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Extracted text: Our goal is to obtain some qualitative behavior of the positive solutions of the difference equation Xn+1 = Bxn-i+axn-k + axn-t n = 0, 1, ..., (1) bxn-t+c' where the parameters 3, a, a, b and c are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, xo are positive real numbers where s = max{1, k, t}. .D TT T T
Extracted text: THEOREM 4.1. Let l, k and t are both odd positive integers then for all B, a, a, b and c are positive real numbers, then Eq. (1) has a prime period two solution if a+B <1 аnd="" c="" (1="" —="" a="" —="" в)="">< а. (7) proof: first, suppose that there exists distinct nonnegative solution p and q, such that ...р, q, р, q, ..., is a prime period two solution of eq.(1). we see from eq. (1) when l, k and t are both odd, then xn+1 = xn-1 = xn-k = xn-t = p. it follows eq. (1) that aq p = bp + ap+ , and q = bq + aq+ o .+09 bp+c therefore, b(1 — а — в) р2 + (с (1 — а — в) — а) р — 0, (8) b (1 — а — в) q + (с (1 — а - b) — а) q — 0, (9) subtracting (9) from (8) gives +q = " a-c(1-a-8) b(1-a-b) (10)
а.="" (7)="" proof:="" first,="" suppose="" that="" there="" exists="" distinct="" nonnegative="" solution="" p="" and="" q,="" such="" that="" ...р,="" q,="" р,="" q,="" ...,="" is="" a="" prime="" period="" two="" solution="" of="" eq.(1).="" we="" see="" from="" eq.="" (1)="" when="" l,="" k="" and="" t="" are="" both="" odd,="" then="" xn+1="xn-1" =="" xn-k="Xn-t" =="" p.="" it="" follows="" eq.="" (1)="" that="" aq="" p="BP" +="" ap+="" ,="" and="" q="BQ" +="" aq+="" o="" .+09="" bp+c="" therefore,="" b(1="" —="" а="" —="" в)="" р2="" +="" (с="" (1="" —="" а="" —="" в)="" —="" а)="" р="" —="" 0,="" (8)="" b="" (1="" —="" а="" —="" в)="" q="" +="" (с="" (1="" —="" а="" -="" b)="" —="" а)="" q="" —="" 0,="" (9)="" subtracting="" (9)="" from="" (8)="" gives="" +q="">
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