×

zbMATH — the first resource for mathematics

Discussion on the differentiable solutions of the iterated equation \(\sum_{i=1}^{n}\lambda_ if^ i(x)=F(x)\). (English) Zbl 0717.39005
The author considers the differentiable solutions of the functional equation (1) \(\lambda_ 1f(x)+\lambda_ 2f^ 2(x)+...+\lambda_ nf^ n(x)=F(x),\) \(x\in [a,b]=I,\) where f: \(I\to I\), \(f^ 0(x)=x\) and \(f^ k=f\circ f^{k-1}\), \(\lambda_ i\in {\mathbb{R}}\). Under suitable assumptions on \(\lambda_ i\), F and constants M and L he proves the existence, uniqueness and stability of the solutions of equation (1) in the following class of functions \(A=\{\phi \in C^ 1[I,I],\quad \phi (a)=a,\quad \phi (b)=b,\quad 0\leq \phi '\leq M\text{ and } | \phi '(x_ 1)-\phi '(x_ 2)| \leq L| x_ 1-x_ 2|,\quad x_ 1,x_ 2\in I\}.\) The proof of the main result relies on the Schauder fixed point theorem.
Reviewer: M.C.Zdun

MSC:
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abel, N.H., Oeuvres complètes, Christiana, Vol. II, 36-39, (1881)
[2] Dubbey, J.M., The mathematical work of charles babbage, (1978), Cambridge University Press · Zbl 0376.01002
[3] Kuczma, M., Functional equations in a single variable, Monografie mat., 46, 383, (1968)
[4] Rice, R.E.; Schweizer, B.; Sklar, A., When is \(⨍ (⨍(z)) = az\^{}\{2\} + bz + c\)?, Am. math. mon., 87, 252-263, (1980) · Zbl 0441.30033
[5] Zhang, Jingzhong; Yang, Lu, Discussion on iterative roots of continuous and piecewise monotone functions, Acta. math. sin., 26, 398-412, (1983), (In Chinese.) · Zbl 0529.39006
[6] Zhang, Jingzhong; Yang, Lu, A criterion of existence and uniqueness of real iterative groups in a single variable, Acta sci. nat. univ. pekin., 6, 23-45, (1982), (In Chinese.)
[7] Dhombres, J.G., Itération linéaire d’ordre deux, Publ. math. debrecen, 24, 277-287, (1977) · Zbl 0398.39006
[8] Mukherjea, A.; Ratti, J.S., On a functional equation involving iterates of a bijection on the unit interval, Nonlinear analysis, 7, 899-908, (1983) · Zbl 0518.39005
[9] Zhao, Liren, A theorem concerning the existence and uniqueness of solutions of functional equation \(λ1⨍(x) + λ2⨍\^{}\{2\}(x) = F(x)\), J. univ. sci. tech. China, special issue of mathematics, 21-27, (1983), (In Chinese.)
[10] Zhang, Weinian, Discussion on the iterated equation \(∑i=1nλi⨍\^{}\{i\}(x) = F(x)\), Kexue tongbao, 32, 1444-1451, (1987) · Zbl 0639.39006
[11] Zhang, Weinian, Stability of the solution of the iterated equation \(∑i=1nλi⨍\^{}\{i\}(x) = F(x)\), Acta math. sci., 8, 421-424, (1988) · Zbl 0664.39004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.