## A global convergence result for a higher order difference equation.(English)Zbl 1180.39003

Summary: Let $$f(z_1,\dots,z_k)\in C(I^k,I)$$ be a given function, where $$I$$ is (bounded or unbounded) subinterval of $$\mathbb R$$, and $$k\in\mathbb N$$. Assume that $$f(y_1,y_2,\dots,y_k)\geq f(y_2,\dots,y_k,y_1)$$ if $$y_1\geq \max\{y_2,\dots,y_k\}$$, $$f(y_1,y_2,\dots,y_k)\leq f(y_2,\dots,y_k,y_1)$$ if $$y_1\leq \min\{y_2,\dots,y_k\}$$, and $$f$$ is non-decreasing in the last variable $$z_k$$. We then prove that every bounded solution of an autonomous difference equation of order $$k$$, namely, $$x_n= f(x_{n-1},\dots,x_{n-k})$$, $$n=0,1,2,\dots$$, with initial values $$x_{-k},\dots,x_{-1}\in I$$, is convergent, and every unbounded solution tends either to $$+\infty$$ or to $$-\infty$$.

### MSC:

 39A10 Additive difference equations
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### References:

 [1] J. Bibby, “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63-65, 1974. · Zbl 0291.40003 [2] D. Borwein, “Convergence criteria for bounded sequences,” Proceedings of the Edinburgh Mathematical Society (2), vol. 18, no. 1, pp. 99-103, 1972/73. · Zbl 0247.40001 [3] E. T. Copson, “On a generalisation of monotonic sequences,” Proceedings of the Edinburgh Mathematical Society (2), vol. 17, no. 2, pp. 159-164, 1970/71. · Zbl 0223.40001 [4] H. El-Metwally, E. A. Grove, and G. Ladas, “A global convergence result with applications to periodic solutions,” Journal of Mathematical Analysis and Applications, vol. 245, no. 1, pp. 161-170, 2000. · Zbl 0971.39004 [5] G. L. Karakostas and S. Stević, “Slowly varying solutions of the difference equation xn+1=f(xn,\cdots ,xn - k)+g(n,xn,xn - 1,\cdots ,xn - k),” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 249-255, 2004. · Zbl 1048.39003 [6] S. Stević, “A note on bounded sequences satisfying linear inequalities,” Indian Journal of Mathematics, vol. 43, no. 2, pp. 223-230, 2001. · Zbl 1035.40002 [7] S. Stević, “A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality,” Indian Journal of Mathematics, vol. 43, no. 3, pp. 277-282, 2001. · Zbl 1034.40002 [8] S. Stević, “A global convergence result,” Indian Journal of Mathematics, vol. 44, no. 3, pp. 361-368, 2002. · Zbl 1034.39002 [9] S. Stević, “A global convergence results with applications to periodic solutions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 45-53, 2002. · Zbl 1002.39004 [10] S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium Mathematicum, vol. 93, no. 2, pp. 267-276, 2002. · Zbl 1029.39006 [11] S. Stević, “Asymptotic behavior of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681-1687, 2003. · Zbl 1049.39012 [12] K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation xn=A+(xn - 2/xn - 1)p,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909-918, 2006. · Zbl 1111.39003 [13] L. Berg, “Inclusion theorems for non-linear difference equations with applications,” Journal of Difference Equations and Applications, vol. 10, no. 4, pp. 399-408, 2004. · Zbl 1056.39003 [14] L. Berg, “Corrections to ‘inclusion theorems for non-linear difference equations with applications’,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 181-182, 2005. · Zbl 1080.39002 [15] L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 217-250, 2005. · Zbl 1104.45001 [16] S. Stević, “On the recursive sequence xn=1+\sum i=1k\alpha ixn - pi/\sum j=1m\beta jxn - qj,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 39404, p. 7, 2007. · Zbl 1180.39006 [17] T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation xn+1=xn - 1/(p+xn),” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 20652, p. 7, 2006. · Zbl 1149.39301 [18] S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence xn+1=f(xn,xn - 1),” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631-638, 2006. · Zbl 1100.39001
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