×

Generic properties of nonlinear functional equations. (English) Zbl 0536.39004

Assume that \((X,\rho)\) is a metric space and let Y be a non-trivial normed space. Fix points \(\xi\in X\) and \(\eta\in Y\) and a continuous function f:\(X\to X\) such that \(\rho\) (f(x),\(\xi)\leq \gamma(\rho(x,\xi))\) for every \(x\in X\), where \(\gamma\) is an increasing and continuous real function defined on an interval I containing the origin, and \(\gamma(t)<t\) for every \(t\in I\backslash \{0\}\). Denote by \(\Phi\) the set of all continuous functions \(\phi\) :\(X\to Y\) such that \(\phi(\xi)=\eta\) and consider the space \({\mathcal H}\) of all continuous functions h:\(X\times y\to y\) such that \(h(\xi,\eta)=\eta\) as a topological space with the compact-open topology. The main results read as follows.
1. If the point \(\xi\) has a compact neighbourhood in X and the space Y is finite dimensional, then the set of all \(h\in {\mathcal H}\) such that equation \((*)\quad \phi(x)=h(x,\phi [f(x)])\) has exactly one solution \(\phi\in \Phi\) and for every \(\phi_ 0\in \Phi\) the sequence of successive approximations starting from \(\phi_ 0\) converges to \(\phi\) uniformly on every compact subset of X is residual in \({\mathcal H}\). 2. If the set \(\{f^ n(x_ 0):n\in {\mathbb{N}}\}\) is infinite for an \(x_ 0\in X\), then the set of all \(h\in {\mathcal H}\) for which equation (*) has no solution in the class \(\Phi\) is dense in \({\mathcal H}\). Problem of the continuous dependence of continuous solutions of equation (*) from the Baire category point of view is also considered.
Linear equations of the form \(\phi [f(x)]=g(x)\phi(x)+F(x)\) are studied from the Baire category point of view in the following papers by the same author: Bull. Acad. Pol. Sci., Sér. Sci. Math. 29, 371-372 (1981; Zbl 0475.39013); Glas. Mat., III. Ser. 17(37), 59-64 (1982; Zbl 0514.39004); ibid. 18(38), 91-102 (1983; Zbl 0519.39004) and ’On linear homogeneous functional equations in the inderminate case’. Fundam. Math. (to appear).
Reviewer: K.Baron

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B12 Iteration theory, iterative and composite equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Baron, K.,Functional equations of infinite order. Uniw. Ślaski w Katowicach Prace Nauk. – Prace Mat., Nr 265 (1978).
[2] Baron, K. andJarczyk, W.,On approximate solutions of functional equations of countable order. Submitted to Aequationes Math.
[3] Bessaga, C. andPełczyński, A.,Selected topics in infinite-dimensional topology. Monografie Mat. Vol.58. Polish Scientific Publishers, Warszawa, 1975. · Zbl 0304.57001
[4] de Blasi, F., Kwapisz, M. andMyjak, J.,Generic properties of functional equations. Nonlinear Anal.2 (1978), 239–249. · Zbl 0426.47043
[5] Dugundji, J.,An extension of Tietze’s theorem. Pacific J. Math.1 (1951), 353–367. · Zbl 0043.38105
[6] Engelking, R.,General topology. Monografie Mat. Vol.60. Polish Scientific Publishers, Warszawa, 1977. · Zbl 0373.54002
[7] Jarczyk, W.,On a set of functional equations having continuous solutions. Glasnik Mat.17 (37), 59–64. · Zbl 0514.39004
[8] Jarczyk, W.,A category theorem for linear functional equations in the indeterminate case. Bull. Acad. Polon. Sci., Ser. Sci. Math.29 (1981), 371–372. · Zbl 0475.39013
[9] Jarczyk, W.,On linear functional equations in the determinate case. Glasnik Mat. (to appear). · Zbl 0519.39004
[10] Jarczyk, W.,On linear homogeneous functional equations in the indeterminate case. Fund. Math. (to appear). · Zbl 0555.39003
[11] Kuczma, M.,Functional equations in a single variable. Monografie Mat. Vol.46, Polish Scientific Publishers, Warszawa, 1968. · Zbl 0196.16403
[12] Myjak, J.,Twierdzenia o kategoriach w teorii równań ró\.zniczkowych i ró\.zniczkowofunkcjonalnych. Zeszyty Nauk. Akad. Górn.-Hutniczej-Mat.-Fiz.-Chem. Nr 687, 1978.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.