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We investigate the periodic nature of the positive solutions of the fuzzy max-difference equationxn+1=maxAn/xn-m,xn-k,n=0,1,…, wherek,m∈{1,2,…},Anis a periodic sequence of fuzzy numbers, andx-d,x-d+1,…,x0are positive fuzzy numbers withd=m,k. We show that every positive solution of this equation is eventually periodic with periodk+1.

We study the following max-type difference equation x n = max⁡{A n/x n−r, x n−k}, n = 1,2,…, where {A n}n=1 +∞ is a periodic sequence with period p and k, r ∈ {1,2,…} with gcd(k, r) = 1 and k ≠ r, and the initial conditions x 1−d, x 2−d,…, x 0 are real numbers with d = max⁡{r, k}. We show that if p = 1 (or p ≥ 2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p ≥ 2 and k being even which has a well-defined solution that is not eventually periodic.

In order to solve the problem that the iterated closest contour point(ICCP) algorithm diverges easily when the initial INS error is large, the terrain contour matching (TERCOM) algorithm is firstly used to reduce the initial INS error, then ICCP algorithm is used to obtain the best matching position. Two matching difference is used as the measurement of Kalman filter, INS error is corrected and the optimal estimate is obtained. The correlative analysis MSD is only introduced in the coarse matching stage, and the sliding window is used in the precise matching stage to improve the algorithm efficiency. Simulations are performed and the results show that the proposed combinational algorithm matching process is more stable and the precision is higher than traditional algorithm.

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