By Gustave MALECOTT
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30) Similarly, we find Now assume that S < f (I, S, I). 30). 13), implies that I = f (S, I, S). 29) we find that L0 = L−2 = I and L−1 = S 22 Dynamics of Third-Order Rational Difference Equations otherwise and because of the strict monotonicity of f , S < f (I, S, I), which is a contradiction. At this point we claim that there exist arbitrarily small positive numbers, 1 and 2 , such that S− 1 = f (I + 2, S − 1, I + 2 ). Assume, for the sake of contradiction and without loss of generality, that for all positive numbers 1 and 2 we have S− By letting 1 1 < f (I + 2, S − 1, I + 2 ).
T} L−n = L−n−2ijp for all n ≥ n0 . Therefore, the sequence {L−q }∞ q=n0 is periodic with periods is1 , . . , isr , 2ij1 , . . , 2ijt . But gcd{is1 , . . , isr , 2ij1 , . . , 2ijt } = 2 and so the sequence {L−q }∞ q=0 is periodic with period two. In fact, it has the following form, . . , I, S, . . When (H1 ), (H3 ), (H4 ), and (H6 ) hold, assume without loss of generality that, for all j ≥ 0, L−2j = I and L−2j−1 = S. At this point we claim that there exist arbitrarily small positive numbers, and 2 , such that S− 1 = f (M1 (I, S) + X1 , .
Mk ) > m. 9) implies (H3 ) : For each m ∈ [0, ∞) and M > m, we assume that either (f (M1 , . . , Mk ) − M)(f (m1 , . . 10) f (M1 , . . , Mk ) − M = f (m1 , . . , mk ) − m = 0. 11) or Preliminaries 13 (H3 ) : For each m ∈ (0, ∞) and M > m, we assume that either (f (M1 , . . , Mk ) − M)(f (m1 , . . 12) f (M1 , . . , Mk ) − M = f (m1 , . . , mk ) − m = 0. 13) or We also define the following sets: S = {is ∈ {i1 , . . , ik } : f strictly increases in xn−is } = {is1 , . . , isr } and J = {ij ∈ {i1 , .