Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this ...Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this paper, we prove the existence of decreasing solutions for a general iterative equation, which was proposed as an open problem in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. （China） 24 （1995） 385-405] （or [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 （2000） 29-36]）.展开更多
In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wa...In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schr¨oder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between k-Motzkin paths and(k-2)-Schr¨oder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.展开更多
基金supported by Zhejiang Provincial Natural Science Foundation of China under Grant No.LY18A010017the National Science Foundation of China(11101105,11301226)
文摘Most known results on polynomial-like iterative equations are concentrated to increasing solutions. Without the uniformity of orientation and monotonicity, it becomes much more difficult for decreasing cases. In this paper, we prove the existence of decreasing solutions for a general iterative equation, which was proposed as an open problem in [J. Zhang, L. Yang, W. Zhang, Some advances on functional equations, Adv. Math. （China） 24 （1995） 385-405] （or [W. Zhang, J.A. Baker, Continuous solutions of a polynomial-like iterative equation with variable coefficients, Ann. Polon. Math. 73 （2000） 29-36]）.
基金The NSF(11601020,11501014)of ChinaOrganization Department of Beijing Municipal Committee(2013D005003000012)Science and Technology Innovation Platform-Business Project 2017(PXM2017_014213_000022)
文摘In this paper, we provide a bijection between the set of underdiagonal lattice paths of length n and the set of(2, 2)-Motzkin paths of length n. Besides, we generalize the bijection of Shapiro and Wang(Shapiro L W, Wang C J. A bijection between 3-Motzkin paths and Schr¨oder paths with no peak at odd height. J. Integer Seq., 2009, 12: Article 09.3.2.) to a bijection between k-Motzkin paths and(k-2)-Schr¨oder paths with no horizontal step at even height. It is interesting that the second bijection is a generalization of the well-known bijection between Dyck paths and 2-Motzkin paths.
