Abstract
Interval oscillation criteria are established for second-orderdigerence equations in the form(k (n)x(n)) +p (n) x (g (n)) +q (n) jx (g (n))jn; n02 N = f0; 1; :::g;1x (g (n)) =e (n) ;(E ) wherenquences of real numbers;k (n) > 0is nondecreasing;g(n)is nondecreasing,limn 1g(n) = 1:Several oscillation criteria are given for equation(E )considered as to separate delay and advanced digerence equationswheng(n) nrespectively. Illustrative examples are included
Highlights
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
Where n n0; n0 2 N = f0; 1; :::g ; > 1; k; p1; p2; q1; q2; e; and are sequences of real numbers, k (n) > 0 is nondecreasing; (n) < n; (n) > n; and are nondecreasing and limt!1 (t) = 1 : Suppose that for any given N 0 there exist a1; a2; b1; b2 ; c1; c2; d1; d2 N such that a1 < b1; a2 < b2 and c1 < d1; c2 < d2: Theorem 4.1
Summary
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). Using Riccatti tecnique, Saker[9] obtained some oscillation criteria for forced Emden-Fowler superlinear di¤erence equation of the form. 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).
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