INTERVAL OSCILLATION CRITERIA FOR SECOND-ORDER DELAY AND ADVANCED DIFFERENCE EQUATIONS

Interval oscillation criteria are established for second-order di¤erence equations in the form (k (n) x(n))+p (n)x (g (n))+q (n) jx (g (n))j 1 x (g (n))=e (n) ; (E ) where n n0; n0 2 N = f0; 1; :::g; > 1; k; p; q; e and g are sequences of real numbers; k (n) > 0 is nondecreasing; g(n) is nondecreasing, limn!1 g(n) =1: Several oscillation criteria are given for equation (E ) considered as to separate delay and advanced di¤erence equations when g(n) < n and g(n) > n respectively. Illustrative examples are included. 1. Introduction We consider second-order di¤erence equations of the form, (k (n) x(n))+p (n)x (g (n))+q (n) jx (g (n))j 1 x (g (n)) = e(n) (E ) where n n0; n0 2 N = f0; 1; :::g; > 1; k; p; q; e and g are sequences of real numbers; k (n) > 0 is nondecreasing; g(n) is nondecreasing, limn!1 g(n) =1: is the forward di¤erence operator de…ned by x(n) = x(n+1) x(n): As is customary, we assume that solutions of (E ) exist on some set fn0; n0+1; :::g: For the theory of existence of solutions of such equations, we refer [1]: A nontrivial solution fx(n)g of (E ) is called oscillatory if for any given ~ n0 n0 there exists an integer n1 ~ n0 such that x(n1)x(n1 + 1) 0, otherwise it is called nonoscillatory. The equation will be called oscillatory if every solution is oscillatory. Taking g(n) as (n) with (n) < n and limn!1 (n) = 1; = , equation (E ) is considered as a delay di¤erence equation (k (n) x(n))+p1 (n)x ( (n))+q1 (n) jx ( (n))j 1 x ( (n))=e (n) (ED) Received by the editors April 16, 2009, Accepted: June. 16, 2009. 2000 Mathematics Subject Classi…cation. 34K11, 34C10. Key words and phrases. Interval oscillation, Second-order, Delay argument, Advanced argument, Oscillatory. c 2009 Ankara University 39 40 A. FEZA GÜVENILIR or taking g(n) as (n) with (n) > n and = ;equation (E ) is considered as an advanced di¤erence equation (k (n) x(n))+p2 (n)x ( (n))+q2 (n) jx ( (n))j 1 x ( (n))=e (n) : (EA) In literature, there isn’t enough work dealing with the oscillation of di¤erence equations (ED) and (EA): Equation (E ); when k(n) 1; p (n) 0 or q (n) 0 and g(n) = n; n+ 1; n has been studied by many authors, see [6; 7; 12; 13; 15] and the references cited therein. Using Riccatti tecnique, Saker[9] obtained some oscillation criteria for forced Emden-Fowler superlinear di¤erence equation of the form x(n)+q (n)x (n+ 1)=e(n) when q(n) and e(n) are sequences of positive real numbers. Zhang and Chen [14] established some oscillation criteria x(n)+q (n) f (x (n+ 1))=0 whenf is nondecreasing and uf(u) > 0 for u 6= 0. The …rst result concerning the interval oscillation of (E ) when g(n) = n + 1; q(n) 0; e(n) 0 has been studied by Kong and Zettl [7]: They have applied the telescoping principle for equation of the form (k (n) x(n))+p (n)x (n+ 1)=0: Recently, Güvenilir and Zafer [4] has presented some su¢ cient conditions about oscillation of second-order di¤erential equation (k(t)x0(t))0+p (t) jx ( (t))j 1 x ( (t))+q (t) jx ( (t))j 1 x ( (t))=e (t) : (1:1) where n 0. Later, in [2] Anderson generalized the results of Güvenilir and Zafer [4] to the dynamic equation (kx ) (t)+p (t) jx ( (t))j 1 x ( (t))+q (t) jx ( (t))j 1 x ( (t))=e (t) (1:2) where n 0 for arbitrary time scales. In this work, our purpose is to derive interval oscillation criteria as discrete analogues of the ones contained [3]: The di¤erence between (E ) and (1:2) is the appearence of both linear and nonlinear terms. Therefore, the results in [2] fails to apply for (E ): For our purpose, we denote D (ak; bk) = fu : u (ak) = u (bk) = 0; k = 1; 2; u (n) 6 0; n 2 N(ak; bk)g ; where N(ak; bk) = fak; ak + 1; :::; bkg: As in [4]; we de…ne P (n) = ( 1) 1 q (n) je (n)j 1= : (1:3) INTERVAL OSCILLATION CRITERIA FOR SECOND-ORDER 41 2. Delay Difference Equations Suppose that for any given N 0 there exist a1,a2,b1,b2 N such that a1 < b1; a2 < b2 and p1 (n) 0; q1(n) 0 for n 2 N( (a1) ; b1) [ N( (a2) ; b2): (2:1) Let e (n) satis…es e (n) 0; for n 2 N( (a1) ; b1) e (n) 0; for n 2 N( (a2) ; b2): (2:2) Theorem 2.1. Suppose that (2:1) and (2:2) hold. If there exist an H1 2 D (ai; bi) ; i = 1; 2; such that bi 1 X n=ai H 1 (n+ 1) (p1 (n) + P (n)) (n) (ai) n+ 1 (ai) ( H1 (n))k (n) 0; (2:3) for i = 1; 2; then (ED) is oscillatory. Proof. To get a contradiction, let us suppose that x (n) is a nonoscillatory solution of equation (ED) : First, assume x (n) > 0, x ( (n)) > 0 for all n n1 for some n1 > 0: We may say F (x) = Ax ( 1) A1= B 1= x+B 0 for x 2 [0;1) (2:4) where A, B are nonnegative constants and > 1; [10]: If we choose A = q1(t), B = e(n) and = in (2:4), we have q1 (t)x ( (n)) e (n) ( 1) 1 q1 (n) 1 je (n)j 1 x ( (n)) : (2:5) for n 2 N( (a1); b1) See also [8; 10]: De…ne w (n) = k (n) x (n) x (n) ; n n1; n1 > 0: (2:6) In view of (ED) ; we see that w (n) = x(n) k(n)x(n+1)w 2 (n) + p1 (n) x( (n)) x(n+1) + [q1 (n)x ( (n)) e (n)] 1 x(n+1) : (2:7) Using (2:1) and (2:5), we see from (2:7) that w (n) x(n) k (n)x(n+ 1) w (n) + [p1 (n) + P (n)] x ( (n)) x (n+ 1) ; n 2 N( (a1) ; b1): Moreover x(n+ 1) = x(n) + x(n); 42 A. FEZA GÜVENILIR x(n+ 1) x(n) = 1 + x(n) x(n) and then x(n) k (n)x(n+ 1) = 1 k (n) w (n) : Therefore w (n) 1 k (n) w (n) 2 (n) + [p1 (n) + P (n)] x ( (n)) x (n+ 1) ; n 2 N( (a1) ; b1): (2:8) Now by the Mean Value Theorem in [1] x(n) x ( (a1)) k ( ) x ( ) k ( ) (n (a1)) for some 2 N( (a1) ; n): From which, for any n 2 N(a1; b1),we have x(n) x (n) (n (a1)); n 2 N(a1; b1) and hence, x (n) x(n) 1 n (a1) ; n 2 N(a1; b1): Moreover, following the arguments in [2], since x(m) x (m) (m (a1)) 0; m 2 N( (n); n+ 1); n 2 N(a1; b1) we have x(m) x (m) (m (a1)) x(m)x(m+ 1) 0: Therefore, ( m (a1) x(m) ) 0: It follows that