faculty of Engineering Cairo University Orman, Giza 2000 Egypt address: Ilgin Sağer Department of Mathematics and Computer Science University of Missouri-St. Louis St. Louis, MO USA address: