- Research Article
- Open Access
© Yuzhen Mi et al. 2009
- Received: 23 March 2009
- Accepted: 6 July 2009
- Published: 4 August 2009
- Continuous Function
- Unknown Function
- Differential Geometry
- Variable Coefficient
- Structure Operator
is a complete space.
Theorem 3.1 (existence).
Theorem 3.2 (Uniqueness).
Suppose that (P1) and (P2) are satisfied, also one supposes that
The existence of (1.2) in is given by Theorem 3.1, from the proof of Theorem 3.1, we see that is a closed subset of , by (3.12) and , we see that is a contraction. Therefore has a unique fixed point in , that is, (1.2) has a unique solution in , this proves the theorem.
This work was supported by Guangdong Provincial Natural Science Foundation (07301595) and Zhan-jiang Normal University Science Research Project (L0804).
- Zhang JZ, Yang L: Disscussion on iterative roots of continuous and piecewise monotone functions. Acta Mathematica Sinica 1983,26(4):398–412.MathSciNetGoogle Scholar
- Zhang WN: Discussion on the iterated equation . Chinese Science Bulletin 1987, 32: 1441–1451.Google Scholar
- Zhang WN: Stability of the solution of the iterated equation . Acta Mathematica Scientia 1988,8(4):421–424.MathSciNetMATHGoogle Scholar
- Zhang WN: Discussion on the differentiable solutions of the iterated equation . Nonlinear Analysis: Theory, Methods & Applications 1990,15(4):387–398. 10.1016/0362-546X(90)90147-9MathSciNetView ArticleMATHGoogle Scholar
- Zhang W: Solutions of equivariance for a polynomial-like iterative equation. Proceedings of the Royal Society of Edinburgh. Section A 2000,130(5):1153–1163. 10.1017/S0308210500000615MathSciNetView ArticleMATHGoogle Scholar
- Kulczycki M, Tabor J: Iterative functional equations in the class of Lipschitz functions. Aequationes Mathematicae 2002,64(1–2):24–33. 10.1007/s00010-002-8028-2MathSciNetView ArticleMATHGoogle Scholar
- Li XP: A class of iterative equation on a Banach space. Journal of Sichuan University. Natural Science Edition 2004,41(3):505–510.MathSciNetMATHGoogle Scholar
- Zhang W, Nikodem K, Xu B: Convex solutions of polynomial-like iterative equations. Journal of Mathematical Analysis and Applications 2006,315(1):29–40. 10.1016/j.jmaa.2005.10.006MathSciNetView ArticleMATHGoogle Scholar
- Xu B, Zhang W: Decreasing solutions and convex solutions of the polynomial-like iterative equation. Journal of Mathematical Analysis and Applications 2007,329(1):483–497. 10.1016/j.jmaa.2006.06.087MathSciNetView ArticleMATHGoogle Scholar
- Li XP, Deng SF: Differentiability for the high dimensional polynomial-like iterative equation. Acta Mathematica Scientia B 2005,25(1):130–136.MathSciNetMATHGoogle Scholar
- Zhang W, Baker JA: Continuous solutions of a polynomial-like iterative equation with variable coefficients. Annales Polonici Mathematici 2000,73(1):29–36.MathSciNetMATHGoogle Scholar
- Si J-G, Wang X-P: Differentiable solutions of a polynomial-like iterative equation with variable coefficients. Publicationes Mathematicae Debrecen 2001,58(1–2):57–72.MathSciNetMATHGoogle Scholar
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